Colorimetry Fundamentals and Applications

Colorimetry Fundamentals and Applications Noboru Ohta

Rochester Institute of Technology, USA

Alan R. Robertson

National Research Council of Canada, Ottawa, Canada (retired)

Colorimetry

Wiley–IS&T Series in Imaging Science and Technology Series Editor: Michael A. Kriss Consultant Editors: Anthony C. Lowe Lindsay W. MacDonald Yoichi Miyake

The Reproduction of Colour (6th Edition) R. W. G. Hunt Color Appearance Models (2nd Edition) Mark D. Fairchild Colorimetry: Fundamentals and Applications Noboru Ohta and Alan R. Robertson

Published in Association with the Society for Imaging Science and Technology

Colorimetry Fundamentals and Applications Noboru Ohta

Rochester Institute of Technology, USA

Alan R. Robertson

National Research Council of Canada, Ottawa, Canada (retired)

Copyright © 2005

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777

Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. This publication is designed to offer Authors the opportunity to publish accurate and authoritative information in regard to the subject matter covered. Neither the Publisher nor the Society for Imaging Science and Technology is engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging in Publication Data Ohta, Noboru. Colorimetry : fundamentals and applications / Noboru Ohta, Alan R. Robertson. p. cm. — (Wiley-IS&T series in imaging science and technology) Includes bibliographical references and index. ISBN-13 978-0-470-09472-3 ISBN-10 0-470-09472-9 (cloth : alk. paper) 1. Colorimetry. I. Robertson, Alan A. II. Title. III. Series. QC495.8.038 2005 535.6—dc22 2005013963 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13 978-0-470-09472-3 (HB) ISBN-10 0-470-09472-9 (HB) Typeset in 10/12pt Bookman by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents About the Authors Series Preface Preface Introduction 1 Light, 1.1 1.2 1.3 1.4

Note 1.2

Vision and Photometry Light Mechanism of the Human Eye Adaptation and Responsivity of the Human Eye Spectral Responsivity and the Standard Photometric Observer Definition of Photometric Quantities Photometric Units Calculation and Measurement of Photometric Quantities Relations Between Photometric Quantities Luminous Exitance, Illuminance, and Luminance of a Perfect Diffusing Plane Light Source Luminance and Brightness

2 Color 2.1 2.2 2.3 2.4 2.5 Note 2.1

Vision and Color Specification Systems Mechanism of Color Vision Chemistry of Color Vision Color Specification and Terminology Munsell Color System Color System Using Additive Color Mixing Colorfulness, Chroma and Saturation

1.5 1.6 1.7 1.8 Note 1.1

3 CIE Standard Colorimetric System 3.1 RGB Color Specification System 3.2 Conversion into XYZ Color Specification System

ix xi xiii xv 1 1 4 7 9 17 21 26 31

34 36 39 39 46 48 52 57 61 63 63 68

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CONTENTS

3.3 X10 Y10 Z10 Color Specification System 3.4 Tristimulus Values and Chromaticity Coordinates 3.5 Metamerism 3.6 Dominant Wavelength and Purity 3.7 Color Temperature and Correlated Color Temperature 3.8 Illuminants and Light Sources 3.9 Standard and Supplementary Illuminants Note 3.1 Derivation of Color Matching Functions from Guild and Wright’s Results Note 3.2 Conversion between Color Specification Systems Note 3.3 Conversion into XYZ Color Specification System Note 3.4 Imaginary Colors [X] and [Z] Note 3.5 Photometric Quantities in the X10 Y10 Z10 Color System Note 3.6 Origin of the Term ‘Metamerism’ Note 3.7 Simple Methods for Obtaining Correlated Color Temperature Note 3.8 Color Temperature Conversion Filter Note 3.9 Spectral Distribution of Black-body Radiation 4 Uniform Color Spaces 4.1 Uniform Chromaticity Diagrams 4.2 Uniform Lightness Scales (ULS) 4.3 CIE Uniform Color Spaces 4.4 Correlates of Perceived Attributes 4.5 Comparing CIELAB and CIELUV Color Spaces 4.6 Conversion of Color Difference 4.7 Color Difference Equations Based on CIELAB Note 4.1 Calculation of Munsell Value V from Luminous Reflectance Y Note 4.2 Modified CIELAB and CIELUV Equations for Dark Colors Note 4.3 Other Color Difference Formulas Note 4.4 Direct Calculation of Hue Difference H * 5 Measurement and Calculation of Colorimetric Values 5.1 Direct Measurement of Tristimulus Values 5.2 Spectral Colorimetry 5.3 Geometrical Conditions for Measurement 5.4 Calculation of Colorimetric Values

71 74 76 78 82 85 92 96 99 101 105 108 109 110 111 113 115 115 122 127 132 134 140 143 144 146 147 150

153 153 156 158 161

CONTENTS

5.5 Colorimetric Values in CIELAB and CIELUV Uniform Color Spaces Note 5.1 Spectral Colorimetry of Fluorescent Materials Note 5.2 Reference Standard for Reflection Measurements

vii

167 172 173

6 Evolution of CIE Standard Colorimetric System 6.1 Additive Mixing 6.2 Subtractive Mixing 6.3 Maximum Value of Luminous Efficacy and Optimal Colors 6.4 Chromatic Adaptation Process 6.5 von Kries’ Predictive Equation for Chromatic Adaptation 6.6 CIE Predictive Equations for Chromatic Adaptation 6.7 Color Vision Models 6.8 Color Appearance Models 6.9 Analysis of Metamerism Note 6.1 Color Mixing Rule Note 6.2 Lambert–Beer Law Note 6.3 Method for Calculating the Maximum Value of the Luminous Efficacy of Radiation Note 6.4 Method for Calculating Optimal Colors Note 6.5 Method for Obtaining Fundamental Spectral Responsivities Note 6.6 Deducing von Kries’ Predictive Equation for Chromatic Adaptation Note 6.7 Application of von Kries’ Equation for Chromatic Adaptation Note 6.8 Application of CIE 1994 Chromatic Adaptation Transform Note 6.9 Theoretical Limits for Deviation from Metamerism

175 176 180

7 Application of CIE Standard Colorimetric System 7.1 Evaluation of the Color Rendering Properties of Light Sources 7.2 Evaluation of the Spectral Distribution of Daylight Simulators 7.3 Evaluation of Whiteness 7.4 Evaluation of Degree of Metamerism for Change of Illuminant

229

184 188 191 194 197 198 204 211 213 214 215 216 221 223 225 226

229 237 242 244

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CONTENTS

7.5 Evaluation of Degree of Metamerism for Change of Observer 7.6 Designing Spectral Distributions of Illuminants 7.7 Computer Color Matching Note 7.1 Computation Method for Prescribed Spectral Distributions

249 255 261 268

Appendix I Basic Units and Terms AI.1 SI Units AI.2 Prefixes for SI Units AI.3 Fundamental Constants AI.4 Greek Letters

271 271 272 272 272

Appendix II Matrix Algebra AII.1 Addition and Subtraction of Matrices AII.2 Multiplication of Matrices AII.3 Inverse Matrix AII.4 Transpose Matrix

275 276 277 277 278

Appendix III Partial Derivatives

281

Appendix IV Tables

285

References Bibliography Index

321 327 329

About the Authors Noboru Ohta Noboru Ohta earned his B.Sci., M.Sci., and Dr.Eng. from the University of Tokyo. In 1968, he joined Fuji Photo Film and from 1973, he spent three years under Gunter Wyszecki at the National Research Council of Canada. He has taught colorimetry and color reproduction at a variety of universities. He joined Rochester Institute of Technology in 1998, and is associated there with the Munsell Color Science Laboratory in the Center for Imaging Science. He has published more than 100 technical papers in Japanese and English, and several books on colorimetry and color reproduction in Japanese, Chinese, and Korean. He has been active in a variety of academic societies, and also in standards organizations such as the Japanese Industrial Standards (JIS), the American National Standards Institute (ANSI), and the International Commission on Illumination (CIE).

x

ABOUT THE AUTHORS

Alan R. Robertson Alan Robertson earned his B.Sc and Ph.D. from the University of London, where he studied under David Wright. He then joined Gunter Wyszecki at the National Research Council of Canada and spent 35 years there before retiring in 2000. He has published over 50 papers in journals and conference proceedings and has given more than 60 invited talks in 10 countries. He is former President of the International Color Association (AIC) and Vice President of the International Commission on Illumination (CIE). In 2005, he received the Godlove Award of the Inter-Society Color Council (ISCC) for long-term contributions in the field of color.

Series Preface How do we define colour? For the most part we are told in our early childhood that an apple is red, a banana is yellow, a lime is green and an orange is orange. Hence, our introduction to colour is a learning process whereby we relate a ‘colour stimulus’ to a descriptive term (which changes with language, of course) supplied by our parents, siblings or teachers. Probably the biggest introduction to colour is the 64 or 128 pack of Crayons® or the 16-colour set of watercolors. Invariably we were told about the primary colours of red, blue, green and yellow or some other set that was popular with those teaching art in elementary schools. There was no concept of additive and subtractive primaries let alone the concepts of hue, chroma and value. Outside of those of us who were artistically gifted or inclined, the biggest issue with colour was matching the paint chips for the living room walls with the choice of fabric for the drapes or the colour of the new car. Our ability to define a colour to others was almost always limited to saying it is sort of sky blue, apple red, lemon yellow or lawn green. In short there was no easy way to define a colour short of having a physical example of what you wanted (and then in a given illuminant). How many times did one buy a matching skirt and blouse or sweater and slacks in a store illuminated by tungsten or fluorescent light sources only to find that the colours shifted (in an undesirable way) in daylight? The science of Colorimetry has evolved to help resolve the above shortcomings of our learned perceptions of colour. The third offering in the Wiley-IS & T Series in Imaging Science and Technology is Colorimetry, Fundamentals and Applications. This text provides a systematic and unambiguous exposition of how colour is defined, measured and seen by human observers under different viewing conditions. In the seven chapters and four appendices of Colorimetry the reader will find both a logical and historic exposition of how colour is physically measured, the methods used to incorporate how

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SERIES PREFACE

the human observer sees the physical stimulus and some practical applications of colorimetry. The authors, Dr Noboru Ohta, of Fuji Photo Film and the Rochester Institute of Technology (RIT), and Dr Alan R. Robertson, National Research Council (NRC) of Canada, have a combined experience in colorimetry of over 70 years. Dr Ohta was a scientist and manager in the Fuji Research Laboratories, where early in his career he focused on the colour aspects of conventional photographic films and image structure. Dr Ohta made the transition to electronic colour imaging systems during the later years at Fuji and has continued his research in electronic colour imaging systems at the Munsell Color Science Laboratory in the Chester F. Carlson Center for Imaging Science at RIT. Dr Robertson has had a long, active and distinguished career in all facets of colorimetry at the NRC, where he rose from a post doctorate fellow to senior research officer to director over his 35 years of service. He has had a major role in the growth of the International Commission on Illumination (CIE) and the International Color Association (AIC), where his efforts resulted in the CIE having the ability to publish colour standards in cooperation with the International Organization of Standardization. In 2005, Dr Robertson received the Godlove Award from the InterSociety Color Council (ISCC) for his long and distinguished service to the colour community. In the early 1970s, Dr Ohta spent several years at the NRC working under the guidance of Dr Gunter Wyszecki. During his stay at the NRC, he met Dr Robertson, thus beginning their long collaboration and friendship. Their combined research in colour, colorimetry and applications of colorimetry and teaching of colorimetry has resulted in this lucid, concise and practical text. When combined with Mark Fairchild’s Color Appearance Models (2nd Edition) and Robert Hunt’s The Reproduction of Colour (6th Edition), the WileyIS&T Series in Imaging Science and Technology provides the student and practitioner of imaging science a comprehensive treatment of colour and colour science. MICHAEL A. KRISS Formerly of the Eastman Kodak Research Laboratories and the University of Rochester

Preface The original version of this book was published in Japanese by the first author. It has had a wide variety of readers in Japan ranging from undergraduate and graduate students in universities to engineers in industries. Due to its success in the Japanese language, Chinese and Korean versions were published soon after the Japanese one. Considering the nature of the book, which covers basic knowledge for novices, practical applications for industrial engineers, and advanced developments for researchers, we have decided to publish an English version to make the content available and, hopefully, useful to a much wider range of readers. However the present book is not merely a literal translation of the Japanese version since that is already more than 10 years old. Instead, we have thoroughly reviewed the content, partly truncated obsolete sections, extensively expanded others, incorporated recent knowledge and added new material that has emerged since the first publication. In writing the book, we employed a variety of materials from books, journals, standards, etc. We gratefully acknowledge the many publishers who kindly gave us citation permission for using their material. We also would like to express our deep appreciation to our friend, the late Dr Heinz Terstiege of the Bundesanstalt für Materialforschung undprufüng, Germany, for the original photographs of color scientists. Finally we would like to express our deepest thanks to Tokyo Denki Daigaku Shuppankyoku, the publisher of the original Japanese version, who kindly gave us permission to translate the Japanese version into the present English one. Noboru Ohta Alan R. Robertson

Introduction Color can be perceived by anyone as long as the person has sound eyesight. Nevertheless, most people find it extremely difficult to explain what ‘color’ is. A typical dictionary definition of color might be ‘a visual perception that enables one to differentiate otherwise identical objects by the intensity and the wavelength of light’. This is not a very satisfactory definition because it describes the physical stimulus that causes the perception rather than describing the perception itself. It would not be very helpful to someone who had never experienced color. Furthermore, the definition can lead to confusion because the word ‘color’ is also used to refer to the physical stimulus itself. In this book, we have accepted the common practice of using the word with both its meanings, but where we think there might be confusion, we have used the more explicit terms ‘perceived color’ and ‘color stimulus’. The difficulty of finding a convincing definition of perceptions without referring to the physical stimuli that evoke them is not confined to color. It occurs with all five senses by which humans detect external stimulation. These five senses are vision (including color), hearing, smell, taste and touch. In the case of the verb ‘smell’, for example, a typical dictionary might define it as ‘to perceive the odor or scent of stimuli affecting the olfactory sense organs of the nose’. Clearly this definition would not mean much to someone who did not have the sense of smell, nor does it offer a distinction between the perception and the stimulus. It is said that 80% of the external stimuli we detect are visual. In fact, there are 217 kanji characters for types of ‘visual perception’, which is a far greater number than the 18 characters that signify ‘auditory perception’. It is clear from this that vision provides a far greater amount of information than the other four senses. This book relates to vision, the most important of the five senses, and, in particular, it describes methods for expressing color

xvi

INTRODUCTION

quantitatively. The quantitative expressions comprise numerical descriptions of color. By analogy, a quantitative expression says; ‘the distance between Tokyo and Osaka is 550 km’, or ‘the distance between Tokyo and Aomori is 740 km’, instead of saying ‘Osaka is far from Tokyo’, or ‘Aomori is very far from Tokyo’. By using such quantitative expressions, the two distances can be accurately compared with the conclusion that: ‘Aomori is 190 km farther from Tokyo than is Osaka’. Similarly, for example, the red circle of the Japanese flag might be described as ‘a red circle with R equal to 16’ instead of using a description with adjectives, such as ‘a slightly vermilion red color, similar to the color of the rising sun’. A Japanese flag must have a length-to-width ratio of 2:3, and the circle must be located at the center, with a diameter equal to 3/5 of the length. Thus the relative dimensions of the flag are always the same. On the other hand, at present, the color of the red circle is not well defined. However, if the color were to be described quantitatively as, for example, R = 16, flags having the same red color could be produced and used at any place as desired. The quantification of color belongs to the field of colorimetry, and that of light belongs to the field of photometry. In this book, the fundamentals of photometry necessary for understanding colorimetry are described first, and then the principles and formulation of colorimetry are explained and recent progress in the field is described. Finally, some fields in industry to which colorimetry is effectively applied are introduced. The book provides a comprehensive background for color engineering, and encompasses the basic principles, development, and application fields of colorimetry. It is intended for students or engineers who are beginners in the field and who want to apply colorimetry, or to those who are already practicing color engineering and want to learn more. The authors hope that those people can acquire the fundamentals of color engineering by reading through the book. For further understanding, important items are described in detail in notes at the end of each section. Noboru Ohta Alan R. Robertson

1 Light, Vision and Photometry 1.1 LIGHT Light is radiation in the form of electromagnetic waves that make vision possible to the human eye. Electromagnetic radiation can be classified by its wavelength or frequency, as shown in Figure 1.1. The wavelength of light is confined to a very narrow range limited by a short-wavelength edge between 360 and 400 nm (1 nm = 10−6 mm; see Appendix I) and a long-wavelength edge between 760 and 830 nm. Infrared radiation and ultraviolet radiation, which are not visible to human eye, are sometimes included in the category of light and referred to as infrared light and ultraviolet light. However, it is better to call these categories infrared radiation and ultraviolet radiation. When it is necessary to distinguish light from radiation not visible to human eye, it is referred to as visible light or visible radiation. Newton (Figure 1.2) showed experimentally that white light, such as sunlight, is composed of various types of colored light. More specifically, he demonstrated the following facts by introducing sunlight into a prism (Figure 1.3). 1. White sunlight incident on a prism is separated into seven components differing in color, as observed in a rainbow. The seven colors are red, orange, yellow, green, blue, indigo, and violet (see Color Plate 1).

Colorimetry: Fundamentals and Applications N. Ohta and A. Robertson © 2005 John Wiley & Sons, Ltd

2

LIGHT, VISION AND PHOTOMETRY

2. The spectrum (i.e., the seven components of light differing in color) can be reunited to give the original white light by focusing the components back through a reversed prism. 3. If one color component alone is incident on a prism, it cannot be further separated into the seven colors. We now know that, when observed in detail, the spectrum includes an infinite range of components of different wavelengths that cannot all be given different color names. The classification into seven named components is based on a simple set of basic

Wavelength (nm)

10

10–6

(nm)

(nm) 380 400

Vacuum ultraviolet

50 Violet

100

P h o t Green o m e t Yellow r Orange y Blue

600

Visible

Infrared

102 1µm

106

1mm 1cm

1

700

108 Near infrared

780

1nm

104

(µ m) Red

1pm

Ultraviolet

Indigo

500

1022

10–4 10–2

100 200

1m 1010

10 15 Far infrared

1012

1km

1018

X-Ray

R a 16 d 10 i o m 1014 e t r y 1012

Light

1010 108 106

1Mm 1016

Gamma ray

Radar

TV

Radio

104

1014

1000

1020

Wireless

100 400

Frequency (Hz) 1024 Cosmic ray

102

Low frequency

100

Figure 1.1 Wavelengths of electromagnetic radiation and light

3

LIGHT

Spectrum White screen

White light

Figure 1.2 Isaac Newton (1643–1727)

Slit Prism Light shield

Dark room

Violet Indigo Blue Green Yellow Orange Red

Figure 1.3 Prismatic dispersion of white light

color names. On detailed examination, other colors can be named. For example, reddish orange colors appear between the pure red and pure orange parts of the spectrum. Monochromatic light is light that cannot be separated into components. White light such as sunlight is polychromatic, i.e., a mixture of monochromatic lights. A spectrum is a band of color observed when a beam of white light is separated into components of light that are arranged in the order of their wavelengths. The approximate correspondence between wavelengths and colors is shown in Figure 1.1. If one or more components is decreased in intensity and the components are recombined, colored light is obtained instead of the original white light. Thus, if an object illuminated with white light reflects the components with differing reflectance depending on the wavelengths, the human eye sees the object as colored. For example, a red object does not reflect much in the range from violet to yellow, but reflects the red component strongly. Thus, it is

4


perceived to have a red color. In general, color is generated whenever white light is modified by reflection or transmission by an object.

1.2 MECHANISM OF THE HUMAN EYE The visual system is very similar to a photographic system in that both respond to light and, in particular, to images. The human eyeball is a sphere about 24 mm in diameter, and its mechanism resembles that of a camera and photographic film. Figure 1.4 shows schematically an eye and a camera. The corresponding components are as follows: Camera Black box Lens Shutter Diaphragm Film

Eye Sclera and choroid Cornea and lens Eyelid Iris Retina

Ear side Vitreous body

Cornea

Optical axis

Fovea centralis

Blind spot

Optic nerve

Nose side

Lens

Photographic lens

Sclera Iris

Choroid Retina

Diaphragm Black box Film

Shutter

Figure 1.4 Structure of eye and camera

MECHANISM OF THE HUMAN EYE

5

Light incident on the eye induces a photochemical reaction in the retina, which corresponds to the photographic film The nerve impulse generated by the reaction is transmitted to the brain to give a visual sensation. The retina covers about two-thirds of the internal surface of the eyeball, and is a transparent film about 0.3 mm in thickness, with a complicated structure comprising several types of cell (Dowling and Boycott 1966). This is illustrated in Figure 1.5. The incident light enters the retina in the direction indicated by the arrows, and reaches the photosensitive neuroepithelial layer. The optic nerve layer, which is located in front of the neuroepithelial layer, performs various types of signal processing. It should be noted that the incident light reaches the neuroepithelial layer after it passes through the transparent optic nerve layer. The photosensitive neuroepithelial layer, which corresponds to the fine photosensitive silver halide (e.g., AgCl, AgBr, or AgI) grains incorporated in a photographic film, consists of two types of cell. These are rods, which perceive brightness or darkness in relatively dark environments, and cones which perceive color in relatively bright environments. The names ‘rods’ and ‘cones’ are derived from the shapes of the cells. There are three types of cone cell, present in the ratio of about 32:16:1, which respond to long-, medium- and short-wavelength light, respectively. Thus, the eye can be thought

[Image not available in this electronic edition.]

Figure 1.5 Structure of retina (Dowling and Boycott 1966). Reproduced by permission of The Royal Society

6


of as being constructed of a high-speed black-and-white film (the rods) and a medium-speed color film (the cones). There are about a hundred million rods and about seven million cones in a human retina. The end of each neuroepithelial cell (the hatched portion in Figure 1.5) is called the external node and contains a photosensitive pigment. The diameter of the external node is between 1 and 2 m for a rod and between 1 and 5 m for a cone. It can be seen therefore that the diameter of the external node is about the same as that of a photographic silver halide grain, which is between 0.05 and 3 m. The human eye has about 60 000 elements per mm2 at the center of the retina, an electronic camera has about 20 000, and a color photograph about 30 000. The distribution of neuroepithelial cells in the retina is shown in Figure 1.6 (Pirenne 1948). The cones are concentrated in the vicinity of the optical axis in the fovea centralis. The fovea centralis is a narrow portion of the retina, about 1.5 mm in diameter, in which approximately 100 000–150 000 cones are concentrated. Maximum resolution is therefore achieved in this narrow portion. In contrast to the cones, rods are rarely found in the vicinity of the fovea centralis, and are distributed over a wide region of the retina. Because the rods, and not the cones, function in dark environments, stars in the sky at night are seen more easily obliquely, i.e., with squinting eyes. The signals generated by the photosensitive pigments in the neuroepithelial cells are processed by various cells, shown in Figure 1.5, and the processed signals are then transmitted to the brain through about one million optic nerves. Because no

Density of neuroepithelial layer cells (mm–2)

×103

Blind spot

160 120 80 40 0

60° 40° Ear side

20°

0°

20°

Fovea centralis

40°

60° 80° Nose side

Figure 1.6 Distribution of rods (solid line) and cones (broken line) (Pirenne 1948)

ADAPTATION AND RESPONSIVITY OF THE HUMAN EYE

7

Figure 1.7 Detection of blind spot

neuroepithelial cell is present in the portion of the retina where the optic nerve penetrates, this portion cannot sense light and is called the blind spot. The blind spot is located at an angle of 15 from the line of sight (optical axis) and is about 5 wide. This can be confirmed readily by a visual experiment using Figure 1.7. If the observer fixates his/her right eye on the cross while closing his/her left eye and adjusting the distance between the eye and the cross to about 20 cm, the solid circle disappears from sight. This occurs because the solid circle is imaged on the blind spot.

1.3 ADAPTATION AND RESPONSIVITY OF THE HUMAN EYE The brightness (more correctly the illuminance) provided by natural and artificial light sources used in daily life ranges widely, as shown in Figure 1.8 (Shoumei Gakkai 1967). The human eye can see an object in direct sunlight where the illuminance is about 100 000 lx, or at night without moonlight at an illuminance of about 0.0003 lx (as described in Section 1.6, the unit of illuminance is the lux, abbreviated lx). To adapt the eye over such a wide range of illuminance, the pupil adjusts the quantity of light reaching the retina by changing its size. Thus, the pupil functions like the diaphragm of a camera. Because the pupil changes its diameter in a range from 2 to 7 mm, the quantity of light adjustable in this way covers a range of only a factor of 12. Thus the change in pupil diameter is insufficient for full control of the quantity of light. Accordingly, the rods and cones share the function by changing the responsivity of the retina. In a relatively bright environment, the cones alone function to give what is

8


1051× 104 103 102 10 1 10–1 10–2 10–3

Figure 1.8 Approximate values of brightness (illuminance) (Shoumei Gakkai 1967). Reproduced by permission of Shoumei Gakkai

called photopic vision. In a relatively dark environment, the rods alone function to realize what is called scotopic vision. In environments having an intermediate brightness between photopic vision and scotopic vision, both the cones and the rods function to provide what is called mesopic vision. Photopic vision is distinguished from scotopic vision by the luminance range in which it operates. Photopic vision occurs for luminances of about 3 cd/m2 or higher (see Sections 1.5 and 1.6 for the definitions of photometric quantities and units), and scotopic vision occurs for luminances of about 0.003 cd/m2 or lower. These numbers depend somewhat on other conditions, such as the color of the stimuli. When one enters a bright environment from a dark one, one’s vision changes from scotopic to photopic via mesopic. This change is completed in about 1 min, and the eye readily adapts to the bright environment. On the contrary, when one enters a dark environment from a bright one, vision changes from photopic to scotopic much more slowly. As shown in Figure 1.9, it takes about 30 min to completely accomplish the adaptation (Chapanis 1947). In photopic vision, the photochemical reaction in the rods saturates, and they become inert to light and the cones alone are left active. The photochemical reaction in the cones continues to an upper limit of about 106 cd/m2 . If this limit is exceeded, the result is a blinding and uncomfortable sensation that can damage the eyes. Referring to Figure 1.9, the detectivity of the eye (the minimum luminance sensed as light) for white light changes from curve A to curve B as the rods take over from the cones upon transfer from light adaptation to dark adaptation.

9

Brightness log (mL)

STANDARD PHOTOMETRIC OBSERVER

–2

A Red light

–4 B

White light

–6 0

10

20 30 Time (minutes)

40

50

Figure 1.9 Progress of dark adaptation

The ordinate of Figure 1.9 is luminance in units of millilambert (mL). These can be converted to cd/m2 by using a conversion factor of 1 mL = 3183=10/cd/m2 . In the initial stage of dark adaptation (about 10 min), cones function to give curve A but, in the later stages, rods with their higher responsivity take over to yield curve B. However, under red light, the portion of curve B in which the rods function does not appear. This is because the rods do not respond to high (red) wavelengths. In scotopic vision, rods are active and exhibit a relatively high response to light. With decreasing luminance, however, even the rods finally become insensitive. Depending on the experimental conditions, the luminance limit at which the rods lose sensitivity is about 10−6 cd/m2 . Taking into account absorption and scattering of light inside the eye and the absorption efficiency of the retina, this limit corresponds to about 5–14 photons incident on the rods. Cones on the other hand require about 100–1000 as many photons before they respond. By comparison, four or more photons are necessary to induce a reaction in the fine silver halide grains of high-speed photographic film. It can be seen that rods have a detectivity that compares well to that of photographic film.

1.4 SPECTRAL RESPONSIVITY AND THE STANDARD PHOTOMETRIC OBSERVER The output of a photodetector divided by the radiant energy input is called its responsivity. The term can be applied to the human eye as well as to physical detectors. For the eye, the output is a brightness response. In the past, the term sensitivity was used, but responsivity is now preferred. The higher the responsivity, the

10


higher is the output for a given input. When the responsivity is expressed as a function of wavelength, the curve is called the spectral responsivity. Equal amounts of radiant energy become less visible to the human eye with decreasing or increasing wavelength on either side of a maximum. Outside the visible region, which extends from about 380 to 780 nm, radiation becomes invisible. Thus the spectral responsivity of the eye is a function of wavelength, decreasing gradually to zero in the ultraviolet and infrared regions. Furthermore, since the manner of transfer from light adaptation to dark adaptation in white light differs from that in red light, it is clear that the spectral responsivity of rods differs from that of cones. In general, the spectral responsivity of a photoreceptor is determined for each wavelength by introducing a monochromatic light of known radiant energy and then measuring the response in the form of a photocurrent, for example. The response of the eye, however, is not determined by a physical measurement, but rather in terms of a brightness sensation. Thus, to obtain the spectral responsivity of the eye, a means such as matching is employed. More specifically, in the matching method, a predetermined reference light having a certain wavelength is used, so that the brightness v of a test light having an arbitrary wavelength may be matched with that of the reference light. By measuring the radiant energy e of the test light in the match, v can be expressed as v = K e

(1.1)

where K is a measure of the brightness per unit radiant energy. Thus, K is the responsivity of the eye. (At this stage, the quantities and units of brightness and radiant energy have not been defined so the units of K are arbitrary. These matters are discussed later in this chapter.) The brightness of two different lights can be matched by any of the following methods: 1. Direct comparison method. This method comprises directly comparing a test light of wavelength 2 with a reference light of wavelength 1 . In principle it is the simplest method, but it is very difficult experimentally to match, for example, the brightness of a red light with that of a blue light. Consequently, the results fluctuate and the method suffers from poor precision. 2. Step-by-step method. Although it is difficult to compare the brightnesses of lights differing greatly in color, those having similar colors can be readily compared. Thus, by using a test light having a wavelength 2 near the wavelength 1 of the reference light,


11

the responsivity for a wavelength 2 can be determined. Then the responsivity for a light having a wavelength 3 near 2 can be obtained by using the previous test light as the reference light. By repeating this process sequentially, the spectral responsivity for the whole spectrum can be obtained, step-by-step. 3. Flicker method. By alternately introducing a reference light having a wavelength of 1 and a test light having a wavelength of 2 into the visual field, the color can be made to flicker, for example, between red and green. On increasing the frequency of repetition, the two colors merge into one at a frequency of about 30 Hz. Above this frequency, no color change is perceived. In the example given, the red and green lights merge to yield yellow. However, if the two lights differ in brightness from each other, the difference in brightness remains as a flicker even if the colors merge into one. On further increasing the frequency to a value higher than 50 Hz, the brightness as well as the color merges to yield a uniform visual field. By utilizing the frequency region in which the flicker attributed to brightness remains, but that for color disappears, the brightness of the test light can be matched with that of the reference light. In the direct comparison method, results cannot be determined with high precision. However, a relatively stable result can be obtained by the step-by-step method or by the flicker method. By setting the reference light sufficiently dark and performing the experiment in scotopic vision, the spectral responsivity of the rods can be measured. On the other hand, if the reference light is set sufficiently bright, photopic vision operates and the responsivity curve of the cones can be measured. The spectral responsivity of the cones can also be obtained by narrowing the observation field to about 2 , because no rods are present in the fovea centralis. As described above, the value of K in Equation 1.1 corresponds to the responsivity of the eye. The value of K is called the luminous efficacy of the radiation. The spectral luminous efficacy, K, can be determined by varying the wavelength, and observing K as a function of . The maximum value, Km , of K is called the maximum luminous efficacy, and the ratio of K to Km , is called the spectral luminous efficiency, V. The maximum luminous efficacies, Km and Km , are related to the spectral luminous efficacies, K and K , by the following equations K = Km V

K =

Km V

(1.2a) (1.2b)

12


where V and V are the spectral luminous efficiencies in photopic and scotopic vision, respectively. Because K ≤ Km and K ≤ Km , the maximum values for V and V are 1.0. Once the spectral responsivity for brightness is known, the brightness of lights differing in color can be treated quantitatively. However, to compare brightness on a worldwide basis, it is necessary for everyone to use the same spectral responsivity function. The Commission Internationale de l’Élairage (CIE) is an international organization that researches and recommends standards related to light. The CIE has established two spectral responsivity curves that are universally used. In 1924, the CIE established the spectral luminous efficiency for photopic vision, V, based on the average observed values from 7 studies involving 251 people with normal color vision. Similarly, in 1951, it established the spectral luminous efficiency for scotopic vision, V (Figure 1.10 and Table 1.1). The responsivities thus established are the average values for a large number of observers. Although a real observer does not necessarily exist with exactly the spectral responsivities illustrated in Figure 1.10, virtual observers with these responsivities are known as the CIE Standard Photometric Observers. The spectral luminous efficiency functions recommended by the CIE were determined by the step-by-step method and the flicker method described above. These experimental methods use special conditions for the evaluation of brightness whereas direct comparison is usually employed in practical situations. However, as is illustrated in Figure 1.11, the fluctuation of the values obtained by direct comparison is too large (Ikeda et al. 1982). Hence, results of the direct comparison method are not used as basic data for establishing spectral luminous efficiency. However, numerous observed

Luminous efficiency

1.0

Scotopic vision V ′(λ)

0.8 Photopic vision V (λ)

0.6 0.4 0.2 0

400

500 600 Wavelength (nm)

700

Figure 1.10 Spectral luminous efficiency in photopic and scotopic vision

13


Table 1.1 Spectral luminous efficiency in photopic and scotopic vision Wavelength (nm) 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780

Photopic vision V

Scotopic vision V

0.0000 0.0001 0.0004 0.0012 0.0040 0.0116 0.0230 0.0380 0.0600 0.0910 0.1390 0.2080 0.3230 0.5030 0.7100 0.8620 0.9540 0.9950 0.9950 0.9520 0.8700 0.7570 0.6310 0.5030 0.3810 0.2650 0.1750 0.1070 0.0610 0.0320 0.0170 0.0082 0.0041 0.0021 0.0010 0.0005 0.0003 0.0001 0.0001 0.0000 0.0000

0.0006 0.0022 0.0093 0.0348 0.0966 0.1988 0.3281 0.4550 0.5670 0.6760 0.7930 0.9040 0.9820 0.9970 0.9350 0.8110 0.6500 0.4810 0.3288 0.2076 0.1212 0.0655 0.0332 0.0159 0.0074 0.0033 0.0015 0.0007 0.0003 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

14


Luminous efficiency (logarithm)

4

3

2

1

400


700

Figure 1.11 Variations (broken lines) and Vb (solid line) for spectral luminous efficiency obtained by the direct comparison method (Ikeda et al. 1982). Reproduced by permission of Optical Society of America

results have been accumulated by the direct comparison method, and the average values Vb have been found to deviate significantly from the values V established by the CIE (see Figure 1.12). It might seem appropriate therefore to replace V by Vb . However V is the prevailing standard in the world, and it is not simple to change it. Furthermore, Vb applies to the direct comparison of brightnesses and is not appropriate for predicting other psychophysical phenomena, such as visual acuity, for which V works better. Thus, the CIE has recommended Vb separately in addition to V (CIE 1988). These issues are discussed in more detail on page 29 and in Note 1.2. The spectral luminous efficiency function for mesopic vision V ∗ , is intermediate between V and V . However, a CIE standard photometric observer for mesopic vision is not yet established. Brightness in mesopic vision cannot be determined unless V ∗ is established, but the problem is not simple because the relative contribution of V and V is complicated. Several methods have been proposed for the determination of V ∗ including one by Sagawa et al. (1992). As the eye’s vision moves from photopic to scotopic via mesopic, the wavelength of maximum luminous efficacy moves to shorter wavelengths because the spectral luminous efficiency changes from

15


Luminous efficiency (logarithm)

0

–1

–2

–3

400


700

Figure 1.12 Spectral luminous efficiency V for the CIE Standard Observer for photopic vision (solid line) and spectral luminous efficiency Vb determined by the direct comparison method (broken line) (CIE 1988)

V to V . Thus, a phenomenon occurs in which two colors having the same brightness in photopic vision, have different brightness in scotopic vision. For example, the brightness of a red color tends to decrease relative to a blue color with the transfer to scotopic vision, even if the red and blue colors are of the same brightness in photopic vision. This phenomenon is called the Purkinje phenomenon after its discoverer. The reason why a darkroom operator wears red eyeglasses or why the illumination in a submarine is often red can be explained by this phenomenon. As can be seen from Figure 1.10, only the cones function in red light, whereas the rods are not stimulated. Accordingly, once the operator returns to the darkroom or observes the dark surface of the sea through a periscope without wearing the red eyeglasses, the rods can be put to work immediately because they have remained dark-adapted. Instead of using matching, the spectral responsivity of the human eye can be measured by a threshold technique. A threshold is the minimum radiant energy that makes light visible. Responsivity can be obtained as the inverse of the threshold. In other words, the higher the responsivity, the lower the threshold. Continuing the analogy with photography, responsivity (sensitivity) there is expressed by film speed as, for example, ISO (ASA) 1000, which is

16


an inverse of a value similar to a threshold value. Figure 1.13 shows the threshold values for rods and cones as a function of wavelength. In theory, the spectral responsivity obtained from thresholds may not agree with the spectral luminous efficiency obtained by matching. However, on deriving a spectral responsivity curve from Figure 1.13 and comparing it with that in Figure 1.10, it is confirmed that the two are approximately the same. In Figure 1.13 two curves are shown that define three regions I, II, and III. Region I corresponds to color vision in which the cones work to sense light and color. Region II is a colorless region in which only light is sensed but not color. In this region, only the rods function. Nothing is sensed in region III because it is below the threshold value for both cones and rods. A light having a wavelength of 650 nm or longer is red-colored if it can be seen at all. No colorless region is present for these wavelengths. This is one of the reasons why a red light is used in important cases such as traffic signals. Although the spectral responsivity of cones differs somewhat from that of rods, they both have a maximum in the middle of the visible wavelength region. With recent progress in artificial illumination, activity at night can be carried out without any problem. However, considering the long history of the human race, activity under artificial illumination accounts for a very short period


Figure 1.13 Spectral responsivity of cones and rods obtained from threshold measurements (Wald 1945). Reprinted with permission from Human vision and the spectrum, Science 101. Copyright (1945) AAAS

17

Relative energy

DEFINITION OF PHOTOMETRIC QUANTITIES

1.5 1.0 0.5 0

0.2

0.4

0.6

0.8 1.0 1.2 1.4 Wavelength (µ m)

1.6

1.8

2.0

Figure 1.14 Spectral distribution of daylight (solid line) and spectral sensitivity (broken lines) of cones (right) and rods (left) (Ikeda 1975b). Reproduced by permission of Asuara Shoten

and, historically, human activity has taken place principally under daylight. Presumably, the evolution of the spectral responsivity of the human eye occurred so as to fully and efficiently utilize daylight. Figure 1.14 shows the spectral distribution of daylight and the spectral responsivity of rods and cones, and it can be seen that the maximum responsivities occur at a wavelengths roughly corresponding to those at which the radiant energy of daylight is at its maximum (Ikeda 1975a,b).

1.5 DEFINITION OF PHOTOMETRIC QUANTITIES In daily life, the term ‘brightness’ is used without thinking carefully about its meaning. For instance, laser radiation is considered ‘bright’, but its brightness is insufficient if it is to be used to illuminate a room. On the other hand, a few fluorescent lamps will provide sufficient light for room illumination although, considered as single lamps, they are not considered to be as ‘bright’ as lasers. To explain this apparent paradox, the brightness of light must be judged by a value normalized by solid angle or area, just as a substance is judged to be light or heavy by considering its mass per unit volume, i.e., its density. Photometry comprises measuring the ‘brightness’ of lights and normalizing the measurements in various ways to obtain photometric quantities. Photometric quantities include quantity of light, luminous flux, luminous intensity, illuminance, luminous exitance, and luminance. Luminous flux v is a quantity obtained by evaluating radiant flux e according to the luminous efficacy of the human eye. Quantity of light Qv is obtained by integrating the luminous flux over time and the other quantities are obtained by various geometric normalizations. Thus, any photometric quantity can be obtained by

18


evaluating the corresponding radiometric quantity according to the luminous efficacy. As noted above, the appropriate value of luminous efficacy depends on the wavelength of the radiation and on the type of vision involved (photopic, scotopic or mesopic). This is discussed in more detail in Section 1.7 after the geometric factors have been explained. A radiometric quantity is a physical quantity measured in units such as joules (J) or watts (W), whereas a photometric quantity is an operationally defined quantity designed to represent the way in which the human visual system evaluates the corresponding radiometric (physical) quantity. Accordingly, it is called a psychophysical quantity. The tristimulus values of color to be described later are also psychophysical quantities. In geometric terms, the definitions of photometric quantities are the same as for the corresponding radiometric quantities. Accordingly, the explanations below are made in terms of photometric quantities, but they apply equally well to radiometric ones. For photometric quantities, a subscript v is normally added to the symbol for each quantity, whereas for radiometric quantities, a subscript e is added. Figure 1.15 illustrates a luminous flux d that passes through two plane elements, A and B. Because the luminous flux passing through the plane elements A and B remains the same (in the absence of any absorption or scattering), it can be expressed as d = I dm = I dm

(1.3)

where, dm and dm are the ‘sizes’ of A and B, respectively, and I and I are the ‘densities’ of luminous flux, respectively. If the light source is infinitesimally small, as shown in Figure 1.16, it becomes a point. Then, because light travels in straight lines, the plane elements are cross sections of the same cone, and the following relation can be obtained by defining the

B

A

dm′ dm

Figure 1.15 Luminous flux passing through plane elements A and B

DEFINITION OF PHOTOMETRIC QUANTITIES

19

dm′

dm

B

A

Figure 1.16 Luminous flux and luminous intensity inside a cone

‘sizes’ of the plane elements by the solid angles d and d that they subtend at the source. Since d = d

(1.4)

the value of I for the point light source can be obtained as I = I = d/d

(1.5)

where I is called the luminous intensity, d is the solid angle and d is the flux within d. If, as is shown in Figure 1.17, the light source is a planar surface, A and if the size of the light source is not negligible with respect to the distance of observation, luminous intensity cannot be defined so easily. However, because a planar light source can be regarded as a group of plane elements, each acting as a point light source, the luminous flux in the plane B increases as the area of the source increases. Thus it is useful to consider the luminous intensity of the light source per unit area. This quantity is called luminance, L and is expressed as L = dI/dS cos

N

(1.6)

B

θ dS A

Figure 1.17 Luminous flux and luminance from a planar light source

20


where dI is the intensity emitted from a plane element of area dS, is the angle between the normal N of the plane A and the light beam AB. The projected area of the plane element, perpendicular to the light beam is dS cos . Combining Equations 1.5 and 1.6 gives L = d/dS cos d

(1.7)

Although the form of Equation 1.7 is not strictly correct from a mathematical point of view because d/(dS d) should be written as a double partial derivative 2 / S , this form of the equation is usually used in photometry because it relates more clearly to the measurement process. Referring to Figure 1.17, if the luminous flux, d at point B changes with in proportion to cos , the luminance, L remains constant because the apparent area of the light source also changes in proportion to cos . This type of light source is called a perfect diffusing planar light source. In the case of luminous intensity, the plane element is defined by the solid angle it subtends at the source. If, instead, the plane element is considered in terms of its area, another quantity, illuminance can be defined as shown in Figure 1.18. Illuminance is the luminous flux incident on a given plane (plane B in Figure 1.18) per unit area. If the luminous flux leaving the plane per unit area is considered, the term luminous exitance is used instead of illuminance but is expressed by the same equation. Thus, illuminance E and luminous exitance M can be expressed by E = d/dS

(1.8a)

M = d/dS

(1.8b)

where, dS is the area of a plane on which luminous flux d is incident (Equation 1.8a) or from which luminous flux d is leaving (Equation 1.8b).

N

dS2 θ2

D

dS2

N

B

N θ1

θ2 D

B

A

A (a)

dS1

(b)

Figure 1.18 Luminous flux and illuminance from a point light source (a) and from a planar light source (b)

21

PHOTOMETRIC UNITS

Figure 1.18(a) illustrates the illuminance E on plane B resulting from a point source, A. By taking the angle between the normal to the plane B and the light beam AB as 2 , the area of the plane B projected perpendicular to AB is expressed as dS 2 cos 2 . This projected plane subtends a solid angle of d at point A expressed by d = dS2 cos 2 /D 2

(1.9)

where D is the distance between A and B. The flux d is obtained from Equation 1.5 as d = I d = I dS2 · cos 2 /D 2

(1.10)

Thus, the illuminance, E incident on plane B is E = d/dS2 = I cos 2 /D 2

(1.11)

In the case of a planar light source, A as is shown in Figure 1.18(b), d can be expressed, from Equation 1.7 as d = L dS1 cos 1 d = L dS1 cos 1 dS2 cos 2 /D 2

(1.12)

Thus, the illuminance E can be expressed by E = d/dS2 = L dS1 cos 1 cos 2 /D 2

(1.13)

Table 1.2 shows the definition of photometric quantities together with the corresponding radiometric quantities. Figure 1.19 schematically illustrates the photometric quantities (Noguchi 1987). In Table 1.2, t represents time in seconds (s), represents solid angle in steradians (sr), S represents area in square meters (m2 ), and represents the angle between the normal of the plane element and the direction of observation. The solid angle is expressed in units of steradian, which is defined as the solid angle that, having its vertex at the center of a sphere of radius r , cuts off an area r 2 in the surface of the sphere. Thus, the whole surface of a sphere yields a solid angle of 4 sr because the surface area of a sphere is 4r 2 .

1.6 PHOTOMETRIC UNITS In the SI (the International System of units), luminous intensity is a base unit. The SI evolved from the MKSA unit system. Three units, the kelvin (abbreviated K) for temperature, the mole (abbreviated

22


Table 1.2 Definition of radiometric and photometric quantities radiometric quantities Quantity

Definition

Unit

radiant energy radiant flux radiant intensity irradiance radiant exitance radiance

Qe e = dQe /dt Ie = de /d Ee = de /dS Me = de /dS Le = de /dS · cos · d

J W (J/s) W/sr W/m2 W/m2 W/(sr · m2 )

photometric quantities Quantity

Definition

Unit

quantity of light luminous flux luminous intensity illuminance luminous exitance luminance

Qv v = dQv /dt Iv = dv /d Ev = dv /dS Mv = dv /dS Lv = dv /dS · cos · d

lm · s lm lm/sr (cd) lm/m2 (lx) lm/m2 lm/(sr · m2 )

(t : time, : solid angle, S: area, and : the angle between the normal of the plane element and the direction of observation)

Radiant flux dΦe (luminous flux dΦv)

Radiant flux dΦe (luminous flux dΦv)

Solid angle element dω Radiant intensity (luminous intensity) Radiant flux dΦe (luminous flux dΦv)

Plane element dS Radiant exitance (luminous exitance)

Plane element dS Irradiance illuminance

Normal θ Radiant intensity dIe (luminous intensity dIv) of a plane element Plane element dS in a given direction Radiance (luminance)

Figure 1.19 Definition of radiometric and photometric quantities. Reproduced by permission of Ohm Sha

PHOTOMETRIC UNITS

23

mol) for amount of substance, and the candela (abbreviated cd) for luminous intensity are added to the four basic units of the MKSA system — meter (m), kilogram (kg), second (s), and ampere (A) — to provide the seven base units of the SI. In addition to the base units, the radian (rad) for planar angle and the steradian (sr) for solid angle are included as derived units. For the radiometric quantities listed in Table 1.2, the joule (J) is the unit of radiant energy and the watt (W) is the unit of radiant flux. For the photometric quantities, the base unit is the candela (cd). Units derived from the candela include the lumen (lm) for luminous flux and the lux (lx) for illuminance. As is shown in Table 1.2, the lumen is defined as the luminous flux emitted into a solid angle of one steradian by a point source having a uniform luminous intensity of one candela. The lux is defined as the illuminance produced by a luminous flux of one lumen uniformly distributed over an area of one square meter. Over time, the standard for the candela has evolved and increased in precision, as is the case with the other base units such as the meter. The unit of length was determined, in earlier times, in accordance with the length of a part of the human body such as one foot. Later a more universal unit, the ‘meter’ was introduced and was originally defined to be one ten-millionth of the quadrant from the equator to the North Pole through Dunkirk. However, developments in the precision of measurements of optical wavelengths led to the adoption of a definition in terms of the wavelength of a spectrum line of krypton. Nowadays, the meter is defined as the length of the path traveled by light in vacuum during a time interval of 1/299792458 s. Similarly, in earlier times, the standard of luminous intensity was provided by burning a specified candle made of whale oil. This is the origin of the unit name ‘candle’ or ‘candela’. There was also an era when a more stable pentane lamp was used in place of the whale-oil candle to provide the standard. Subsequently, further precise standards were established. In 1948, the candela was defined as the ‘luminous intensity in the direction perpendicular to a surface of area 1/60 cm2 of a black body at the temperature of the freezing point of platinum (2042 K) under a pressure of 101325 Pa (one atmosphere)’. Figure 1.20 illustrates the mechanism of a standard which realized the candela according to the above definition (Wyszecki and Stiles 1982). In 1979, the standard was revised, and the candela was re-defined as the ‘luminous intensity in a given direction of a source that emits monochromatic radiation of 540×1012 Hz and that has a radiant intensity in that direction of

24

Photoreceptor

LIGHT, VISION AND PHOTOMETRY Lens Prism Diaphragm Ceramic crucible Thoria powder Thoria crucible Platinum Thoria powder High frequency heating coil

Figure 1.20 Standard for luminous intensity (Wyszecki and Stiles 1982). Reproduced by permission of John Wiley & Sons, Ltd

1/683 watt (W) per steradian’. (540 × 1012 Hz is equivalent to a wavelength of about 555 nm, which is where V is unity.) As described above, the standard for photometric units started from a concrete concept such as a candle, but is now defined precisely in terms of the radiometric quantity of monochromatic light. However, it should be re-emphasized that photometric quantities are psychophysical. At frequencies other than 540 × 1012 Hz, they must be linked to radiometric quantities by a convention (the V curve) that represents the response of the human visual system. For practical implementation in the industrial field, a standard luminous intensity is provided by a standard photometric light source calibrated by the national metrology institute in each nation because of the difficulty in realizing the above definition directly. Among the photometric quantities, illuminance and luminance are widely used in the industrial field. The simplest system of units for these and other photometric quantities is the SI. In this system, illuminance is measured in lm/m2 (usually called lux, abbreviated lx), and luminance is measured in cd/m2 (sometimes called nits, abbreviated nt). However, various other units are often used depending on needs and on history. The conversion factors for some of the most common of these units are listed in Table 1.3 (Ohyama 1969). However there are still other units that are not given in the table. The ‘equivalent’ units shown in Table 1.3 are introduced to relate illuminance with luminance. For example, the luminance of a perfect diffusing plane under an illuminance of 1 phot is 1 equivalent phot. In the measurement of sensitivity in photography, the illuminance unit meter-candle (m cd) is used conventionally. One meter-candle is equivalent to one lux.

Lux (lm/m ) Phot Milliphot Footcandle

Nit (cd/m2 ) Stilb Apostilb Lambert Millilambert Footlambert Candela/ft2 Candela/in2

1 10000 03183 3183 3183 3426 1076 1550

Nit (nt)

00001 1 000003183 03183 00003183 00003426 0001076 0155

Stilb (sb)

1 1000 10 1076

Lux (lx)

3142 31420 1 10000 10 1076 3382 4869

Apostilb (asb)

00001 1 0001 0001076

Phot (ph)

1 equivalent phot =1 L 1 equivalent lux =1 blondel =1 asb 1 equivalent fcd =1 fL

Notes: (1) 1 foot (ft) equals 0.3048 m (2) The coefficient 3.142 in the table is the value of (3) In luminance, the following equivalent units are sometimes used:

1 1 1 1 1 1 1 1

(b) Luminance

1 1 1 1

2

(a) Illuminance

00003142 3142 00001 1 0001 0001076 0003382 04869

Lambert (L)

03142 3142 01 1000 1 1076 3382 4869

Millilambert (mL)

01 1000 1 1076

Milliphot (mph)

02919 2919 00929 929 0929 1 3142 4524

Footlambert (fL)

00929 929 0929 1

Footcandle (fcd)

Table 1.3 Units for illuminance and luminance (Ohyama 1969). Reproduced by permission of Seishin Shobo

26


In another special case, retinal illuminance is used for expressing illuminance on the retinal surface by taking into consideration the diameter of the pupil of the eye. Retinal illuminance is used principally in the field of physiological optics and has the unit troland (td). When the eye views a surface of uniform luminance, the number of trolands is equal to the product of the area in mm2 of the limiting pupil (natural or artificial) and the luminance of the surface in cd/m2 . Thus, the retinal illuminance Er when a surface having a luminance of L cd/m2 is viewed with a pupil of r mm diameter can be written as Er = r 2 L

(1.14)

Taking into consideration that 1 mL = 10 / cd/m2 , a surface having a luminance of 1 mL provides a retinal illuminance of 10 td through a pupil having a radius of 1 mm.

1.7 CALCULATION AND MEASUREMENT OF PHOTOMETRIC QUANTITIES As described above, radiometric quantities can be converted into photometric quantities and vice versa by the equation (photometric quantity) = K× (radiometric quantity). One lumen corresponds by definition to 1/683 W of a monochromatic light of m = 555 nm, which is the wavelength where V has its maximum value of unity. Thus, Km , the maximum luminous efficacy of radiation for photopic vision is 683 lm/W. In the case of scotopic vision, on the other hand, V = 0.40175 at = 555 nm as can be obtained by interpolating the values in Table 1.1. For monochromatic light having a radiant intensity of 1/683 W at = 555 nm to yield a luminous flux of 1 lm in scotopic vision, Km , the maximum luminous efficacy for scotopic vision must be set to 683/0.40175 = 1700 lm/W. This maximum occurs at m = 507 nm. Thus, Equations 1.2a and 1.2b can be written explicitly as K = Km V = 638V

K =

Km

V = 1700V

(1.15a) (1.15b)

Figure 1.21 shows the comparison between the spectral luminous efficacies K and K obtained through Figure 1.10 and Equations 1.15a and 1.15b. In general, photometric quantities for photopic vision are used more frequently than those for scotopic

27

PHOTOMETRIC QUANTITIES

Km' = 1700 lm/W

Luminous efficacy (lm/W)

1600 Scotopic vision K(λ) = Km′ ⋅ V ′(λ)

1200

800

Km = 683 lm/W Photopic vision K(λ) = Km ⋅ V(λ)

400

0

λm = 555 λm' = 507 400


700

Figure 1.21 Spectral luminous efficacyK and K for photopic and scotopic vision respectively

vision. Accordingly, the following explanation is given using the symbols for photopic vision. In the above description, only monochromatic light is treated. However, in daily life, monochromatic light is rare, and, in general, polychromatic light, such as sunlight or light emitted from an artifical lamp with a continuous or partly continuous polychromatic spectrum, is used. By finely dividing a polychromatic light as is shown in Figure 1.22, it can be considered to consist of a number of near-monochromatic components. The quotient X of a radiometric or photometric quantity dX in an elementary range of wavelength d at a wavelength, by that range, i.e., dX /d, is called the spectral concentration, and the spectral concentration expressed as a function of wavelength is called the spectral distribution and denoted by X . In particular, the spectral concentration of radiant flux, e = de /d is called spectral radiant flux, and the spectral distribution of radiant flux is denoted by e . By finely dividing the spectral distribution e into n sections of wavelength width , the i th radiant flux (with a wavelength of i can be written as e i . Thus, the total radiant flux e for polychromatic light with a spectral distribution of e can be found by summation as follows e =

n i =1

e i

(1.16)

28


Relative power

10

Φe,λ(λi)⋅∆λ

8 6 Φe,λ(λi)

4 2 0

i=1 2 3 4 400

n – 1n 500 ∆λ 600 Wavelength (nm)

700

Figure 1.22 Approximation of spectral distribution e by narrow-band near-monochromatic components

where the summation is over the entire wavelength range of the distribution e . In the case of a physical quantity such as radiant flux, the additivity implied in Equation 1.16 holds true in a strict sense. However, in the case of luminous quantities, it is difficult to predetermine theoretically whether it is correct to apply additivity. Experimentally, however, it can be confirmed that additivity holds for luminous flux within a certain precision. That is, two radiant fluxes e 1 and e 2 in narrow bands of wavelength , at wavelengths 1 and 2 , can be safely converted into luminous fluxes v 1 and v 2 , and added together. According to Equations 1.1 and 1.2, the individual luminous fluxes are written as v 1 = Km e 1 V1

(1.17a)

v 2 = Km e 2 V2

(1.17b)

The fact that additivity is valid means that, when the two radiant fluxes are viewed together, the total luminous flux is v = v 1 + v 2 = Km e 1 V1 + e 2 V2

(1.18)

This is known as Abney’s Law. By extending Equation 1.18 to n radiant fluxes, the luminous flux of polychromatic light can be obtained as v = Km e 1 V1 + e 2 V2 + · · · + e n Vn (1.19)

29

PHOTOMETRIC QUANTITIES

By taking limits → 0 and n → , the following equation is obtained v = Km e Vd (1.20) vis

where the integral is taken over the entire visible wavelength region. In practice, Abney’s law of additivity holds very well when the luminous quantities are used to predict psychophysical phenomena such as the perception of fine detail, reaction times and heterochromatic flicker photometry. However, it holds less well for predicting the relative perceived brightness of stimuli of different colors. (see Note 1.2) Photometric methods can be classified into two types, visual and physical. Visual photometry is simple, but has the following disadvantages: 1. The spectral luminous efficiency V for the CIE standard observer is an average value, and hence it does not agree exactly with the spectral luminous efficiency for most individual observers. 2. It is extremely difficult to compare quantitatively two radiations differing in brightness. The only viable method is the null method in which one of the two radiations is adjusted until the two are equal to each other. 3. Even with the null method, it is difficult to compare the brightness of two radiations that differ in color (e.g., red and green), and thus the results have a large uncertainty. 4. The results are influenced by the degree of fatigue of the observer and by the experimental environment. Disadvantages 2 can be overcome to a certain degree by using an optical attenuator (for example, an optical bench). However, an optical bench requires an extremely large space (typically an area of about 3 × 14 m for 10 m bench). Disadvantage 3 can be overcome by using the flicker method but, although the flicker method enables results with higher precision than the direct comparison method, it requires special measuring conditions, and the results are still of low reliability compared with other physical measurements. Accordingly, in the derivation of the V curve, an average value had to be determined by repeating the measurements a number of times. In the light of these drawbacks of visual photometry, physical photometry is usually used in practice. Physical photometry can be performed in the following two ways.

30


1. Measurement of the corresponding spectral radiant quantity, usually by measuring a ratio at successive wavelengths with the value obtained with a standard photometric light source, followed by calculation of the photometric quantity using Equation 1.20. 2. Direct measurement of the photometric quantity by using a photoelectric detector such as a photoelectric cell, a photoelectric tube, or a photomultiplier in combination with a suitable optical filter to provide a photodetector having a relative spectral responsivity approximately equal to V. Such a detector is known as a V photodetector.

Spectral responsivity

The spectral method 1 comprises a comparative measurement for each wavelength. Accordingly, the spectral concentration of the radiometric quantity as well as the photometric quantity can be measured with high precision, and the relative spectral responsivity of the photoreceptor need not be the same as V. An instrument for measuring photometric quantities by means of a V photoreceptor (method 2) is called a photometer. The key to this method is the accuracy of the approximation to V by the relative spectral responsivity of the photometer. Photometers capable of approximating V with a sufficiently high precision for most applications have been developed and are commercially available. An example is shown in Figure 1.23. Such photometers are useful for simple photometry.

1.0

0.5

0

400


700

Figure 1.23 Spectral responsivity (solid line) of a photometer and V (broken line)

RELATIONS BETWEEN PHOTOMETRIC QUANTITIES

31

1.8 RELATIONS BETWEEN PHOTOMETRIC QUANTITIES The photometric quantities described in Section 1.5 have different dimensions and hence cannot, in general, be converted one into another. However, if the quantities are obtained under a light source with certain well-defined characteristics, some quantities can be converted to others by simple relations. For example, the values of photometric quantities obtained under a perfect diffusing planar light source or a point light source can be converted in the following manner.

1.8.1 Perfect Diffusing Planar Light Source From Equation 1.6, the luminance, Ln normal to a plane element of area dS on a surface, and the luminance, L in a direction making an angle of with respect to the normal, are Ln = dIn /dS

and

L = dI /dS cos

(1.21)

where dI n and dI are the luminous intensities normal to the plane element and in a direction making an angle, with respect to the normal, respectively. In the case of a perfect diffusing planar light source, the luminance is the same for all directions so L = Ln dI = dIn cos

(1.22a) (1.22b)

This is known as Lambert’s cosine law. If the plane surface satisfying Lambert’s cosine law is a reflecting surface instead of a light source, the reflecting surface is called a uniform reflecting diffuser. In particular, a uniform reflecting diffuser is called a perfect reflecting diffuser if it has a reflectance = 1.0. In daily life, a surface of gypsum or white chalk is close to a perfect reflecting diffuser. For a perfect reflecting diffuser, the luminous exitance M (lm/m2 and the illuminance E (lm/m2 can be related to luminance L (cd/m2 by the following equations (see Note 1.1) E = L

(1.23a)

M = L

(1.23b)

32


Thus, if the illuminance E on a perfect reflecting diffuser is known, the luminance L of the light reflected is L = E/

(1.24)

In the case of a uniform reflecting diffuser with < 1.0, the luminous exitance M = E, where E is the illuminance, and hence L = E/

(1.25)

Table 1.3 shows that 1 mL = 3.183 cd/m2 . Accordingly, a luminance of 1 mL for a perfect reflecting diffuser corresponds to an illuminance of 10 lx since, from Equation 1.23a, E = L = 3142 × 3183 = 10.

1.8.2 Uniform Point Light Source The total luminous flux of a point light source having the same luminous intensity I in all directions can be obtained from Equation 1.5 as = 4I

(1.26)

This relation is based on the fact that the total solid angle with respect to a point light source is 4 sr.

1.8.3 Correlation with Brightness Illuminance, luminance, and retinal illuminance can be roughly related with each other in a typical situation as shown in Figure 1.24. The luminance is obtained according to Equation 1.25 by taking = 05. The conversion from luminance to retinal illuminance assumes a pupil size of 15 mm2 at a luminance of 1 cd/m2 and 44 mm2 at a luminance of 0.001 cd/m2 . These values represent typical values of the eye’s natural pupil at these luminances. The question arises which photometric quantity correlates best with the sensation of ‘brightness’. For the sake of simplicity, let us consider the case of a 100 W incandescent lamp. The efficiency of an incandescent lamp is about 15 lm/W. Accordingly, a luminous flux, of 100 ×15 = 1500 lm is emitted from the lamp. Thus, from Equation 1.26, the luminous intensity I is I = /4 =119 lm/sr. The illuminance E at a distance D of 40 cm from the incandescent lamp

33

(cd/m2)

Retinal illuminance

(td)

105

Classification

Vision

(lx)

Luminance

Photoreceptor

Illuminance

Spectral luminous efficiency

RELATIONS BETWEEN PHOTOMETRIC QUANTITIES

103

Cones

6 × 103

102

Photopic vision

104

V(λ)

101 1.5 × 101

10–1 10–2 10–3 10–4

V*(λ)

V'(λ)

4.4 × 10–2 Rods

6 × 10–3

Mesopic vision

100

Scotopic vision

6 × 100

10–5

Figure 1.24 Illuminance and corresponding luminance (for = 0.5)

can be obtained from Equation 1.11 as E = I /D 2 = 119/(0.4)2 = 744 lx when 2 = 0. Thus, a 100 W lamp can supply an illuminance suitable for reading a book, but the question remains whether or not illuminance is suitable as a quantity for representing brightness as sensed by the human eye. Let us consider the case further. Although the entire page of the book is exposed to light of the same illuminance, the white portion of the paper is brighter, and the printed black portion darker. As an example, let the reflectance, of the white portion be 0.9, and that of the black portion 0.03. Because the luminous exitance M is found by M = E, the luminous exitance of the white portion is M = 09 × 744 = 670 lm/m2 , and that of the black portion is M = 003 × 744 = 22 lm/m2 . These values are in good agreement with the brightness sensed by the eye. However, the light is reflected from the page in all directions, and only a small portion of it enters the eye. If the reflectance varies with the direction of viewing, as

34


it does for a glossy ink for example, exitance would not be a good indicator of brightness because it measures the total reflected flux. The quantity that shows the variation with angle is luminance. Hence luminance is the photometric quantity corresponding best to the brightness perceived by the eye (see Note 1.2). In our example, the luminance L is obtained as L = 670/ = 213 cd/m2 for the white portion, and as L = 22/ = 7 cd/m2 for the black portion.

NOTE 1.1 LUMINOUS EXITANCE, ILLUMINANCE, AND LUMINANCE OF A PERFECT DIFFUSING PLANE LIGHT SOURCE Referring to Figure 1.25, a plane element of area dS 1 is considered at point A on a perfect diffusing planar light source with a luminance L . The luminous exitance of dS 1 can be derived as follows. A cone with a height of D0 and a base plane of radius R is drawn in the direction normal to dS 1 , and a plane element of area dS 2 is considered on this base plane. The position of dS 2 is defined by polar coordinates r and in the base plane. Then, the solid angle d subtended by dS 2 with respect to A is d = dS2 cos /D 2

(1.27)

In the case of a perfect diffusing planar light source, the luminance is the same for all directions. Accordingly, the following equation can be derived from Equation 1.7 d = L dS1 cos d = L dS1 cos2 dS2 /D 2

(1.28)

dr θ

B r

D θ d S1 A

D0

dS2 dφ N R

D02 + R2

Figure 1.25 Luminous exitance for a perfect diffusing planar light source

35

PERFECT DIFFUSING SOURCE

By simple geometry, it can be seen from Figure 1.25 that the following equations hold D 2 = D02 + r 2

(1.29a)

dS2 = r dr d

(1.29b)

D0 = D cos

(1.29c)

Thus, the following relation can be obtained from Equation 1.28 by substitution d = L d S1 D02 r dr d/D02 + r 2 2

(1.30)

Thus, the luminous flux, inside the cone is =

R

0

2 0

L dS1 D02 r d dr/D02 + r 2 2

= 2L dS1 D0

R 0

r dr/D02 + r 2 2

= L dS1 R 2 /D02 + R 2

(1.31)

By taking R → = L dS1

(1.32)

The luminous exitance, M can then be obtained as M = /dS1 = L

(1.33)

to yield the relation between luminous exitance and luminance for a perfect diffusing planar light source. Next, referring to Figure 1.26, the illuminance is considered at point B at a distance D0 from the center of a disk of radius R on a perfect diffusing planar light source of luminance L . A plane

dr A r R

O

dS1 dφ

θ

Ν

D θ

D0 2

2 R D0 +

dS2 B

Figure 1.26 Illuminance for a perfect diffusing planar light source

36


element of area dS 1 is considered at point A (defined by polar coordinates r and ) on the light source. Then, by substituting 1 = 2 = into Equation 1.13, the following expression can be obtained for the illuminance dE produced by dS 1 . dE = L dS1 cos2 /D 2

(1.34)

As in the previous example, the following geometrical relationships exist D 2 = D02 + r 2

(1.35a)

dS1 = r dr d

(1.35b)

D0 = D cos

(1.35c)

Thus, by substitution, the following relation can be obtained from Equation 1.34 dE = L D02 r dr d/D02 + r 2 2

(1.36)

Then, the luminance for the whole area of the plane light source is E=

0

R

2 0

L D02 r d dr/D02 + r 2 2

= 2 L D02

R 0

r dr/D02 + r 2 2

= L R 2 /D02 + R 2

(1.37)

E = L

(1.38)

By taking R →

It can be seen from Equations 1.33 and 1.38 that the illuminance E produced by an infinite planar light source is equal to the luminous exitance, M of the source. This indicates that the maximum illuminance obtained by infinitely increasing the area of a perfect diffusing planar light source depends only on its luminous exitance.

NOTE 1.2 LUMINANCE AND BRIGHTNESS Among the photometric quantities, the one that corresponds best to the sensation of brightness is luminance. Thus, lights having the same luminance might be expected to have the same brightness,

37

LUMINANCE AND BRIGHTNESS

irrespective of color. If the luminances of a red and a blue light were the same, for example, the brightness of the red light might be expected to appear the same as that of the blue light. However, in fact, the brightnesses are not the same. More specifically, if a white light is taken as the reference light and a red light is taken as the test light, the red light appears brighter than the white light when the luminance is the same for both. That is, the ratio of brightness to luminance B/L , where L is the luminance and B is the brightness, is found to vary with the color. This phenomenon is known as the Helmholtz–Kohlrausch effect. Possible reasons for this variation of B/L are: 1. The function V is not correct. 2. Abney’s law expressed by Equation 1.18 (the additivity of luminous fluxes) does not hold in a strict sense. With respect to the function V, it has been shown by numerous experiments that the function used at present is too low in the short-wavelength region. Hence, Judd proposed to revise V as shown in Figure 1.27. Judd’s modified function was adopted by the CIE in 1988 as a supplemental spectral luminous efficiency VM to be used in those conditions where luminance measurements

1.0 0.5

Luminous efficiency

0.1 0.05

0.01 0.005

0.001 0.0005

0.0001 400


700

Figure 1.27 V of CIE 1924 (solid line) and VM of CIE 1988 (broken lines) (CIE 1990)

38


0.8 1.3 1.2 1.1 0.6

1.0 0.9 0.836

y

0.4

0.2

0

1.4 1.5 1.6 1.7 0.2

0.4 x

0.6

Figure 1.28 Correction factor 10f for brightness as a function of chromaticity

of short wavelengths consistent with the color vision of normal observers is desired (CIE 1990). However, even if the modified VM or any of the other proposed V functions are used, a constant B/L still cannot be obtained. Moreover, a number of experimental results indicate that Abney’s law does not hold in a strict sense for brightness judgments. Accordingly, the reason for the variation of B/L is that both effects listed above occur at the same time. The evaluation of brightness is a particularly important practical problem. Thus, until an analytical solution is found, the CIE has recommended that, to compare the brightness of two objects of different color, one with a luminance L1 and another with a luminance L2 , the following experimentally deduced equation be used (Kaiser 1986): logL1 + f1 = logL2 + f2

(1.39)

where f is a correction factor determined by f = 0256 − 0184 y − 2527 xy + 4656 x 3 y + 4657 xy 4

(1.40)

and x y are chromaticity coordinates (see Chapter 3). If the equation holds true, it can be concluded that L1 and L2 are of the same brightness. Figure 1.28 shows the correction factor 10f in graphical form.

2 Color Vision and Color Speciﬁcation Systems 2.1 MECHANISM OF COLOR VISION The mechanism of how colors are perceived has long been an object of interest, and numerous hypotheses have been proposed. Referring back to Figure 1.3, we know that the human eye can distinguish the various colors of the spectrum dispersed by a prism. Accordingly, one simple hypothesis assumes the presence of a different type of photoreceptor for each of the spectral colors. Because we can distinguish spectral colors differing from each other by only a few nanometers (nm), this hypothesis implies the existence of about 100 types of photoreceptor with different spectral responsivities. Thus it is somewhat unrealistic. Other polychromatic hypotheses propose the presence of six or seven types of photoreceptor, but the two most convincing hypotheses are the trichromatic theory originally proposed by Thomas Young and Hermann von Helmholtz, and the opponent-color theory originally proposed by Ewald Hering. The trichromatic theory was proposed by Young (Figure 2.1) in 1802. The theory was expressed quantitatively and extended by


40

COLOR VISION AND COLOR SPECIFICATION SYSTEMS

Figure 2.1 Thomas Young (1773–1829)

Helmholtz (Figure 2.2) in 1894. It postulates that the retina comprises three types of photoreceptor (cones) which sense red, green, and blue colors respectively, and that all colors are characterized by the degree of response of these photoreceptors. For example, the sensation of yellow is generated by the simultaneous response of the red and green photoreceptors. The trichromatic theory is simple and readily understood because it assumes only three photoreceptors. Figure 2.3 shows the spectral responsivity of the three photoreceptors according to modern knowledge (Judd and Wyszecki 1975). Note that the peaks do not really correspond to red, green and blue spectral colors, so it is more accurate to describe the photoreceptors as long-, medium- and short-wavelength photoreceptors, respectively. Trichromatic theory is based on the experimental result that

Figure 2.2 Hermann von Helmholtz (1821–1894)

41

MECHANISM OF COLOR VISION

10.0

Spectral responsivity (relative value)

B 8.0

6.0

G

4.0 R 2.0

0

400


700

Figure 2.3 Spectral responsivities of red (R), green (G), and blue (B) photoreceptors in the trichromatic theory (Judd and Wyszecki 1975). Reproduced by permission of John Wiley & Sons, Ltd

almost all colors can be reproduced by properly mixing three lights, usually red, green and blue. Color TV sets, color photography, color printing, etc., are all based on the trichromatic theory, and their color reproduction can be excellent. Accordingly, the trichromatic theory is a realistic and convincing hypothesis. On the other hand, Hering (Figure 2.4) proposed a hypothesis in 1878, which says that the retina comprises three types

Figure 2.4 Ewald Hering (1834–1918)

42


of photoreceptor, which respond to red–green, yellow–blue and white–black opponencies respectively, and that all colors are characterized by the degree of response of these photoreceptors. For example, the first type of photoreceptor responds in a positive fashion to red and in a negative fashion to green. This theory is called the opponent-colors theory, and is based on empirical facts showing that there can be a yellowish red and a bluish red, but no greenish red, and that therefore green and red are opponent colors. Figure 2.5 shows the spectral responsivities of the assumed opponent-color photoreceptors. In the figure, positive and negative values have no particular significance, but for example, show that red and green are opponent colors (Judd and Wyszecki 1975). Because four colors, i.e., red, green, yellow, and blue are regarded as fundamental in the opponent-colors theory, the theory is also known as the four-color theory. The trichromatic theory and the opponent-colors theory are each empirically based, and both can explain various color vision phenomena without facing any contradictions. Accordingly, it is difficult to decide which is the actual phenomenon that is occurring in the retina. However, the use of microscopic techniques has enabled the measurement of spectral absorption spectra for single cones in the retina. For example, in 1964 Brown and Wald reported the

W–K

Spectral responsivity (relative value)

1.0

Y–B

0.5

0 R–G –0.5

400


700

Figure 2.5 Spectral responsivities of red–green (R–G), yellow–blue ( Y–B), and white–black (W–K) photoreceptors in the opponent-colors theory (Judd and Wyszecki 1975). Reproduced by permission of John Wiley & Sons, Ltd

43



Figure 2.6 Spectral absorption in human cones (Brown and Wald 1964). Reprinted with permission from Visual pigments in single rods and cones of the human retina, Science 144. Copyright (1964) AAAS

results shown in Figure 2.6 (Brown and Wald 1964). The results clearly indicate the presence of three types of cone having peaks at wavelengths of about 450, 525 and 555 nm respectively, as suggested by the trichromatic theory. Furthermore, thanks to progress in electrophysiology, a method of directly measuring the electrical response of cones to colored lights has been employed by placing a microelectrode about 0.1 m in size in the retina. Figure 2.7 shows

B

G Response

R

[2mV 400

500 600 700 Wavelength (nm)

Figure 2.7 Trichromatic response of carp cones (Tomita 1967). Reprinted from Vision Res. 7. T. Tomita, A. Kaneko, M. Murakami and E. L. Pautler, Spectral response curves of single cones in the carp, 519–531(1967), with permission from Elsevier

44


the measured results obtained by Tomita on carp in 1967, and it also clearly shows a trichromatic response similar to Figure 2.6 (Tomita et al. 1967). It seems from these results that the trichromatic theory explains the mechanism of color vision. However, by inserting microelectrodes into the retina, Svaetichin (1953) had already found an opponent-type spectral response known as the S potential 1953. The S potential, shown in Figure 2.8, supports the opponent-colors theory and was initially believed to be the response of cones, but Tomita revealed by detailed measurements that it came from a region several tens of m distant from the cones. From these results it has emerged that a trichromatic response is present in the cones and then the electric signals generated there undergo processing which obeys the opponent-colors theory in the horizontal cells, amacrine cells, etc., to be finally transmitted to the brain via ganglion cells. The spectral absorption spectra of the rods functioning in scotopic vision was measured by Crescitelli and Dartnall (1953) to give the results shown in Figure 2.9. As expected, the absorption spectrum is very similar to V , the luminous efficiency function for scotopic vision. In conclusion, it is now known that the retina comprises rods and three types of cone, and that they work cooperatively to realize vision which responds to a vast range of brightness and color from 0.0003 to 100 000 lx, an operating range of about 8 decades.

Wavelength (nm) 400

500

600

700

W–K

Y–B

Response

10mV R–G

Figure 2.8 S potential in the carp retina (Tomita 1963). Reproduced by permission of Optical Society of America

45



Figure 2.9 Absorption spectrum of rods and V in scotopic vision (Crescitelli and Dartnall 1953). Reprinted with permission from Nature, Human visual purple, Macmillan Magazines Limited

A model that explains the phenomena of color vision by combining the four types of photoreceptor is called a color vision model. Numerous models have been proposed. The most convincing of these are stage theories that are based on the results of psychological experimentation, of microscopic spectral measurement, and of electrophysiological measurement. Such models assume a trichromatic response at the cone level and an opponentcolors response in later stages. Figure 2.10 is the color vision model proposed by Vos and Walvaren (1971). In this model, the brightness response of the rods and the red (R), green (G), and blue (B) response of the cones are assumed for the first step. The response

Scotopic vision

Photopic vision

V

V'

CYB

CRG

Opponent colors response Y–B R–G Y

Trichromatic response Rod

Cone B

G

R Photoreceptor

Figure 2.10 Color vision model based on stage theory (Vos and Walraven 1971). Reprinted from Vision Res., 11, J. J. Vos and P. L. Walraven, On the derivation of the foveal receptor primaries, 799–818. Copyright (1971) with permission from Elsevier

46


in the rods is directly related to the brightness response in scotopic vision, V . The response from the three types of cones, R, G, and B, provide two opponent-color responses (R–G) and (Y–B) by synthesizing a yellow color (Y) from R and G, and then, as shown in Figure 2.10, by effecting subtraction of each of the signals. The brightness response V in photopic vision is assumed to be synthesized by combining the outputs R, G, and B. It is believed that the rods do not function in photopic vision. However, according to a recent report, rods are found to be active to a luminance of about 500 cd/m2 . Thus, color vision models based on the stage theory may see further progress in the future (Berman et al. 1987).

2.2 CHEMISTRY OF COLOR VISION As previously described, the spectroscopic and psychophysical characteristics of vision are known, at least in outline. Whereas the mechanism of a phototube’s response to light is entirely electronic, i.e., it is based on the photoelectric effect, the mechanism of the eye’s response to light is photochemical, i.e., it is similar to photography. In photography, a photosensitized silver halide (AgCl, AgBr, or AgI) is converted into a silver image by a development process. Numerous chemicals are known to be suitable for the development. For example, if hydroquinone (C6 H6 O2 , H2 Q) is used as the chemical, silver ions Ag+ combine with electrons e− to yield metallic silver Ag according to a chemical reaction expressed by H2 Q → Q + 2H+ + 2e− Ag + + e− → Ag

(2.1)

In photography, a film, once developed, is no longer photosensitive. That is, a photographic film cannot be reused. However, an eye can be photosensitized repeatedly. The chemical reaction occurring in vision is far more complicated than that taking place in photography, and is distinguished in that it is a reproducible chemical reaction. The photosensitive substance in vision (corresponding to silver halide in photography) is known as a visual pigment. One visual pigment was identified about 100 years ago as a substance whose color fades when subjected to light. It was named visual purple according to its color. Visual purple is found not only in human vision. Similar pigments have been found in other animals. Figure 2.11 shows, for instance, the spectroscopic absorption for rods and cones of an ape, and Table 2.1 gives the absorption maxima of the visual pigments of other animals (Bowmaker

47

CHEMISTRY OF COLOR VISION

415

Absorbance

100

500

B

535 567

G

R

80 60 40 20 0

400


Figure 2.11 Spectral absorption for rods (broken lines) and cones (solid line) of an ape (Bowmaker et al. 1983). Reprinted from Colour Vision: Physiology and Psychophysics, J. K. Bowmaker, J. D. Mollon and G. H. Jacobs. Copyright (1983), with permission from Elsevier

Table 2.1 Absorption maxima of visual pigments (nm)

Human Ape Cock Goldfish

Rods

Short

Cones Medium

Long

498 498 507 522

440 447 413 455

534 535 463 535

564 577 562 620

et al. 1983). The observed values in Figure 2.11 and Table 2.1 are obtained through different experiments, and are slightly different from each other. However, it is clear that visual pigments exhibit similar properties among different animals. Rhodopsin (from rhodon, the Greek word for rose) is the visual pigment that is present in the rods, and is different from iodopsin (from iodos, the Greek word for violet), the visual pigment present in cones. However, both are based on 11-cis-retinal combined with a protein (opsin). More specifically, the retina contains 4 types of visual pigment, all based on 11-cis-retinal, but each having a different opsin; one type is present in the rods, and the other three types are present in the cones. Detailed experimental study has revealed that the chemical reaction for vision occurs through the process shown in Figure 2.12. When light is incident on rhodopsin, 11-cis-retinal-opsin dissociates into a full trans-retinal and opsin, and the structural change in opsin affects the action potential of the membrane of the cell containing the visual pigment. This change in potential is the signal that mediates vision. The terms cis and trans

48


H3C CH3 CH3

11

Rhodopsin (11-cis-retinal opsin)

12 CH3 CH3

14 C O H

Light

(11-cis-retinal) Opsin dark CH3 CH3 CH3

CH3 CH2OH

CH3 CH3 CH3

CH3

O C

Reduction (full trans-retinol, Vitamin A)

H

(full trans-retinal)

Figure 2.12 Chemical reaction of visual pigments

show that the direction of the molecular chain changes from the 11-position, from the ‘same side’ (cis) to the ‘opposite side’ (trans). The full trans-retinal is reduced to full trans-retinol (vitamin A), and is converted again to 11-cis-retinal. It can therefore be seen that vitamin A plays an important role in vision. Even before the mechanism of vision was understood, it was known that nyctalopia (night blindness) occurs from a shortage of vitamin A, and that, in severe cases, a lack of vitamin A is detrimental to vision. In comparison with photography, vision can be regarded, as described in Section 1.2, as composed of a high-speed black and white photography (rods) and a medium-speed color photography (cones). That is, three types of photosensitive layer, corresponding to red, green, and blue colors, function in color photography. Similarly, three types of cone operate in the human eye. In color photography, the exposed amount of each of the photosensitive layers expresses the color of the object. Similarly, the color observed by the eye is expressed quantitatively by the degree of photosensitization of the three types of cone. This is the principle behind color specification based on a color mixing system, as described in the following section.

2.3 COLOR SPECIFICATION AND TERMINOLOGY Color specification is a means of expressing color quantitatively, and the quantitative values are called color specification values. A system of color specification, comprising a series of definitions, is called a color system. The classification of color systems and the associated terminology are described in this section.

49

COLOR SPECIFICATION AND TERMINOLOGY

Color systems include color appearance systems and color mixing systems. Color appearance systems are based on color perception, but are defined and/or illustrated in terms of material standards (for example, color chips) and their appearance under specified conditions. Color mixing systems are based on the amounts of mixed colored light necessary to obtain a color match to a test color in a color mixing experiment. Color appearance systems and color mixing systems are compared in Table 2.2. Color appearance systems are subjective, because they are based on a psychological impression, called color perception. The attribute of colors specified in this way is called perceived color. The specification may include additional sensory elements such as the texture of the surface, or the condition of the environment. Let us consider a color appearance system in which, for example, 1000 color chips differing in color are viewed under controlled conditions and classified in an ordered manner. First, they can be classified according to the kind of color, i.e., red, green, blue, etc. This is known as a classification by hue. In the next step, it is found that each hue (e.g., red) includes bright reds and dark reds. Thus, the chips can be classified according to a relative sense of bright and dark, which is called lightness. Furthermore, it is found that red colors having the same lightness can be still distinguished Table 2.2 Comparison between color mixing systems and color appearance systems Color mixing systems

Color appearance systems

Category Basis Principle

Psychophysical color Color sensation Color mixing of light

Object of Expression Typical Example Specification Values

Color of light

Perceived color Color perception Color appearance of material standards (e.g. color chips) Color of object

CIE Colorimetric values (e.g., tristimulus values) Converting color stimulus function into psychophysical values using color matching functions

Munsell Color appearance values (e.g., hue, lightness, chroma) Visual color matching with material standards or converting from colorimetric values

Process for Color Expression

50


from each other because some are vivid, and some are dull. Thus, there is another classification, which we call chroma (see Note 2.1). Thus, hue, lightness, and chroma constitute the three attributes of color in this system. It is possible to use different sets of three attributes but hue, lightness and chroma are commonly used. Color specification values in a color appearance system are called color appearance values, and a proper series of values of hue, lightness and chroma comprise a complete color appearance system. The Munsell color system, which utilizes color chips, is typical of such color appearance systems. The overall quantity of light in a color can be expressed by brightness or lightness. More specifically, brightness is a property related to a simple visual perception running from dark to bright, and is used mainly, though not exclusively, for the colors of light sources. Lightness is relative brightness and is used principally for the colors of reflecting objects. Thus, the two terms must be distinguished from each other. For example, let us consider the case of a white object with high reflectance and a gray object with low reflectance. If the white object was observed under a low illuminance while the gray object was observed under a high illuminance, the gray object would have a higher ‘brightness’, even though it had a lower reflectance. The psychophysical quantity corresponding to brightness is luminance. However, if the white object and the gray object were to be observed side by side under the same illumination, the white object would be brighter. This attribute is called ‘lightness’ and the corresponding psychophysical quantity is luminous reflectance. In a color mixing system we deal with color without considering factors such as the surround that contribute to the total perception. Thus, only a most simple attribute of perception is involved. We call it color sensation. A uniform area of color viewed through a dark aperture can be regarded simply as a color sensation. It is also possible to obtain such a color sensation from other perceived colors, depending on the observing conditions. For example, the perception obtained by observing a sheet of colored paper with the eyes focused on a small area can be considered as color sensation. The light that causes the color sensation in the eyes is called a color stimulus. A color match for an arbitrary color stimulus can be obtained by mixing three colored lights in such a manner that the same response is obtained in the three types of cone. Thus, the color stimulus can be specified indirectly by the quantities of the three mixed colored lights instead of directly by the response of the three types of cone. Such a system does not specify the appearance of a stimulus, but simply specifies, in physical terms, the nature

51

COLOR SPECIFICATION AND TERMINOLOGY

of a defined stimulus that produces an identical appearance under identical conditions. Color mixing systems are based on these basic characteristics of a color stimulus whereby it can be matched by a mixture of three reference stimuli, usually red, green and blue. The color stimulus is specified by the amounts of the three reference stimuli in the mixture and the system is thus known as a trichromatic system. The three amounts are known as tristimulus values. The spectral distribution of the color stimulus is known as the color stimulus function. Because the color stimulus function, which is a physical quantity, is combined with a psychological spectral response of the eye, the tristimulus values are known as psychophysical values. In general, any three values, such as the tristimulus values, that express the color stimulus, are called colorimetric values. The attribute of a color stimulus that is expressed by colorimetric values is called psychophysical color. The tristimulus values of a color mixing system can be related to the specification in a color appearance system under a given condition of observation. Accordingly, to make the tristimulus values more directly understood, they are sometimes converted into correlates of hue, lightness, and chroma. This is possible because, in either of the systems, three independent values are sufficient for the specification of a color. Table 2.3 summarizes various types of colors that are treated in color appearance systems and color mixing systems. More simply, color appearance systems treat real perceived colors expressed by color appearance values, usually based on material standards in specified conditions, whereas color mixing systems are concerned with abstract psychophysical colors, independent of viewing context, expressed by colorimetric values such as tristimulus values. Because psychophysical colors relate to an abstract (i.e., out of context) color stimulus function, they do not depend on whether the color stimulus comes from a light source or an object.

Table 2.3 Classification of colors

psychophysical color color light source color luminous color perceived color

surface color

object color non-luminous color

transmitted color aperture color

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On the other hand, perceived colors can be classified, according to the object with which they are associated, as object colors, illuminant colors, and aperture colors. An object color is a color that appears to be inherent in an object, and is sometimes classified into surface color and transmitted color. An illuminant color (sometimes called a light-source color) is the color of a light that appears to be emitted from a light source. An aperture color is a color observed when, for example, one looks at blue sky through an aperture. Thus, aperture color refers to a case when the object emitting the light is unknown. Object colors can be further classified into ‘luminous colors’ and a ‘non-luminous colors’, depending on whether the light appears to be emitted from the object or the light seems to be reflected or transmitted by the object. This recognizes the fact that an object that appears to emit light is often a primary light source, but not always. For example, a bright reflecting area on a dark background may appear to emit light and thus be categorized as a luminous color. When a color is observed in relation with another color, it is sometimes referred to as a related color, and is distinguished from the case in which the color is observed isolated from other colors, which is referred to as an unrelated color.

2.4 MUNSELL COLOR SYSTEM The Munsell color system is a commonly used color appearance system based on painted color chips. The system was developed originally by an American artist, Albert Munsell (Figure 2.13) in 1905, and the scale was refined and renotated by the Colorimetry

Figure 2.13 Albert Henry Munsell (1858–1918)

53

MUNSELL COLOR SYSTEM

Committee of the Optical Society of America (OSA) in the late 1930s (Newhall et al. 1943). The original color system proposed by Munsell was called the Munsell color system, and that corrected by the OSA was called the Munsell renotation system. In practice, however, only the new system is currently employed and, unless there is particular need to distinguish between the systems, it too is referred to simply as the Munsell color system. As shown in Figure 2.14, the color chips are arranged and notated using cylindrical coordinates, with Munsell value V as the ordinate (value is the name given by Munsell to the attribute more commonly known as lightness), Munsell hue H as the circumferential angle, and Munsell chroma C as the radius (Wyszecki and Stiles 1982) (see Color Plate 2). A color having no Munsell hue, such as white, gray, or black is called an achromatic color, and a color having hue is called a chromatic color. It can be seen that achromatic colors are arranged along the central axis from V = 0 (ideal black) to V = 10 (ideal white). As shown in Figure 2.15, the scale of chroma is expressed by the distance from the central axis, taking the chroma of achromatic colors to be C = 0. Referring to Figure 2.16, Munsell hues are placed circumferentially in an arrangement known as a Munsell value

10Y 5BG 8/ Reproducible color range of colored material

6/ 4/ Cylinder of constant chroma

Constant hue plane 5R

2/

10PB

/2

Munsell hue /6

/10 Munsell chroma

Figure 2.14 Color solid for the Munsell color system (Wyszecki and Stiles 1982). Reproduced by permission of John Wiley & Sons, Ltd

54


10/ White 9/

Reproducible color range of colored material

8/

Munsell value

7/ 6/ /2

5/

/4

/6

/8

/10

4/ 3/ 2/ 1/ 0/ Black Munsell chroma

Figure 2.15 Arrangement of Munsell color chips in a hue plane ( Wyszecki and Stiles 1982). Reproduced by permission of John Wiley & Sons, Ltd

10RP

5R

10R

12

/10

5RP

5YR 6

/8

10P

Hue 34

7

8

/6

9

10YR

/4 /2

5P 10PB

5Y /2

/4

5PB

/2

/4

10Y /6

/6 /10

/8 /1

0

5GY

/8

10B

10GY

Chroma 5G

5B 10BG

5BG

10G

Figure 2.16 Arrangement of Munsell color chips of equal Munsell value (Wyszecki and Stiles 1982). Reproduced by permission of John Wiley & Sons, Ltd

hue circle. The whole three-dimensional arrangement of color chips is called a color solid. Munsell selected five hues, red (R), yellow ( Y), green (G), blue (B), and purple (P) as the principal hues of his system. These five hues

MUNSELL COLOR SYSTEM

55

are spaced equally around the hue circle, and YR, GY, BG, PB, and RP are added as intermediate hues. Furthermore, as is shown in Figure 2.16, other hues between the principal hues and the intermediate hues are expressed by dividing the range between each principal and intermediate hue into ten equal steps and providing numbers 1 through 10. In this way, 100 hues in total are established. The five principal and five intermediate hues are numbered 5 (5R, 5YR, 5Y, etc.) and the intermediate steps are numbered from 1 to 10. The hues 10R, 10YR, 10Y, etc., correspond to 0YR, 0Y, 0GY, etc., but the expressions 0YR, etc., are not used. If necessary, the hue circle can be further subdivided by using decimal fractions such 5.65R. Color chips representing the Munsell system are available in various forms and the Munsell notation of any given specimen can be obtained by finding the color chip whose color appearance most closely matches that of the specimen. If no color chip matches exactly, the notation can be determined by interpolation between neighboring chips. Conventionally a Munsell notation is given in the order hue H , value V , and chroma C, in the form HV/C. For instance, a person’s skin color might yield H = 1.6YR, V = 6.3, and C = 39. Thus, it would be written 1.6YR 6.3/3.9, and read as ‘1.6YR, 6.3 over 3.9’ or ‘1.6YR, 6.3 slash 3.9’. Achromatic colors are expressed by the capital letter of ‘Neutral’, and are written, for example, N 5. Two types of color chips are available, one with a glossy surface and the other with a matte surface. Both types are available at intervals of H = 25 V = 1, and C = 2. The former type includes about 1550 colors, and the latter about 1250 colors. The perceived difference corresponding to a difference in Munsell value of V = 1 corresponds to a difference in chroma of about C = 2 and a difference in hue of about H = 3 at C = 5 (Nickerson 1936). Having such a set of color chips is of great help in the practical understanding of color specification, because the color corresponding to a particular HV/C specification can be visually observed immediately. The notation assigned to Munsell color chips is based on observation under a standard light, known as CIE illuminant C, which is similar to average daylight. Thus the spacing is most uniform under this light or, to a slightly lesser degree, under real daylight, or similar bright illumination, preferably having an illuminance of 1000 lx or higher. For visual comparison of a specimen with a color chip, it is best if the specimen is illuminated perpendicularly, and observed from a direction making an angle of 45 with respect to the specimen. Alternatively, the specimen can be illuminated from a direction making an angle of 45 , and observed in a perpendicular direction. In general, an achromatic area with V = 5–7 is best for

56


the surrounding area. However, greater precision can be obtained for colors with a high Munsell value (V ≥ 8) if a white mask is used. Similarly, a black mask may be used for colors with V ≤ 3. The HV/C notation thus obtained can be converted into colorimetric values, such as the CIE tristimulus values described in Chapter 3, by using a table or a diagram. For some applications, the Munsell color system is inconvenient in that the number of color chips is limited. The color of a specimen rarely matches exactly with a Munsell color chip. Thus, in most cases, the Munsell notation of the specimen color must be evaluated by interpolating between neighboring color chips at the expense of precision of the resulting values. Furthermore, recently many colored materials have been produced whose chroma exceeds the highest chroma of the Munsell color chips. Figure 2.17 shows the range of chroma of Munsell color chips and the theoretical limit for colored materials (see Section 6.3) on a 5B – 5YR hue plane. It can be seen that the region of Munsell color chips is quite limited (Agoston 1987). Extrapolation must be made to obtain the Munsell notation of colors outside the region covered by the chips, but again with a loss of precision of the resulting values. Other examples of color appearance systems using color chips include the Ostwald color system proposed by Wilhelm Ostwald (Figure 2.18) in 1923, the DIN color system adopted as a German Standard in 1955, and the OSA color system introduced by the Optical Society of America in 1978. Special series of color chips for the colors of teeth, skin, hair, flowers, and leaves, etc., are also available.

Value 5B

10 8

5YR

6 4 2

16

12

8

4

Chroma

0 0

4

8

12

16

20

Chroma

Figure 2.17 Chroma of Munsell color chips (•) and theoretical limit of colored materials. Color Theory and its Applications in Art and Design, G. A. Agoston, 1987. Copyright Springer-Verlag, Berlin

COLOR SYSTEM USING ADDITIVE COLOR MIXING

57

Figure 2.18 Wilhelm Ostwald (1853–1932)

Finally, a simple color specification system can be constructed by using color names. Color specification methods employing color names include the use of common objects such as skin, sky, peach, etc., and systematic names obtained by combining modifiers (dark, pale, deep, etc.) with basic color names (red, green, blue, etc.) to give names such as deep yellowish red, pale purplish blue, etc. The use of color names is advantageous in that the specification can be directly and readily understood by an untrained person, but it compares unfavorably with the Munsell and other systems in terms of its precise quantification.

2.5 COLOR SYSTEM USING ADDITIVE COLOR MIXING As described above, experiments show that, within certain limits, any desired color can be synthesized by mixing red [R], green [G], and blue [B] lights in the correct ratio. The symbol [ ] is used to denote a color stimulus. Color matching is the process of mixing the stimuli [R], [G], and [B] and controlling their intensities in such a manner that the mixed light matches a test light. Figure 2.19 shows schematically a typical color matching experiment (Hunt 2004). If stimulus [F1 ] matches stimulus [F2 ], and [F3 ] matches [F4 ], we can write F1 = F2

and

F3 = F4

(2.2)

where = signifies that the left-hand-side stimulus matches the righthand-side stimulus. Such equations are called color equations.

58


F

Test light

White board

B

Observers' eye

R

G

Screen

Colored lights

Figure 2.19 Typical color matching experiment (Hunt 2004). Reproduced by permission of John Wiley & Sons, Ltd

Figure 2.20 Hermann Grassmann (1809–1877)

Grassmann (Figure 2.20) summarized the results obtained in numerous color matching experiments into simple laws that can be summarized as follows. 1. Proportionality. Color matching holds when the intensities of all the components are multiplied by a constant . That is, if Equation 2.2 hold, the following equations also hold

F1 = F2

and

F3 = F4

(2.3)

2. Additivity. Color matching holds for stimuli obtained by adding color-matched stimuli. That is, the following equations hold F1 + F3 = F2 + F4

and

F1 + F4 = F2 + F3

(2.4)

COLOR SYSTEM USING ADDITIVE COLOR MIXING

59

Grassmann’s laws are easy to understand, because they allow the results obtained by increasing, decreasing, or adding a color stimulus in a mixture to be treated in a manner completely equivalent to commonly used mathematical equations. In general, when a test stimulus [F] is matched by amounts R G and B of stimuli [R], [G], and [B] , the color matching equation can be written as F = RR + GG + BB

(2.5)

The values of R G and B depend on which type of [R], [G], and [B] stimuli are used. Accordingly, in order to be able to compare the results of color mixing experiments, the characteristics of [R], [G], and [B] must be fixed beforehand. First of all, the colorimetric values of [R], [G], and [B] must be fixed, and the relative values of their luminances must be defined. In general, the mutual relation of [R], [G], and [B] is defined by normalizing their luminances L R LG , and L B to be equal for a match of specified standard stimulus. The color stimulus that functions as the standard is called the basic stimulus, and a white stimulus is usually chosen. The [R], [G], and [B] stimuli are called reference color stimuli or primary stimuli, and their unit quantities, expressed by photometric quantities L R LG , and LB , are called luminous units. Then, if a stimulus [F] is matched by luminances PR PG , and PB , of [R], [G], and [B] respectively, the normalized values, R = PR /LR G = PG /LG , and B = PB /L B , yield the values R G and B of Equation 2.5. These values are called tristimulus values. Thus it follows, as a matter of course, that R = G = B = 1 for the basic stimulus. Let us consider a test stimulus [F ] of a single wavelength with a predetermined radiant power (e.g., 1 W). Such a stimulus is known as a monochromatic stimulus or spectral stimulus. However, a monochromatic stimulus in a strict sense cannot exist, and, in practice, there must be a finite range of wavelengths, albeit of limited width such as 1 nm. When the color stimulus [F ] is matched by mixing reference color stimuli [R], [G], and [B], the color equation is written as F = r¯ R + g¯ G + b¯ B

(2.6)

where r¯ , g¯ , and b¯ are known as color matching coefficients. The color matching coefficients are the tristimulus values of the spectral stimulus and are sometimes called spectral tristimulus values. The color matching coefficients, each obtained as a function of wavelength, i.e., r¯ , g¯ and b¯ , are called color matching

60


functions. For convenience, the subscript is usually omitted and ¯ ¯ the symbols become r¯ , g and b, respectively. Then, by matching a monochromatic light [F1 ] having a wavelength 1 and a radiant power f1 , and another monochromatic light [F2 ] having a wavelength 2 and a radiant power f2 , the following color equations can be obtained. f1 F1 = f1 ¯r1 R + g¯ 1 G + b¯ 1 B f2 F2 = f2 ¯r2 R + g¯ 2 G + g¯ 2 B

(2.7)

The color [F] obtained by mixing the two monochromatic lights above can be expressed according to Grassmann’s laws by adding each of the left-hand-side values and each of the right-hand-side values to yield F = f1 F1 + f2 F2 = f1 ¯r1 R + g¯ 1 G + b¯ 1 B + f2 ¯r2 R + g¯ 2 G + b¯ 2 B = f1 r¯1 + f2 r¯2 R + f1 g¯ 1 + f2 g¯ 2 G + f1 b¯ 1 + f2 b¯ 2 B

(2.8)

Thus, it can be seen that the tristimulus values of [F] are ( f1 r¯1 + f2 r¯2 , ( f1 g¯ 1 + f2 g¯ 2 , and ( f1 b¯ 1 + f2 b¯ 2 . In general, a color [F] obtained by mixing n monochromatic lights [F1 ], [F2 ], …, [Fn ], with radiant powers f1 , f2 , …, fn , respectively, can be expressed by a color equation as follows: F = f1 r¯1 + f2 r¯2 + · · · + fn r¯n R + f1 g¯ 1 + f2 g¯ 2 + · · · + fn g¯ n G + f1 b¯ 1 + f2 b¯ 2 + · · · + fn b¯ n B =

n i =1

fi · r¯i +

n

fi g¯ i +

i =1

n

fi b¯ i

(2.9)

i =1

Thus, the tristimulus values R, G, and B of the color [F] can be obtained as R=

n

fi r¯i

i =1

G=

n

fi g¯ i

i =1

B=

n i =1

fi b¯ i

(2.10)

COLORFULNESS, CHROMA AND SATURATION

61

As described before, a light having a spectral distribution P can be regarded as a group of lights each having a narrow band of wavelengths of width . Furthermore, a light having a power of P W/nm in a narrow wavelength band of has a total power of P in that wavelength band. Therefore, the tristimulus values R G and B can be obtained according to Equations 2.10 by substituting P for fi : n

R=

P¯ri

i =1 n

G=

Pg¯ i

(2.11)

i =1 n

B=

Pb¯ i

i =1

When n → , Equations 2.11 become integrals: R= G= B=

vis

vis

vis

P¯r d ¯ Pgd

(2.12)

¯ Pbd

where the integral is taken over the entire visible wavelength region. By using the tristimulus values R G, and B thus obtained, the color of a stimulus of any arbitrary spectral distribution can be determined uniquely. For an object color, the stimulus function is defined by the multiplication of the spectral distribution of the illuminating light by the spectral reflectance (or transmittance) of the object. Then, the same derivation holds for the resulting spectral distribution. The CIE color system, which is described in detail in Chapter 3, is the major example of a color system based on additive color mixing.

NOTE 2.1 COLORFULNESS, CHROMA AND SATURATION The attribute that expresses the vividness of a color (in lay terms, the ‘intensity’) can be classified into the following concepts depending on the method of comparison. The term ‘colorfulness’ expresses

62


the vividness of a color, corresponding to the most intuitively understood concept. For instance, when a red paper is observed under an illumination with changing illuminance, its redness is observed more clearly with increasing illuminance. Thus, colorfulness (redness in this example) changes with illuminance, and expresses the vividness of the color in an absolute way. However, when this colored paper is observed against a white background, the brightness of the white background increases as the colorfulness of the color paper increases with increasing illuminance. Thus, when the change of colorfulness of the red paper is considered in comparison with the background, the relative vividness of the paper is judged to be unchanged. The colorfulness of an object judged in comparison with the brightness of a white object under the same illuminating condition is called chroma. ‘Chroma’ in the Munsell color system corresponds to this definition. Finally, let us consider the appearance of a yellow filter. When the illuminance decreases, the vividness decreases to exhibit a darker yellow or even brownish appearance. After a passage of time, the eyes adapt to clearly perceive the original yellow color. We know by experience that there is no such thing as brown light and can recognize a relative colorfulness that has not changed. This attribute that is judged in comparison with the brightness of the object is called saturation. Saturation is a concept that is applied principally to the color of light sources, but can also be used to describe the appearance of an object seen in increasing degrees of shade. Although the colorfulness of such an object decreases, the saturation remains the same.

3 CIE Standard Colorimetric System As described in Chapter 2, there are two types of color specification system. One, exemplified by the Munsell system, is based on color appearance. The other, exemplified by the CIE system, is based on additive color mixture. The Munsell color system can be understood intuitively because actual color chips form the basis of the system. However, additional operations such as interpolation or extrapolation are necessary for the specification of an arbitrary color, and the results are thus of relatively low precision. On the other hand, high precision can be obtained in the CIE system by using spectrophotometric colorimetry, and color can be specified precisely for any arbitrary color stimulus. Accordingly, the CIE system is commonly used in industrial and other quantitative applications. In this chapter, the specification of the CIE standard colorimetric system is described in detail.

3.1 RGB COLOR SPECIFICATION SYSTEM In a color mixture system, once the color matching functions have been determined, the tristimulus specification of any arbitrary color stimulus can be determined easily. However, standardization is


64

CIE STANDARD COLORIMETRIC SYSTEM

necessary in order to be able to compare results, because the color matching functions depend on the basic and reference stimuli. Accordingly, the Commission Internationale de l’Eclairage (CIE) established standard color matching functions in 1931 based on the following principles (CIE 1986, 2004a). 1. The reference stimuli [R], [G] and [B] are monochromatic lights of wavelength R =700.0 nm, G =546.1 nm, and B =435.8 nm, respectively. 2. The basic stimulus is the white color stimulus of the equienergy spectrum. The amounts of the reference stimuli, [R], [G] and [B], required to match the basic stimulus are in the ratio 1.0000:4.5907:0.0601 when expressed in photometric units, and 72.0966:1.3791:1.0000 when expressed in radiometric units. Thus an equi-energy white light of 10000 + 45907 + 00601 = 56508 lm can be matched by additive color mixing of 1.0000, 4.5907 and 0.0601 lm of the reference stimuli [R], [G] and [B], respectively. By dividing each of these amounts by their respective luminous efficiencies, the ratio of the radiometric quantities of the three stimuli can be obtained, i.e., 243783 466333 338134 = 72096 13791 10000. In establishing the color matching functions, the CIE adopted an average of the data reported by Guild (Figure 3.1) obtained from seven observers (Guild 1931) and those by Wright (Figure 3.2) from ten observers (Wright 1928– 1929). The color matching functions thus obtained are assumed to be representative of people having normal color vision. The color matching functions are shown in Figure 3.3. Note 3.1 gives details of the procedure by which these functions were derived from Guild and Wright’s results.

Figure 3.1 John Guild (1889–1979)

65

RGB COLOR SPECIFICATION SYSTEM

Figure 3.2 William David Wright (1908–1998) 0.4 r(λ)

g(λ) 0.2

500 r(λ)

600 700 Wavelength (nm) 700.0

400 435.8

0.0

546.1

Tristimulus values

b(λ)

Figure 3.3 Color matching functions r , g and b of the CIE 1931 RGB color specification system

As described before, the color matching functions are the amounts of the reference stimuli [R], [G] and [B] needed to match monochromatic stimuli of each wavelength. However, as shown in Figure 3.3, the color matching functions include a negative portion. At first sight, this seems strange because a negative value signifies that a negative amount of the reference stimulus is needed to make the match. This happens because of the following fact. Even though an observer may try various ways to mix the three reference stimuli, he or she will find it impossible to match any monochromatic light stimulus [F ], using monochromatic reference stimuli [R], [G] and [B], because the test light is always too saturated, except in the trivial case where the test stimulus is identical to one of the reference stimuli. Thus, in a practical color matching experiment, the saturation of [F ] is lowered by mixing it with a certain amount of

66


[R], for example, and then matching the mixture by a combination of [G] and [B]. The color equation in this case is expressed by F + RR = GG + BB

(3.1)

Assuming that Grassmann’s Laws apply, the equation can be transformed into the following form F = −RR + GG + BB

(3.2)

Thus, the first term −R is negative, and this is the reason why partly negative color matching functions are obtained. This color equation can be regarded as a vector equation in threedimensional space by taking [R], [G] and [B] as vector components. The three-dimensional space so constructed is used for the geometrical expression of colors and is called a color space. Thus, as shown in Figure 3.4, any color [F] can be located in the space at the point defined by the matching amounts of [R], [G] and [B], i.e., R G and B. The intersection (r g b) of the vector [F] and the unit plane R + G + B = 1 is commonly used to express color [F] according to the following equations r = R/R + G + B g = G/R + G + B

(3.3)

b = B/R + G + B It is apparent that r + g + b = 1. Thus, two coordinates out of the three, for example (r g), are sufficient to locate the color [F] in the unit plane. The coordinates (r g b) determined in this manner are called chromaticity coordinates, and a diagram showing two g

[B] [F] (R, G, B)

(0, 0, 1)

1.0 r + g = 1.0 g

r

g (0, 1, 0)

[R]

[G]

[F] r

1.0

r

(1, 0, 0)

Figure 3.4 Three-dimensional expression of color [F] and chromaticity diagram

67

RGB COLOR SPECIFICATION SYSTEM

chromaticity coordinates in a plane is called a chromaticity diagram. The chromaticity coordinates of color [F] define a point in the chromaticity diagram known as a chromaticity point. The psychophysical property defined by the chromaticity coordinates is called the chromaticity of [F]. The chromaticity coordinates for monochromatic stimuli are called spectral chromaticity coordinates, and the curve obtained by connecting the chromaticity points of the monochromatic stimuli in wavelength order is known as the spectrum locus. The straight line connecting the two ends of the spectrum locus is called the purple boundary, and it represents the additive color mixture of the monochromatic stimuli (blue and red) located at the ends of the visible spectrum. Along the purple boundary, the color changes continuously from blue to red via various shades of purple. The system for expressing color established by the definitions and normalization above is called the CIE 1931 RGB Color Specification System. Figure 3.5 shows the rg chromaticity diagram of this system and the chromaticity point WE (1/3, 1/3) for the equi-energy white color. The chromaticity points for all real colors, i.e., those that can exist in practice, are located inside the region enclosed by the spectrum locus and the purple boundary. However, colors outside this region, for example a chromaticity point with r = 10 and g = 10, can be considered from a mathematical point of view. These colors cannot exist in practice and are therefore called imaginary colors. However, as is described in Section 3.2, imaginary colors are used for converting the RGB color specification system into a convenient XYZ color specification system. g 2.0

0

0

52

1.6

53

0

51

1.2

0.4

+ WE

0

0.8

56

55

0

54

0

500

4

–1.0

–0.6

0 47 50 4

40

–1.4

0

0 48

0.4

58 59 0 60 0 6 0 70 10 0

57

0

90

0.8

1.2

r

Figure 3.5 Chromaticity diagram for the CIE 1931 RGB color specification system and chromaticity point WE for the equi-energy white color

68


3.2 CONVERSION INTO XYZ COLOR SPECIFICATION SYSTEM Figure 3.3 shows that the color matching functions in the RGB system have some negative values. In earlier times, when tristimulus values had to be calculated manually, the presence of both negative and positive values made the calculations complicated. The color matching functions for a different set of reference stimuli can be obtained by a simple conversion of primaries (see Note 3.2). This allowed the CIE to introduce the 1931 XYZ color system in addition to the RGB color system by establishing reference stimuli [X], [Y] and [Z] in such a manner that the color matching functions have allpositive values. Because the XYZ system is based on the 1931 RGB system and thus on color matching experiments at a viewing angle of 2 , it is called the CIE 1931 Standard Colorimetric System (CIE 1986, 2004a) or, alternatively, the CIE 2 Colorimetric System The color matching functions x , y and z of the system are given in Figure 3.6 and Table A1 in the Appendix. The virtual observer having these color matching functions is called the CIE 1931 Standard Colorimetric Observer or the CIE 2 Colorimetric Observer. Another characteristic of the XYZ color system is that, in accordance with a proposal of Judd (Figure 3.7), y is set so that it is identical to the spectral luminous efficiency function V. This is convenient because it means that the tristimulus value Y directly

z (λ)

Tristimulus values

1.5

1.0

y (λ)

x (λ)

0.5 x (λ)

0 400


700

Figure 3.6 Color matching functions x , y and z in the CIE 1931 XYZ color system

CONVERSION INTO XYZ COLOR SPECIFICATION SYSTEM

69

Figure 3.7 Deane Brewster Judd (1900–1972)

expresses a photometric quantity. Another feature of the XYZ system is that the straight line connecting the reference stimuli [X] and [Y] in the chromaticity diagram is tangential to the spectrum locus at the long wavelength end with 650 nm. Because of this, z = 0 for 650 nm, and the tristimulus value Z can be obtained with fewer steps of calculation. Moreover, as described in Note 6.5, another advantage is that z matches approximately one of the basic spectral responsivities of the human color vision system. The RGB system can be related three-dimensionally to the XYZ system as is shown in Figure 3.8 (Ikeda 1980). [Y]

[G]

[R] [Z] [B]

Non-luminous line Non -lum in e ous pl ane Unit plan (L = 0)

[X]

Figure 3.8 Positional relationship between RGB color system and XYZ color system (Ikeda 1980). Reproduced by permission of Asuara Shoten

70


The RGB and XYZ systems can be mutually converted according to the following relations (Note 3.3). Tristimulus values R G and B can be converted into tristimulus values X Y and Z by      X 27689 17517 11302 R      (3.4)  Y  =  10000 45907 00601   G  Z 00000 00565 55943 B Conversely, tristimulus values X Y and Z can be converted into R G and B by    −1   R 27689 17517 11302 X        G  =  10000 45907 00601   Y  B 00000 00565 55943 Z    04185 −01587 −00828 X    = −00912 02524 00157   Y  (3.5) 00009 00025 01786 Z Because the color matching functions x y and z as well as r g and b are the tristimulus values of spectral colors, the relation expressed by Equations 3.4 and 3.5 are also applicable to the color matching functions. Furthermore, the chromaticity coordinates x y and z can be obtained from r g and b according to the following equations x =049000r + 031000g + 020000b /066697r + 113240g + 120063b y =017697r + 081240g + 001063b /066697r + 113240g + 120063b

(3.6)

z =000000r + 001000g + 099000b /066697r + 113240g + 120063b Similarly, r g and b can be calculated from x y and z. The tristimulus value Y can be obtained from Equation 3.4 as Y = 10000R + 45907G + 00601B

(3.7)

This confirms that the tristimulus value Y is a photometric quantity because the coefficients for R G and B (1.0000, 4.5907, 0.0601) are in agreement with the luminous units for the reference stimuli [R], [G] and [B]. Equation 3.7 defines the plane in the RGB color space on which lie all colors of the same luminance Y .

X10 Y10 Z10 COLOR SPECIFICATION SYSTEM

71

From Equation 3.5, the R G and B coordinates of the reference stimuli [X] and [Z] can be obtained as

X = 04185−00912 00009

Z = −00828 00157 01786 (3.8) The luminances of [X] and [Z] are zero because we obtain Y = 0 by putting the values of Equation 3.8 into Equation 3.7. That is, the reference stimuli [X] and [Z] are colors with no luminance and are thus imaginary colors that cannot exist in practice (see Note 3.4). The following equation can be obtained by putting Y = 0 in Equation 3.7. 10000R + 45907G + 00601B = 0

(3.9)

This represents a plane containing all colors with a luminance of zero. As with the reference stimuli [X] and [Z], a color on this plane is a virtual color that cannot be realized in practice. The plane is called the non-luminous plane, and it intersects the unit plane R + G + B = 1 in a straight line given by 09399r + 45306g + 00601 = 0

(3.10)

This straight line is called the non-luminous line. The nonluminous line is located completely outside the chromaticity area enclosed by the spectrum locus and purple boundary. The nonluminous plane and the non-luminous line are sometimes called the alychne.

3.3 X10 Y10 Z10 COLOR SPECIFICATION SYSTEM As mentioned above, the XYZ color system is based on color matching experiments with a viewing angle of 2 . The reasons why the color matching experiments were conducted with such a narrow viewing angle are: 1. The region in the retina having the highest visual acuity, the fovea, has a viewing angle of about 2 (see Figure 1.4). 2. Because the central portion of the retina is covered with a yellow pigment called the macular pigment, the color matching functions for this central portion differ from that of the peripheral portion in fields larger than 4 . Because of adaptation, one is not conscious of the presence of this yellow pigment in ordinary life. However, when a color matching

72


experiment is performed in a large viewing field, a spot-like image of the pigmented area (known as Maxwell spot) appears in the center of the visual field. The Maxwell spot appears clearly when there is a large difference between the spectral distributions of the two lights being matched. This can disturb the color matching experiment. For these reasons, the XYZ color system established in 1931 was based on color matching experiments with a viewing angle of 2 , but was expected to be applicable to a visual field of any desired size. However, it was reported thereafter that there was sometimes a visible color difference between a pair of specimens that the system predicted to be a match (Jacobsen 1948, Judd 1949,1950). It can be seen from Figure 3.9 that the observation condition of 2 in viewing angle is quite narrow. If a distance of 250 mm is supposed for observation, the radius for a viewing angle of 2 is about 4.4 mm, which is too small for most color matching situations. Accordingly, the observed mismatches were believed to be due to spectral responsivity changes across the retina. In the light of these circumstances, a color matching experiment with an enlarged visual field of 10 diameter was performed by Stiles (Figure 3.10) and Burch of the UK, and by Speranskaya of the Soviet Union (Russia) (Stiles and Burch 1959, Speranskaya 1959). In the experiments, the Maxwell spot was either neglected (Stiles and Burch) or hidden by a disk (Speranskaya). In 1964, the CIE employed a weighted average of the experimental results of Stiles and Burch for 49 persons, and those of Speranskaya for 18 persons (which was later increased to 27 persons), and recommended a new system, the CIE 1964 Standard Colorimetric System, applicable to a viewing angle of 4 or greater (CIE 1986, 2004a). Because the CIE

r = 4.4 mm

r = 21.9 mm

r

1°

2°

4°

XYZ color system

10° X10 Y10 Z10 color system

Figure 3.9 Size of visual field from a distance of 250 mm

73

X10 Y10 Z10 COLOR SPECIFICATION SYSTEM

Figure 3.10 Walter Stanley Stiles (1901–1985)

1964 system is based on color matching experiments with a visual field of 10 , it is also called the CIE 10 Colorimetric System. The CIE 1931 and 1964 systems are applicable to different viewing angles with respect to the observer’s eye. More specifically, the former is recommended when the viewing angle is 1–4 , and the latter is recommended for a viewing angle exceeding 4 . The manner in which the color matching functions were determined in the X10 Y10 Z10 system differs from that in the XYZ system, but the CIE defined the new reference stimuli [X10 ], [Y10 ] and [Z10 ] in such a manner that the color matching functions of the systems resemble each other. Figure 3.11 shows the color matching functions

Color matching functions

2.0

1.5

1.0

0.5

0

400


700

Figure 3.11 Color matching functions for the XYZ system (filled circles) and the X10 Y10 Z10 system (open circles)

74


x 10 y 10 , and z 10 for the new system in comparison with the XYZ color matching functions, and Table A1 in the Appendix gives the numerical values. The virtual observer having the color matching functions of the CIE 1964 system is called the CIE 1964 Standard Colorimetric Observer or the CIE 10 Colorimetric Observer. As shown in Figure 3.11, the differences between the 2 color matching functions and the 10 color matching functions are not large, but they are nonetheless significant in some applications. When the new color matching functions were introduced, numerous field tests were performed (Stiles and Wyszecki 1962, Wyszecki 1964). However, the breakdowns in color matching pointed out by Jacobsen were not completely overcome by the new functions. According to further examination by Judd (1968), the variation of color matching functions among observers must also be taken into consideration. The problem pointed out by Judd is now judged properly by the method established by the CIE for evaluating the variation of color matching function (see Section 7.5).

3.4 TRISTIMULUS VALUES AND CHROMATICITY COORDINATES The tristimulus values X Y and Z of a color stimulus can be obtained by first calculating tristimulus values R G and B by Equation 2.12, and then converting them into the tristimulus values X Y and Z using Equation 3.4. However, in general, they are obtained directly using the color matching functions x y and z according to the following equations X =k Y =k Z =k

x d vis

y d

(3.11)

vis

z d vis

where, k is a constant, and the integral is taken in the visible wavelength region. For a reflecting object, the color stimulus is = RP, and for a transparent object it is = TP, where P is the spectral distribution of the illuminating light, R is the spectral reflectance of the reflecting object, and T is the spectral

TRISTIMULUS VALUES AND CHROMATICITY COORDINATES

75

transmittance of the transmitting object. For example, the tristimulus values X Y and Z of a reflecting object can be expressed as X =k Y =k Z =k

R P x d

vis

R P y d

(3.12)

vis

R P z d vis

where, the constant k is k = 100/

P y d

(3.13)

vis

The constant k is selected such that the tristimulus value Y yields a value of 100 for a perfect reflecting object (R = 1 for all ). In general, R < 1 for any real object color, and Y is therefore < 100. The tristimulus value Y of a reflecting (transmitting) object is called the luminous reflectance (transmittance), and, as described in Section 2.3, is roughly correlated with the lightness of the object color. To calculate the tristimulus values X10 Y10 and Z10 of the 1964 system, the color matching functions x 10 y 10 and z 10 are used in Equation 3.11 in place of x y and z. Similarly, in the case of object colors, the 1964 color matching functions can be used in Equations 3.12 and 3.13. In this case, however, although the tristimulus value Y10 roughly represents brightness, the CIE has not adopted y 10 officially as the spectral luminous efficiency for a 10 visual field (see Note 3.5). As with the RGB system, the chromaticity coordinates x and y are established by the intersection of the color vector (X Y Z ) with the unit plane X + Y + Z = 1 as follows x = X/X + Y + Z y = Y/X + Y + Z

(3.14)

The chromaticity coordinates x10 and y10 in the X10 Y10 Z10 system can be defined in exactly the same manner. Figure 3.12 shows the spectrum locus and purple boundary in the xy chromaticity diagram and in the x10 y10 chromaticity diagram. The xy chromaticity diagram given in Color Plate 3 is not reproduced correctly from the viewpoint of colorimetry, but it illustrates schematically how colors correspond to chromaticity coordinates, except for saturated colors that cannot be reproduced with printing inks.

76


0.8

520

540

520

540 500 560 560

0.6

y, y10

500 580 580

0.4

600 600 650 650 480

0.2 480

0

450 380

450 380

0.2

0.4 x, x10

0.6

0.8

Figure 3.12 xy chromaticity diagram of the XYZ color specification system (filled circles) and x10 y10 chromaticity diagram of the X10 Y10 Z10 color specification system (open circles)

A two-dimensional xy chromaticity diagram is often used to plot colors. However, since three pieces of information are needed to specify color, a third must be added to x and y for a complete specification. Any of the tristimulus values X Y and Z could be used, but in general, the photometric quantity Y is chosen and colors are expressed by (x y Y ).

3.5 METAMERISM As is clear from Equation 3.12, two objects will match in color not only if they have the same R, but also if one has a reflectance R that is different from R but satisfies the following equations vis

vis

vis

R P x d = R P y d = R P z d =

vis

vis

R P x d R P y d R P z d

vis

(3.15)

77

METAMERISM

However, if the illumination is changed from P to a different distribution, P , in general Equation 3.15 no longer holds, and becomes R P x d = R P x d vis vis R P y d = R P y d (3.16) vis vis R P z d = R P z d vis

vis

Thus, the two objects no longer match. Ostwald denoted the color matching that occurs ‘under specified observation conditions’ when two color stimuli have different spectral distributions as metamerism (Note 3.6). The two color stimuli for which metamerism occurs are called metamers. The specified conditions include the color matching functions of the observer, the field size of observation, and, in the case of object colors, the spectral distribution of the illuminant. In conventional color photography, printing, television and computer displays, the reflectances R of the object and R of the reproduced color are generally not equal. Thus, the color reproduction is based on metamerism. Figure 3.13 shows the spectral reflectance of an actual skin color and that of the same skin color reproduced by a photographic color film under black-body radiation at a color temperature (see Section 3.7) of 4800 K (Ohta 1973). Figure 3.14 shows another example of metamerism, that of the

100

Reflectance

80 60 40 20 0

400

500

600 Wavelength (nm)

700

800

Figure 3.13 Actual skin color (broken lines) and skin color reproduced by a photographic color film (solid line) (Ohta 1973)

78


Radiation power

60

40

20

0 400


700

Figure 3.14 Actual skin color (broken lines) and skin color reproduced in color television (solid line) (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

spectral radiant power of an actual skin color under daylight and that of the same skin color reproduced by color television (Wyszecki and Stiles 1982). Although these metamers give the same tristimulus values, there is a large difference in the spectral characteristics. The CIE has recommended evaluation methods for the degree of metamerism exhibited when there is a change in the spectral distribution of the illuminant or in the color matching functions of the observer (see Sections 7.4 and 7.5)

3.6 DOMINANT WAVELENGTH AND PURITY As shown in Figure 3.15, the distance and direction from a specified chromaticity point W in the xy chromaticity diagram are sometimes used to specify chromaticity instead of x and y. The point W is known as the white point and represents an achromatic stimulus [W] (a color stimulus seen as achromatic under normal observation conditions). If the intersection of the spectrum locus with the straight line through the white point W and the test point F1 is D, the color [F1 ] represented by F1 can be obtained by properly mixing the white color stimulus [W] and the monochromatic light stimulus [D] represented by point D. The ratio of distances WF1 /WD is a scale that indicates how near [F1 ] is to the monochromatic stimulus [D], and is called the excitation purity, pe of [F1 ]. The wavelength of the monochromatic stimulus at the intersection D is called the dominant wavelength of [F1 ], and is denoted by the symbol d . The excitation purity pe can be expressed in terms of the chromaticity coordinates xw and yw of the white point

79

DOMINANT WAVELENGTH AND PURITY

0.8

P'

520

540

560 0.6 500 580 D (xd, yd)

y 0.4 F1 (x1, y1)

600

W(xw, yw)

620 650 770 nm

F2 (x2, y2)

0.2 480

0

P (xp, yp)

470 450

380 0.2

0.4

0.6

x

0.8

Figure 3.15 Dominant wavelength and excitation purity

W, the chromaticity coordinates x1 and y1 of the test point F1 , and the chromaticity coordinates xd and yd of the intersection point D pe = WF 1 /WD = x1 − xw /xd − xw = y1 − yw /yd − yw

(3.17)

The equations involving x and y are equivalent in Equations 3.17. The one having the larger divisor is recommended to obtain a result with higher precision. When the color is in the purple region defined by broken lines in Figure 3.15, as is the case with color [F2 ], the cross point P is not on the spectrum locus, but on the purple boundary. In such a case, the excitation purity pe can be obtained as follows pe = WF 2 /WP = x2 − xw /xp − xw = y2 − yw /yp − yw

(3.18)

where xp and yp are the chromaticity coordinates of point P. The wavelength of the monochromatic stimulus [P ], where P is the point that the straight line extended in the direction of W crosses the spectrum locus, is used instead of the dominant wavelength. This wavelength is called the complementary wavelength, and is denoted by the symbol c .

80


For light sources, the white point is usually set at xw = yw = 1/3, whereas for object colors it is usually set at the illuminant point. Then, instead of using chromaticity coordinates, the dominant wavelength or complementary wavelength can be used in combination with the excitation purity to define the color stimulus. Because, roughly speaking, the dominant wavelength expresses hue and the excitation purity expresses chroma, this method of specification is helpful in directly understanding the appearance of a color stimulus. Let the X tristimulus values of the stimuli [W], [F1 ] and [D] be Xw , X1 and Xd , respectively, let the sums (stimulus sums) of the three tristimulus values be Sw , S1 and Sd , respectively, and the x chromaticity coordinates be xw , x1 and xd respectively. Then, the following relations can be obtained xw = Xw /Sw

x1 = X1 /S1

X1 = Xw + Xd

S1 = S w + S d

xd = Xd /Sd (3.19)

By substituting these relations into Equations 3.17, the following can be obtained pe = x1 − xw /xd − xw = X1 /S1 − Xw /Sw /Xd /Sd − Xw /Sw = Xw + Xd /Sw + Sd − Xw /Sw /Xd /Sd − Xw /Sw = Sd /Sw + Sd = Sd /S1

(3.20)

Thus, the excitation purity pe is the ratio of the stimulus sum of the monochromatic stimulus [D] and the stimulus sum of the test stimulus [F1 ] In general, when an additive mixture of an achromatic stimulus and a monochromatic stimulus match a test stimulus, the mixing ratio is called the purity of the color stimulus. Excitation purity pe above is one type of purity, but purity can be expressed differently by employing another definition using the ratio of the luminances of the stimuli [F1 ] and [D]. This is called the colorimetric purity pc , and is defined by the following equation pc = Yd /Y1

(3.21)

where Y1 and Yd are the values of Y for the stimuli [F1 ] and [D] respectively. Because the following relations hold, Y1 = y1 S1

Yd = yd Sd

(3.22)

81

DOMINANT WAVELENGTH AND PURITY

Equation 3.22 can be substituted into Equation 3.21, and from Equation 3.20, the following relation can be obtained pc = Yd /Y1 = yd Sd /y1 S1 = yd pe /y1

(3.23)

As an example showing the use of purity, the just noticeable difference (jnd) in colorimetric purity pc has been measured as shown in Figure 3.16 (Wyszecki and Stiles 1982). This result was obtained for monochromatic light with a wavelength of 650 nm, but a similar result can be obtained for other wavelengths. Figure 3.17 12 10 8 ∆pc

6 4 2 0

20

40 60 Colorimetric purity (%)

80

100

Figure 3.16 Just-noticeable difference pc of colorimetric purity (at = 650 nm and white point of 4800 K) (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

Number of steps

20 15 10 5 0

500

600 Wavelength (nm)

700

Figure 3.17 Number of just-noticeable steps between white light (4800 K) and monochromatic light (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

82


shows the distinguishable number of colored lights between white light and monochromatic light derived from the result illustrated in Figure 3.16. Color specification by means of dominant wavelength and purity has been used frequently because it can be readily understood, but recently chromaticity coordinates have been preferred. In particular, colorimetric purity has a discontinuity in the purple region, and is therefore used less because of this inconvenience.

3.7 COLOR TEMPERATURE AND CORRELATED COLOR TEMPERATURE As described above, a color stimulus can be specified in three dimensions by tristimulus values, and in two dimensions by chromaticity coordinates or by dominant wavelength and excitation purity. The variables for the methods are (X Y Z ), (x y) and (d , pe ), respectively. Of course, the two-dimensional specifications can only distinguish among a restricted set of colors that all have the same luminance. To go one step further, for stimuli with the spectral power distribution of an ideal black body (see Section 3.8), the absolute temperature and the spectral power distribution of the radiation (blackbody radiation) are in 1 to 1 correspondence. (Absolute temperature is a temperature scale first introduced by Lord Kelvin in 1848 and thus is sometimes called ‘Kelvin temperature’. Originally, the unit for this scale was the ‘degree Kelvin’ (abbreviated K), but it is now known simply as the kelvin (abbreviated K). The lowest theoretically possible temperature (absolute zero) is set at 0 K, and the temperature (001 C) of the triple point of water (a state in which ice, water, and water vapor co-exist) is defined as 273.16 K. Thus the absolute temperature can be obtained by adding 273.15 to the Celsius temperature ( C) used in everyday life). Thus, the colors of these stimuli can be specified by only one variable (absolute temperature of the black body). The color specification based on this concept is called color temperature or correlated color temperature. Color temperature (usually denoted by Tc ) expresses the chromaticity of a given radiation by the temperature of the black body having the same chromaticity as that of the radiation. For radiation whose chromaticity is not exactly equal to that of a black body, correlated color temperature Tcp is defined as the temperature of the black body whose chromaticity is nearest to that of the radiation. The absolute temperature scale (in kelvin) is used for expressing these temperatures.

83

CORRELATED COLOR TEMPERATURE

Color temperature Tc indicates that the chromaticity of the given radiation corresponds to the chromaticity of the radiation from a black body of absolute temperature Tc . However, it does not necessarily indicate that the light source itself is heated to this temperature. This also applies to correlated color temperature Tcp . For example, some fluorescent lamps have a Tcp of 6000 K or more, but their actual temperature is not nearly so high. The value of Tcp signifies only that such a lamp radiates light having a color most resembling that of a black body heated to an absolute temperature of 6000 K. The line connecting the chromaticity points of the series of absolute temperatures of black bodies is called the Planckian locus. The color temperature corresponding to these chromaticities can be obtained immediately as the corresponding absolute temperature on the Planckian locus. For chromaticities not on the Planckian locus, the correlated color temperature can be obtained on the CIE 1960 uv chromaticity diagram (see Section 4.1) by drawing a line from the chromaticity point of the radiation in such a manner that it crosses the Planckian locus at a right angle, and then determining the temperature corresponding to the cross point (Note 3.7). A line crossing the Planckian locus at a right angle in the uv diagram is called an isotemperature line. These lines can be obtained for a series of correlated color temperatures and converted into xy coordinates. The results are shown in Figure 3.18. The correlated color temperature for a given radiation can be obtained

0 60

66

0

50 0

40 0

300

0.500

200

0.400 y 100

2500 2000

1515K

3333

0.300 5000 0

10 000

0.200

∞K

0.200

0.300

0.400 x

0.500

0.600

Figure 3.18 Planckian locus (thick line) and iso-temperature lines (fine lines) (numerals on upper side indicate reciprocal color temperature in units of MK−1 )

84


geometrically from Figure 3.18. However, it is not appropriate to extend the isotemperature lines, and obtain, for example, a correlated color temperature of a saturated green light. The applicable chromaticity coordinates should be limited to a range of about 0.05 units on either side of the Planckian locus in the uv diagram. The correlated color temperature of light sources generally encountered in everyday life is given in Table 3.1. The value obtained by dividing color temperature or correlated color temperature into 106 is called reciprocal color temperature or reciprocal correlated color temperature. Originally, the unit for these values was called the ‘mired’, an abbreviation of ‘microreciprocal-degree’. Recently, it has been proposed to use ‘mirek’ (for micro-reciprocal-kelvin) instead of ‘mired’, but this name has not been widely accepted. The correct SI unit is the reciprocal megakelvin (MK−1 ). One advantage of using this reciprocal scale is that color temperatures or correlated color temperatures are discriminable by the human eye if there is a difference of about 5.5 MK−1 , regardless of the value of Tc or Tcp . Reciprocal color temperature is widely used to express the performance of a color temperature conversion filter (a filter that converts color temperature from one value to another). Such filters are often used in color photography. For example, a filter of 35 MK−1 converts Table 3.1 Color temperature of light sources (Yamazaki 1979). Reproduced by permission of Sougen Sha Color temperature (K)

Specific examples of colored light

800 1000 1200 1900 2400 2740 2860 2920 3200

Red nichrome wire Yellow fire in furnace White fire in furnace Flame of paraffin candle or kerosene lamp 20 W light bulb, acetylene lamp 40 W gas-filled light bulb 100 W gas-filled light bulb 500 W gas-filled light bulb 200 W or higher light bulb for movie cameras, flash bulbs Carbon arc, acetylene oxygen flame Medium-size flash bulb Full moon Sun observed 2 h after sun rise Light from the sun on a slightly cloudy day Light in the middle of a fine day with clear air Light from blue sky on a fine day

3700 3800 4100 4400 5300 6000 20000–25000

85

ILLUMINANTS AND LIGHT SOURCES

Color temperature T2 (K) 10 000 9000 8000 7000 6000 5000

Reciprocal color temperature (MK–1)

Color temperature T1 (K)

400

2000

350 300 250 200

2500

150 4000

100 50

3000

0 3000

–50 –100

4000

–150 2500

–200

5000

–250 6000 –300 7000 –350 2000

–400

8000 9000 10 000

Figure 3.19 Conversion from color temperature T1 to T2 using a color temperature conversion filter

the light of an incandescent lamp of 2856 K to a light of 3200 K. The filter is effective with any type of illuminating light, and it can also convert a light of 6500 K to a light of 8400 K (See Note 3.8). Figure 3.19 is a useful conversion chart for obtaining the converted color temperature when such a filter is used. The color temperature T2 of the converted illuminant can be obtained by connecting the color temperature T1 of the unfiltered illuminant with the value (MK−1 ) of the reciprocal color temperature of the filter, and then reading out the value on the extended line.

3.8 ILLUMINANTS AND LIGHT SOURCES Light sources used for illumination in daily life include natural light sources such as the sun and artificial light sources such as incandescent lamps. People have used sunlight from ancient times, and,

86


as described in Section 1.4, human vision has evolved in a way that utilizes the spectral distribution of daylight most effectively. Because daylight plays such an important role, numerous measurements of its spectral distribution have been reported and it has been found to fluctuate depending on various factors such as the place, the time of day, the season, and the weather. In the early 1960s, the CIE collected and analyzed results (622 cases) of daylight measurements. First, the chromaticity coordinates xD , yD of the measured daylights were plotted on the xy chromaticity diagram, and the locus that best approximated the chromaticity points was determined. The locus is called the daylight locus, and was fitted by a polynomial curve of second order expressed by yD = −3000xD2 + 2870xD − 0275

(3.24)

Figure 3.20 shows the daylight locus (D), which is approximately parallel to the Planckian locus (P). The spectral distributions were then analyzed by a statistical method (principal component analysis) to obtain the three basic functions (eigenvectors) S0 , S1 and S2 that constitute the spectral distribution of daylight (see Figure 3.21 and Table A2 in the Appendix). By a linear combination of the three eigenvectors, the spectral distribution of daylight SD at any desired correlated color temperature Tcp can be obtained by the following equation SD = S0 + M1 S1 + M2 S2

(3.25)

5500 4800

6500

4000 K

7500

0.350

5000 6000

10 000

y 0.300

0.250

7000 8000 9000 10 000

D

20 000

P

40 000

∞

0.250

0.300

0.350

0.400

x

Figure 3.20 Daylight locus (D) and Planckian locus (P). Fine lines are isotemperature lines

87

ILLUMINANTS AND LIGHT SOURCES 140

S0(λ), S1(λ) and S2(λ)

120 S0(λ)

100 80 S1(λ)

60 40 20

S2(λ)

0 –20 300

400

500

600

700

800

Wavelength (nm)

Figure 3.21 Three eigenvectors S0 , S1 and S2 constituting daylight

where M1 =−13515 − 17703xD + 59114yD /00241 + 02562xD − 07341yD M2 =00300 − 31442xD + 300717yD

(3.26)

/00241 + 02562xD − 07341yD In Equation 3.26, the value for yD is obtained from Equation 3.24, and the value for xD is obtained from the correlated color temperature Tcp as follows: When 4000 K ≤ Tcp ≤ 7000 K xD = − 46070 × 109 /Tcp 3 + 29678 × 106 /Tcp 2 + 009911 × 103 /Tcp + 0244063

(3.27)

and when 7000 K < Tcp ≤ 25000 K xD = − 20064 × 109 /Tcp 3 + 19081 × 106 /Tcp 2 + 024748 × 103 /Tcp + 0237040

(3.28)

The spectral power distributions thus obtained are called CIE daylight illuminants and represent average daylight as a function of correlated color temperature Tcp . The measured results that provided the basis of these illuminants were obtained at an interval of 10 nm and linear interpolation is recommended when values with

88


an interval less than this are needed. As mentioned above, the spectral distribution of daylight fluctuates, depending on the conditions, but the CIE recommends a daylight illuminant at a correlated color temperature of approximately 6500 K as an overall representative average. Illuminants at approximately 5000, 5500, and 7500 K are also recommended for use depending on the circumstances. A typical artificial light source is an incandescent lamp, which uses the light emitted by a heated tungsten filament. In general, any non-flammable object begins to emit red light when it is heated, but as it reaches a higher temperature, the color of the object turns from red to yellow, and then to white. For a perfectly black body, the light thus obtained is called black-body radiation. Figure 3.22 shows the spectral power distribution of a black body at various absolute temperatures normalized to be unity at 560 nm. Planck analyzed black-body radiation theoretically, and found that the spectral radiant exitance Me (in W m−3 at an absolute temperature T can be expressed by Me = c1 −5 expc2 /T − 1−1

(3.29)

where represents the wavelength and c1 and c2 are constants with the following values c1 = 2 h c 2 = 37418 × 10−16 W m2 c2 = h c/k = 14388 × 10−2 m K

Radiant power

3

(3.30)

1 00 000 1000

2 10 000 9000 8000 7000

1

6000

5000

5000 4000 3000 2000

0 400

1000

500

600 Wavelength (nm)

10 000 1 00 000

700

Figure 3.22 Spectral distribution of black-body radiation of various absolute temperatures (normalized to 1 at 560 nm)

89


where c is the velocity of light in vacuum, h is Planck’s constant, and k is Boltzmann’s constant (see Appendix I). This is known as Planck’s law of radiation (see Note 3.9). By using Equation 3.29, the spectral radiant exitance can be obtained in absolute units (W m−3 , but in the calculation of the colorimetric values of object colors, relative values are sufficient. An object that obeys Plank’s law is called a black body, a full radiator, or a Planckian radiator. In general, however, a practical object does not always completely obey Planck’s law. The ratio of the radiance of a practical object to that of a black body at the same temperature is called the spectral emissivity. In the case of tungsten, the spectral emissivity is lower at longer wavelengths as shown in Figure 3.23 (Larrabee 1959). Accordingly, the radiation from tungsten yields a color temperature slightly higher than that of a blackbody at the same temperature, but its spectral distribution is, nevertheless, very close to that of a black body. Other artificial light sources commonly used in daily life are fluorescent lamps. A fluorescent lamp uses the discharge in mercury vapor sealed inside a glass tube under saturated vapor pressure and at a tube wall temperature of about 44 C. Ultraviolet radiation with a wavelength of 254 nm is generated efficiently by the discharge, and induces light emission from phosphors applied to the inner wall of the glass tube. Various types of fluorescent lamps with different spectral power distributions can be obtained by properly selecting the phosphors. Fluorescent lamps in common use have a wide range color and spectral power distribution. Traditionally, broadband halophosphate phosphors have been used, but

Spectral emissivity

0.500

1600 K

0.450

0.400

0.350 300

2400 K

400

500

600

700

800

Wavelength (nm)

Figure 3.23 Spectral emissivity of a real object (tungsten) (Larrabee 1959). Reproduced by permission of Optical Society of America

90


Radiant power

recently, to make the color rendition and energy efficiency better, a new type of a fluorescent lamp, the triphosphor type, emitting radiant flux predominantly in three wavelength regions (445–475, 525–555, and 595–625 nm) has been developed and is widely used. Figure 3.24 shows spectral power distributions of various types of fluorescent lamp.

400

400

300

300

200

200

100

100

0

400

500

600

700

0

400

Wavelength (nm)

Radiant power

(a) Cool white color 400

300

300

200

200

100

100

400

500

600

700

0

400

Wavelength (nm)

Radiant power

400

300

300

200

200

100

100

0 600

Wavelength (nm) (e) Daylight color

500

600

700

(d) Daylight white color

400

500

700

Wavelength (nm)

(c) White color

400

600

(b) Warm white color

400

0

500

Wavelength (nm)

700

0

400

500

600

700

Wavelength (nm) (f ) Three-wavelength type, daylight white color

Figure 3.24 Approximate spectral power distributions for various types of fluorescent lamps (Hirota 1987)

91


Other types of artificial light sources include mercury lamps, which take advantage of the emission lines of the discharge with added red fluorescent light emission; low-pressure sodium lamps, which generate monochromatic light with a wavelength of about

400

500

Cadmium

404.7

480.0

Mercury

407.8

435.8

600 508.6 546.1

400

690.8

579.1

577.0

589.6

486.1

434.1

800

643.8

589.0

Sodium Hydrogen

700

656.3

500

700

600

800

Wavelength (nm)

Radiant power

Figure 3.25 Light emission spectrum of representative discharge light sources

400

400

300

300

200

200

100

100

0

400

500

600

700

0

400

600

700

(b) ScI3–NaI 200

400

Radiant power

500

Wavelength (nm)

Wavelength (nm) (a) NaI–T1I–InI3

300 100

200 100 0

400

500

600

Wavelength (nm) (c) DyI3–T1I–InI3

700

0

400

500

600

700

Wavelength (nm) (d) SnI2

Figure 3.26 Spectral distribution of metal halide lamps with various additives (Shimogaki 1987). Reproduced by permission of Ohm Sha

92


589 nm; high-pressure sodium lamps, which emit a more continuous spectrum by extending the width of the emission lines of the low-pressure sodium lamp; and metal halide lamps. Figure 3.25 shows the wavelengths emitted by the discharge from mercury, sodium, and other gases. Metal halide lamps, which can have prescribed spectral distributions according to their intended use are now employed for general illumination in addition to their professional uses. Figure 3.26 shows the spectral distribution of typical metal halide lamps for use in general illumination (Shimogaki 1987).

3.9 STANDARD AND SUPPLEMENTARY ILLUMINANTS The tristimulus values X Y and Z which define an object color, are obtained by Equation 3.12, and it is clear that the values depend on the spectral distribution P of the illuminant. In other words, even for the same object color, various tristimulus values can be obtained depending on the type of illuminant. This is inconvenient in quantitatively expressing colors. Thus, the CIE established several types of standard illuminants (CIE standard illuminants) and artificial light sources (CIE standard sources) for realizing the standard illuminants (CIE 1998a, 2004a). A standard illuminant is a light specified by a (relative) spectral distribution P, and a standard source is an artificial apparatus, such as an incandescent light bulb, that realizes the standard illuminant in practice. Thus, the terms each have a different meaning and must be distinguished from each other. A standard illuminant has its spectral distribution specified numerically in a table, and is used only for the calculation of the tristimulus values. The word ‘illuminant’ is derived from the old English word ‘illume’ (light up) by adding a noun suffix ‘-ant’ (who does ), and it signifies a light-radiating body. The CIE has selected incandescent lamp light and daylight as standard illuminants, representing the most common illuminating lights in daily life, and has recommended representative spectral distributions. Standard illuminant A represents the light emitted from an incandescent light bulb and has a black-body distribution with a correlated color temperature of approximately 2856 K. Standard illuminant D65 , which has a correlated color temperature of approximately 6500 K, is selected from the series of CIE daylight illuminants defined above. An older CIE illuminant, known as CIE illuminant C, is still used in some applications to represent daylight, although it is not as good a representation as D65 , especially

93

STANDARD AND SUPPLEMENTARY ILLUMINANTS

in the ultraviolet region of the spectrum. Although the use of fluorescent lamps is very common in daily life, no fluorescent lamp is yet designated as a standard illuminant. In addition to the standard illuminants, the CIE has also defined some supplementary illuminants for use in colorimetry. There are three types of supplementary illuminants, known as D50 , D55 , and D75 , with correlated color temperatures of 5000, 5500, and 7500 K, respectively. There is also another older standard illuminant, known as standard illuminant B, which is now deprecated. If necessary, other black-body radiations or CIE daylight illuminants may be used as well, but the standard or supplementary illuminants are preferred for simplicity. The spectral distributions of the standard and supplementary illuminants are given in Figures 3.27 and 3.28, and also in Table A3 in the Appendix. The colorimetric values of these illuminants, calculated for a wavelength interval of 5 nm in a wavelength range of 380–780 nm, are given in Table A4 in the Appendix. Figure 3.29 shows the chromaticity points of the standard and supplementary illuminants together with the Planckian locus. It can be seen that that these illuminants represent a wide range of typical illuminating lights.

250

200

Radiant power

A

150 C B 100

C D65 D65

B

50

A 0 300

400


700

800

Figure 3.27 Spectral distribution of CIE illuminants A, B, C, and D65

94

CIE STANDARD COLORIMETRIC SYSTEM 160 140

Radiant power

120 100 80

D50 D55 D65 D75

60

D75 D65

40

D55

20

D50

0 300

500

400

700

600 Wavelength (nm)

800

900

Figure 3.28 Spectral distribution of CIE standard illuminant D65 , and supplementary illuminants D50 , D55 , and D75 (normalized to 100 at a wavelength of 560 nm) 0.8

520 540

560

0.6 500

580

y

A

0.4

2000

3500 2856 D65 B 1500 K 4500 C 6500 10 000

∞

0.2

600 620 650 770 nm

480

0

470 450

380

0.2

0.4 x

0.6

0.8

Figure 3.29 Planckian locus (open circles) and chromaticity points for CIE standard and supplementary illuminants (filled circles)

The CIE recommendations for illuminants and sources are as follows. 1. Standard illuminant A and standard source A. Standard illuminant A represents an incandescent light bulb. It has the spectral

STANDARD AND SUPPLEMENTARY ILLUMINANTS

95

distribution specified in Table A3 in the Appendix, and a correlated color temperature of about 2856 K. To realize standard illuminant A, a gas-filled tungsten lamp with a colorless transparent bulb is recommended as standard source A. 2. Standard illuminant D 65 and daylight simulator D 65 . Standard illuminant D65 represents average daylight. It has the spectral distribution specified in Table A3, and a correlated color temperature of about 6500 K (more accurately, 6504 K; the correlated color temperature of standard illuminant D65 was originally 6500 K, but a change in the international temperature scale changed the value of the constant c2 of Planck’s law of radiation from 14380 × 107 to 14388 × 107 nm K. Thus, the temperature of standard illuminant D65 was changed to 6500·1.4388/1.4380 6504 K in order to maintain the same spectral distribution. Similarly, the color temperature of standard illuminant A, which was originally set at 2848 K when the value of c2 was 1.4350×107 nm K, is now 2848·1.4388/1.4350 2856 K). No standard source D65 is yet developed, but lamps such as xenon lamps are often used as simulators to provide an approximate standard source. 3. Illuminants D 50 , D 55 , and D 75 . Illuminants D50 , D55 and D75 represent daylight with correlated color temperatures of about 5000, 5500 and 7500 K, respectively (more accurately, 5003, 5503 and 7504 K). Their spectral distributions are specified in Table A3 in the Appendix. As with D65 , no standard source is formally recommended to realize these illuminants. However, daylight simulators have been developed that provide approximate standard sources. 4. Illuminant C and source C. Illuminant C represents average daylight. It has the spectral distribution specified in Table A3 in the Appendix, and a correlated color temperature of about 6774 K. To realize illuminant C, standard source A covered with a color temperature conversion filter is recommended. A so-called Davis-Gibson filter, comprising two chemical solutions is used as the color temperature conversion filter. A Davis–Gibson filter has poor durability because it is a chemical solution, and it requires a complicated procedure to reproduce it. Accordingly, solid filters are often used instead. Compared with real daylight at the same correlated color temperature, illuminant C has a smaller relative spectral distribution in the ultraviolet wavelength region. Thus, it cannot be used to express the color of an object that emits fluorescent light when excited by ultraviolet radiation. Accordingly, illuminant C is being replaced in most applications by standard illuminant D65 .

96


5. Illuminant B. CIE illuminant B was previously recommended as a representation of direct sunlight. It has the spectral distribution specified in Table A3 in the Appendix, and a correlated color temperature of about 4874 K. To realize illuminant B, source A together with another Davis–Gibson filter was recommended. As with illuminant C, CIE illuminant B has less power in the ultraviolet wavelength range than does direct sunlight of the same correlated color temperature. Thus, it should not be used for object colors that emit fluorescent light when excited by ultraviolet radiation. Accordingly, illuminant B is replaced by illuminant D50 in most applications and the CIE has already discarded its recommendation for illuminant B. A daylight simulator is a light source that realizes a D illuminant approximately. Simulators can be produced by using an incandescent lamp with a suitable filter or by other sources with or without filters. However, as can be seen from Figure 3.28, the spectral distribution of CIE daylight is not smooth and has many indentations. Thus, in a strict sense, there is little possibility of establishing a series of standard D sources. Accordingly, daylight simulators with good (but not perfect) approximation must be used. As is described in Section 7.2, the CIE has specified a method for evaluating the spectral distribution of such simulators. On the other hand, there is also a proposal that accepts the impossibility of realizing the current CIE daylight illuminants, and instead proposes replacing them with more practical sources. For instance, Hunt (1992) proposes employing a tungsten light bulb, a xenon lamp, and a filtered fluorescent lamp as standard illuminants DT , DX , and DF , respectively.

NOTE 3.1 DERIVATION OF COLOR MATCHING FUNCTIONS FROM GUILD AND WRIGHT’S RESULTS When color matching a monochromatic light [F ], the following relation holds from Equation 2.6 F = r R + gG + bB

(3.31)

However, since the quantitative relation between [F ] and the reference stimuli [R], [G] and [B] was not known, Guild and Wright’s experiments were actually performed according to the following relation F = Kr R + gG + bB

(3.32)

97

DERIVATION OF COLOR MATCHING FUNCTIONS

where K is a constant. By dividing both sides of this equation by Kr + g + b, we obtain F /Kr + g + b = rR + gG + bB

(3.33)

where r = r /r + g + b g = g/r + g + b

(3.34)

b = b/r + g + b are the chromaticity coordinates of monochromatic light (spectral chromaticity coordinates). In Guild and Wright’s experiments, spectral chromaticity coordinates, as illustrated in Figure 3.30, were obtained instead of color matching functions. The chromaticity coordinates in Figure 3.30 were obtained as an average value for many observers. Figure 3.31 shows how the chromaticity coordinates varied among observers (Wright 1928–1929). Furthermore, in Wright’s experiment, the units of [R], [G] and [B] were set so that a mixture of equal amounts of [R] and [G] matched a monochromatic light of wavelength 582.5 nm, and a mixture of equal amounts of [G] and [B] matched a monochromatic light of wavelength 494 nm. This method has the advantage that, if the spectral responsivity of the cones of the observers is equal, their spectral chromaticity coordinates will also be equal because the method compensates for any differences among the observers in pre-receptoral filtering. This method is named the WDW method after the proposer.

r (λ)

g (λ)

b (λ) 1.0

650

530

0

582.5

494

0.5

460

Chromaticity coordinates

1.5

–0.5 400


700

Figure 3.30 Spectral chromaticity coordinates, r, g, b

98


b(λ)

g(λ)

r(λ)

1.0 0.8 0.6

0

–0.2 406

431

530

0.2

650

582.5

494

0.4

460


1.2


596

747

Figure 3.31 Variation of spectral chromaticity coordinates for ten observers (Wright 1928–1929)

The spectral luminous efficiency V is used to obtain the color matching functions r , g and b from the chromaticity coordinates r, g and b. That is, suppose that V can be expressed by V = LR r + LG g + LB b

(3.35)

where, LR , LG , and LB are the luminances of unit amounts of the reference stimuli [R], [G] and [B], respectively. From Equation 3.34, we obtain r = S r g = S g

(3.36)

b = S b where S = r + g + b. By substituting Equation 3.36 into Equation 3.35 we obtain V = S LR r + LG g + LB b

(3.37)

If we measure the luminances LR , LG and LB , S can be calculated from Equation 3.37 because V is known. Thus, the color matching functions r , g and b can be obtained from Equation 3.36. In the X10 Y10 Z10 system, on the other hand, because V10 was not known, the photometric quantities for the reference stimuli and the monochromatic stimuli were measured directly to obtain the color matching functions.

99

CONVERSION BETWEEN COLOR SYSTEMS

Guild and Wright measured the luminances LR , LG and LB by the flicker method. In addition, Wright devised an ingenious method as follows, which enables one to obtain the color matching functions without measuring LR , LG and LB . The following relation can be obtained from Equations 3.36 and 3.37 r = S r = r V/ LR r + LG g + LB b

(3.38)

Similarly, the following relations can be obtained g = g V/ LR r + LG g + LB b b = b V/ LR r + LG g + LB b

(3.39)

By taking the white color stimulus of the equi-energy spectrum as the basic stimulus for which the three tristimulus values are equal, the following relation is obtained. vis

r d =

vis

gd =

bd

(3.40)

vis

By substituting Equations 3.38 and 3.39 into Equation 3.40, we obtain = =

vis

vis

vis

r V/ LR r + LG g + LB bd g V/ LR r + LG g + LB bd b V/ LR r + LG g + LB bd

(3.41)

The relative luminances of the reference stimuli can thus be obtained by determining the ratios of LR , LG , and LB that satisfy Equation 3.41. The color matching functions can then be calculated.

NOTE 3.2 CONVERSION BETWEEN COLOR SPECIFICATION SYSTEMS The conversion between different color systems (for example, from RGB to XYZ ) is performed in the manner described below. The

100


tristimulus values of reference stimuli [X], [ Y], and [Z] are expressed in the RGB color system as X = RX

GX

BX

Y = RY

GY

BY

Z = RZ

GZ

BZ

(3.42)

This can be written in color matching equations as follows X = RX R + GX G + BX B Y = RY R + GY G + BY B

(3.43)

Z = RZ R + GZ G + BZ B Equations 3.43 can be expressed in matrix form as follows     R X R X GX B X       Y  =  RY GY BY   G  B BZ GZ BZ Z 

(3.44)

The following relations hold if we introduce stimulus sums SX SY and SZ , and chromaticity coordinates (rX gX bX , (rY gY bY , and (rZ gZ bZ for the reference stimuli [X], [ Y], and [Z] in the RGB color system, RX = SX rX

GX = SX gX

BX = SX bX

RY = SY rY

GY = SY gY

BY = S Y b Y

RZ = SZ rZ

GZ = SZ gZ

BZ = SZ bZ

(3.45)

Equations 3.45 can be expressed in matrices as follows     SX 0 0 rX gX bX RX GX BX       RY GY BY  =  0 SY 0   rY gY bY  0 0 SZ RZ GZ BZ rZ gZ bZ 

(3.46)

Thus, Equation 3.44 becomes      rX gX bX R X SX 0 0        Y  =  0 SY 0   rY gY bY   G  0 0 SZ B rZ gZ bZ Z 

(3.47)

101

CONVERSION INTO XYZ SYSTEM

On the other hand, the following relation holds for a color [F] having tristimulus values R,G and B in the RGB system and tristimulus values X ,Y and Z in the XYZ system F = RR + GG + BB = X X + Y Y + Z Z

(3.48)

Equation 3.48 can be expressed in matrices as follows 

F = R

G

 R   B  G  = X B



Y

 X   Z  Y  Z

(3.49)

By substituting Equation 3.47 into Equation 3.49, we obtain 

R

G

 R   B  G  = X B



Y

  rX gX bX R    0   rY gY bY   G  0 SZ rZ gZ bZ B

SX 0  Z  0 XY 0

0



(3.50) Thus 

R

G

B = X

Y

  0 0 rX gX bX    Z  0 SY 0   rY gY bY  0 0 SZ rZ gZ bZ

(3.51)

−1  −1 SX 0 0 rX gX bX     B  rY gY bY   0 SY 0  0 0 SZ rZ gZ bZ

(3.52)

SX

and, by inversion 

X

Y

Z = R

G

NOTE 3.3 CONVERSION INTO XYZ COLOR SPECIFICATION SYSTEM The conversion from the RGB color system to the XYZ color system is obtained by the method in Note 3.2. As shown in Figure 3.32, the CIE established the reference stimuli [X], [Y] and [Z] to include the

102

CIE STANDARD COLORIMETRIC SYSTEM [Y] 2.5 g 2.0

510 520 530 500

1.5 540

1.0 G550 560

490

0.5

–1.5

[Z] –1.0 –0.5

480 470

B

+ W

400 –0.5

570 580 590 600 700 R

0.5

1.0

r [X]

Figure 3.32 Location of primaries [X], [Y], and [Z] in rg chromaticity diagram

entire spectrum locus, and such that all the following conditions are satisfied: 1. An equi-energy white color stimulus is used as the basic stimulus. 2. The luminous units of the reference stimuli [X] and [Z] are set at 0, and that of the reference stimulus [Y] is set at 1. Accordingly, y = V. 3. The straight line connecting the reference stimuli [X] and [Y] is brought into tangential contact with the spectrum locus at the long wavelength end. Accordingly, z = 0 for ≥ 650 nm. 4. The straight line connecting the reference stimuli [Y] and [Z] is brought approximately into contact with the spectrum locus at the chromaticity point of = 504 nm, at which the curvature of the spectrum locus is minimum. The chromaticity coordinates of the reference colors [X], [Y] and [Z] satisfying these conditions were selected (Ikeda 1980) as X

rX = 12750

gX = −02778

bX = 00028

Y

rY = −17392

gY = 27671

bY = −00279

Z

rZ = −07431

gZ = 01409

bZ = 16022

(3.54)

103

CONVERSION INTO XYZ SYSTEM

First, for simplification, let X = Y = Z = 1 when R = G = B = 1. By substituting Equations 3.54 into Equation 3.51, we obtain    SX 0 0 12750 −02778 00028    1 1 1 = 1 1 1  0 SY 0  −17392 27671 −00279 

= SX

SY

0 0 SZ −07431 01409 16022   12750 −02778 00028   SZ −17392 27671 −00279  (3.55) −07431 01409 16022

By multiplying both sides of Equation 3.55 above by the inverse of the second matrix on the right-hand side, and exchanging leftand right-hand sides, we obtain  −1 12750 −02778 00028   SX SY SZ = 1 1 1 −17392 27671 −00279  −07431 01409 16022   09088 00912 00000   = 1 1 1  05749 04188 00063  03709 00055 06236 = 18546 05155 06299

(3.56)

Then, by substituting the values of Equations 3.54 and 3.56 into Equation 3.51, we obtain   18546 0 0   05155 0  R G B = X Y Z  0 06299  12750 −02778 −00028   ×  −17392 27671 −00279  −07431 01409 16022   23646 −05152 00052   = X Y Z  −08966 14264 −00144  (3.57) −04681 00888 10092 

0

0

Equation 3.57 can be rewritten as      R 23646 −08966 −04681 X       G  = −05152 14264 00888   Y  B 00052 −00144 10092 Z

(3.58)

104


Then, by inversion    −1   X R 23646 −08966 −04681  Y  = −05152 14264 00888   G  Z B 00052 −00144 10092    04900 03100 02000 R    G = 01770 08124 00106 00000 00100 09900 B

(3.59)

The conversion above was made for the particular case in which X = Y = Z = 1 when R = G = B = 1. However, to make the luminances of test stimuli the same in both the RGB system and the XYZ system, the coefficients of the matrix must be multiplied by a constant factor. Luminance L can be expressed by L = LR R + LG G + LB B = LX X + LY Y + LZ Z

(3.60)

where, LR LG , and LB are the luminances of the primaries in the RGB system and LX LY , and LZ are the luminances of the primaries in the XYZ color system. Thus, if the basic stimulus (1 1 1) in the RGB system is the same as the basic stimulus (K K K ) in the XYZ color system, the following relation holds LR + LG + LB = KLX + LY + LZ

(3.61)

The luminous units of the two systems are LR = 1

LG = 45907

LB = 00601

LX = 0

LY = 1

LZ = 0

(3.62)

By substituting these values into Equation 3.61, we obtain Equation 3.63 K = 1 + 45907 + 00601 = 56508

(3.63)

Thus, the conversion formulae can be obtained by multiplying Equation 3.58 by 1/K , and multiplying Equation 3.59 by K . From Equation 3.58 we obtain      R 23646 −08966 −04681 X  G  = 1/56508 −05152 14264 00888   Y  B 00052 −00144 10092 Z    04185 −01587 −00828 X    Y = −00912 02524 00157 (3.64) 00009 00025 01786 Z

105

IMAGINARY COLORS

From Equation 3.59 we obtain 

    X 04900 03100 02000 R       Y  = 56508  01770 08124 00106   G  Z 00000 00100 09900 B    27689 17517 11302 R    =  10000 45907 00601   G  00000 00565 55943

(3.65)

B

Because the conversion of chromaticity coordinates from (r g b) to (x y z) is unaffected by the size of the stimulus sum, x y and z can be derived from either Equation 3.59 or Equation 3.65. From Equation 3.59 we obtain: x = 04900r + 03100g + 02000b/ 04900 + 01770 + 00000r +03100 + 08124 + 00100g + 02000 + 00106 + 09900b = 04900r + 03100g + 02000b/06670r + 11324g + 12006b y = 01770r + 08124g + 00106b/06670r + 11324g + 12006b z = 00000r + 00100g + 09900b/06670r + 11324g + 12006b (3.66) Because color matching functions are the tristimulus values of monochromatic stimuli (spectral tristimulus values) as a function of wavelength, Equations 3.64 and 3.65 are applicable to monochromatic light. Accordingly, from Equation 3.65, we obtain 

x





27689 17517 11302



r



      y  =  10000 45907 00601   g  z 00000 00565 55943 b

(3.67)

NOTE 3.4 IMAGINARY COLORS [X] AND [Z] Reference stimuli [X] and [Z], which have no lightness, can be derived theoretically, but cannot be understood by intuition. To make these imaginary colors more readily understood, let us imagine a virtual world in which pure sugar [S] is not available, but exists only in three types of syrup in which it is dissolved in water [W] and alcohol [A]. Water and alcohol also are not available, in pure form. An inhabitant of this virtual world must mix the three

106


types of syrup [D], [M] and [T] to obtain any desired sweet taste. Sweetness is an additive quantity and can be determined by the mixing ratio of the three types of syrup. Thus, the sweetness K of the mixture can be obtained by the following equation K = 80d + 50m + 10t

(3.68)

where the unit sweetnesses for [D], [M] and [T] are 80, 50 and 10, respectively, and the amounts of each syrup in the mixture are d m and t respectively. Because sweetness is independent of the quantity of syrup, only the ratios of [D], [M] and [T] are important, and d m and t can normalized by d +m +t =1

(3.69)

This discussion parallels the case of color. Sweetness K corresponds to lightness Y , and syrups [D], [M] and [T] correspond to reference stimuli with lightness coefficients of 80, 50 and 10, respectively. In each case, the sum of the amounts (of sweetness or lightness) of the three reference stimuli yields the amount (of sweetness or lightness) in the mixture. Figure 3.33 shows a plane SWA which expresses sweetness K in mixtures of [D], [M] and [T]. The sweetnesses that can be obtained by mixing syrups [D], [M] and [T] fall inside the range defined by the triangle DMT. Let us consider a case in which only syrups [D] and [M] are mixed. This signifies that t = 0 in Equation 3.68, and we obtain K = 80d + 50m = 80d + 501 − d = 30d + 50

(3.70)

S D

M

T R

A

Q P W

Figure 3.33 Plane (SWA) expressing sweetness K obtained by mixing syrups [D], [M] and [T]

107

IMAGINARY COLORS

Thus, the sweetness of pure [D] can be obtained as K = 80 when d = 1. Figure 3.34 shows this in a two-dimensional td plane. As one moves from D towards M, d decreases and thus K decreases until K = 50 is obtained at d = 0 as the sweetness for pure [M]. Although it may be difficult for the inhabitants of the virtual world to understand this, an imaginary syrup with K = 0 (no sweetness) can be obtained by extrapolating the straight line DM beyond point M and further decreasing d to the point P (d = −5/3). This imaginary syrup having no sweetness contains [D], [M] and [T] in the proportions (−5/3):(8/3):0. It is in fact a mixture of water and alcohol with no sugar, but the inhabitants, whose world does not contain pure sugar, pure water or pure alcohol, consider this to be an imaginary rather than a real syrup, just as we have to consider [X] and [Z] to be imaginary colors. In the case of a mixture containing only [D] and [T], sweetness K can be obtained by putting m = 0 in Equation 3.68, yielding K = 80d + 10t = 80d + 101 − d = 70d + 10

K=

80

K=

10

0

d 2.0

(3.71)

10 K =0

K

=5

0

S

K=

1.0 D

, A ( 13 8 T

M

1 2

)

R ( 45 , 0 )

1.0 Q ( 8 , – 1 ) 2.0 t 7 7

–1.0

+

t=

1. 0

P (0, –

5 3

3

d–

4

t+

–1.0

3

5

=0

d

)

–2.0 W( –

5 8

,–

5 2

)

Figure 3.34 Two-dimensional diagram for the mixture of syrups [D], [M] and [T]

108


Thus, sweetness K = 0 can be obtained at the point Q (d = −1/7). Similarly, point R can be obtained as a point with K = 0 for a mixture of [M] and [T]. For the inhabitants of the virtual world, points P, Q, and R, each having no sweetness, are all imaginary syrups, even though they are all available in our world. The straight line connecting the points P, Q and R is expressed by d − 4t/3 + 5/3 = 0

(3.72)

This straight line is the tasteless line expressing a syrup with no sweetness and corresponds to the alychne in colorimetry. From Equation 3.68, equi-sweetness lines can be obtained as lines parallel to the tasteless line K = 30d − 40t + 50

(3.73)

The sweetest syrup [D] with K = 80 can be obtained at d = 1 and t = 0 as above. If we further increase d, a virtual syrup S (pure sugar) with sweetness K = 100 can be found at d = 3/2 and t = −1/8. Water and alcohol are positioned on the tasteless line, more specifically, at W (−5/8, −5/2) and A (13/8, 1/2). [A] and [W] correspond to imaginary colors [X] and [Z], respectively, and can be determined arbitrarily so long they are located on the straight line PQ.

NOTE 3.5 PHOTOMETRIC QUANTITIES IN THE X10 Y10 Z10 COLOR SYSTEM Photometric quantities can be derived from radiometric quantities by using the spectral luminous efficiency functions V or V . Because y = V in the XYZ color system, the tristimulus value Y conveniently corresponds to the photometric quantity. In the X10 Y10 Z10 system, however, the tristimulus value Y10 does not officially correspond to a photometric quantity. Although the tristimulus value Y10 roughly represents brightness, the CIE has not adopted y 10 officially as the spectral luminous efficiency for a 10 visual field. However, it is possible that it will be so adopted in the future. Considering that the X10 Y10 Z10 color system is more frequently used, it is convenient that the tristimulus value Y10 is roughly related to a photometric quantity, even though the relationship is not exact. When a reflecting object is evaluated by using y and y 10 , the maximum difference Y10 between the tristimulus values Y and Y10 is shown in Figure 3.35 as a function of Y for gray colors.

109

METAMERISM

4.0 2.0 ∆Y10 0.0

0

20

40

60

Y

80

100

–2.0 –4.0

Figure 3.35 Difference Y10 between tristimulus value Y and Y10 (Ohta 1992). Reproduced by permission of Wiley

The same tendency can be observed for other colors. This shows that the range of errors would be approximately ±4.0 if tristimulus value Y10 were to be used in place of the photometric quantity (e.g. luminous reflectance) (Ohta 1992).

NOTE 3.6 ORIGIN OF THE TERM ‘METAMERISM’ The term ‘metamerism’ is said to be have been first proposed by Wilhelm Ostwald, a German chemist. In the field of chemistry, the word metamerism is sometimes used to describe structural isomerism. For example, organic compounds expressed by the empirical formula C2 H6 O include ethanol and dimethyl ether. However, they are clearly distinguished from each other because their rational formulas are CH3 CH2 OH and (CH3 2 O, respectively, and their structures are H

H

H — C — C — OH

H

H

H

H

H—C—O—C—H

H

H

Considered in relation to the CIE color system, the molecular formula corresponds to the trichromatic color system; atomic species C, H, and O correspond to the tristimulus values X Y and Z ; and

110


the number of atoms 2, 6, and 1 correspond to the values of the tristimulus values. Probably, the use of the word ‘metamerism’ in color was proposed because of the similarity in which the same molecular formula (tristimulus values) yields completely different structural formulas (spectral properties).

NOTE 3.7 SIMPLE METHODS FOR OBTAINING CORRELATED COLOR TEMPERATURE Correlated color temperature is defined by the CIE in terms of the CIE 1960 UCS chromaticity diagram (uv chromaticity diagram) (see Section 4.1). The chromaticity coordinates u and v are obtained from tristimulus values X Y and Z by u = 4X/X +15Y + 3Z v = 6Y/X +15Y + 3Z

(3.74)

This diagram is no longer recommended by the CIE, except for obtaining correlated temperature and color rendering index. Figure 3.36 shows the Planckian locus in the uv chromaticity diagram. Correlated temperature can be obtained graphically from a straight line (isotemperature line) that crosses the Planckian locus

0.4

520

540

560

580

600

5000

500 10 000

0.3

620

770

2000 1515 K

3333

640 660

∞

v 0.2 480

0.1

0

470 460 450 440 420

0.1

0.2

400

0.3

0.4

0.5

0.6

u

Figure 3.36 Planckian locus and iso-temperature lines in the uv chromaticity diagram

COLOR TEMPERATURE CONVERSION FILTER

111

at a right angle in this diagram. Robertson (1968) has proposed a simple computer algorithm for implementing this graphical method. Alternatively, McCamy (1992a) proposes the following formula for a direct calculation method for obtaining correlated color temperature T from chromaticity coordinates x and y. T = −437n 3 + 3601n 2 − 6861n + 552431

(3.75)

n = x − 03320/y − 01858

(3.76)

where,

The error in this approximation is said to be negligible in practice.

NOTE 3.8 COLOR TEMPERATURE CONVERSION FILTER The spectral transmittance of a filter that converts the color temperature T1 of a black-body radiation P T1 into a color temperature T2 of another black-body radiation P T2 can be expressed by = k P T2 /P T1

(3.77)

where k is a constant. When c2 /T is sufficiently large, Planck’s law of radiation can be approximated to P T = c1 −5 · exp−c2 /T

(3.78)

This is known as Wien’s law. By substituting Equation 3.78 into Equation 3.77, the following can be obtained log 1/ = K − c2 log e106 /T1 − 106 /T2 / = K − 0006245106 /T1 − 106 /T2 /

(3.79)

where K is a constant, and the wavelength, is expressed in m. From Equation 3.79, the optical density log 1/ plotted against 1/ yields a straight line. The gradient, tan of the straight line can be expressed by tan = −0006245106 /T1 − 106 /T2

(3.80)

112


The term (106 / T1 − 106 / T2 on the right-hand side of the equations above shows the degree of color temperature conversion, and is called the color temperature altering power of the filter. For a given filter material, the optical density is proportional to the thickness. Thus, the following relation holds when a filter of thickness d2 converts color temperature from T1 to T2 , and a filter of thickness d3 converts from T1 to T3 d2 106 /T1 − 106 /T2 = d3 106 /T1 − 106 /T3

(3.81)

This shows that the color temperature altering power increases linearly with increasing thickness of the filter. Figure 3.37 shows an example of two color temperature conversion filters made from the same material. Filter 1 is 1 mm in thickness, and has a color temperature altering power of about 35 MK−1 . Accordingly, when T1 = 2856 K, the light can be converted into a light having a color temperature of about 3200 K because 106 /2856 − 106 /T2 = 35 T2 = 3173K

(3.82)

Since filter 2 has a thickness of 6.5 mm, it converts light of 2856 K into light of about 8200 K because 106 /2856 − 106 /T3 = 35 × 65 T3 = 8154K

(3.83)

1.6 2 log 1/ τ (λ)

1.2

0.8

0.4 0

2.6

2.4 0.400

2.2 0.450

2.0 1/ λ 1.8 0.500

1.6

1.4

0.550 0.600 0.650 0.700 0.750

Wavelength (µ m)

Figure 3.37 Spectral properties of two color temperature conversion filters made from different thicknesses of the same material (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

113

BLACK-BODY RADIATION

NOTE 3.9 SPECTRAL DISTRIBUTION OF BLACK-BODY RADIATION Figure 3.22 shows the spectral distribution of black-body radiation P T for the visible wavelength region alone, and the entire structure cannot be readily understood. Figure 3.38 shows P T for a wider range in wavelength (Burnham et al. 1963). Thus, as shown by the sloping broken line in Figure 3.38, the peak wavelength m of P T moves to shorter wavelengths with increasing temperature T . The relation between T and m can be expressed by m T = 289789 × 106 nm K

(3.84)

Thus, for T = 6500 K for example, the peak occurs at m = 446 nm. This relation is known as Wien’s displacement law and is used for the measurement of high temperatures. The luminance of black-body radiation can be obtained in accordance with Equations 3.29 and 1.20, as illustrated in Figure 3.39.

108 20 000 K

Spectral radiant exitance (W m2 nm–1)

107 106

10 000

105 5000

104 103 2000

102 10

1000

1 500

10–1


105

Figure 3.38 Spectral distribution of black-body radiation (the region defined between broken lines correspond to the visible wavelength range) (Burnham et al. 1963). Reproduced by permission of Wiley

114


log luminance (cd cm–2)

4.0

2.0

1.778

= 60 cd ⋅cm–2 2042 K Freezing point of platinum

0.0

–2.0

–4.0 1000

3000 2000 Absolute temperature (K)

4000

Figure 3.39 Variation of luminance of black-body radiation with absolute temperature (K). Reproduced by permission of Wiley

A luminance of 60 cd/cm2 is obtained at the melting temperature (T = 2042 K) of platinum, and the luminous intensity of a plane having an area of 1/60 cm2 is thus 1 cd. This was the basis of the 1948 definition of the candela, now superseded by the 1979 definition (see Section 1.6).

4 Uniform Color Spaces 4.1 UNIFORM CHROMATICITY DIAGRAMS As described in Chapter 3, the Commission Internationale de l’Eclairage (CIE) recommended, in 1931, a standard colorimetric system including color matching functions for a standard colorimetric observer, standard illuminants, standard light sources, and a chromaticity diagram. A second standard observer was added in 1964 for use in situations involving large fields of view. Both systems have been used widely in various fields, but are incomplete in some aspects. In particular, the fact that the chromaticity diagram is not uniform has been found to be a serious problem in practice. That is, color differences, represented by equal distances in the xy chromaticity diagram (or the x10 y10 diagram), are not perceived as being equal. More specifically, for example, let [A], [B], [C] and [D] be four colored lights having the same luminance. Let the distance in the xy chromaticity diagram between [A] and [B] be the same as that between [C] and [D]. One might expect that the perceived differences between the colors would be the same because the distances are the same. However, in fact, the differences vary greatly, depending on the position in the diagram. Since the chromaticity diagram is regarded as a ‘map of colors’, it is extremely inconvenient if greatly different perceived differences result from distances that are equal. Since the question of uniformity in the chromaticity diagram is extremely important, numerous experimental results have been


116

UNIFORM COLOR SPACES

reported. For example, Wright (1941) reported pairs of colors having the same small perceived difference at the same luminance. His results are given in Figure 4.1. The two ends of the short straight lines (called Wright lines) drawn in the figure represent the chromaticity coordinates for such pairs of colors. It can be seen that the xy chromaticity diagram is extremely nonuniform on the spectrum locus as well as in the inner region. The Wright lines located on the spectrum locus correspond to perceptible differences in the wavelength of monochromatic light. As is shown in Figure 4.2, Wright and

520

0.8

530 540

510

550 560

0.6

570 500 580

y

590

495

0.4

600 610 620 630 650 nm

490

0.2 480 470 460

0.0

380

0.2

0.4 x

0.6

0.8

Figure 4.1 Pairs of colors having the same perceived difference (located at the ends of each straight line) (Wright 1941)

10

Viewing field λ

8 ∆λ (nm)

6

λ ± ∆λ

2°

Ca 70td

4 2 400


700

Figure 4.2 Just-noticeable difference in the wavelength of monochromatic light (Wright and Pitt 1934)

UNIFORM CHROMATICITY DIAGRAMS

117

Pitt (1934) measured the just-noticeable wavelength interval, known as the just-noticeable difference (jnd) for monochromatic light. Their results substantially agree with the results shown in Figure 4.1, and give further evidence for the nonuniformity of the xy chromaticity diagram. MacAdam (Figure 4.3) conducted a more detailed experiment on the nonuniformity of the xy chromaticity diagram (MacAdam 1942). Instead of measuring differences in predetermined directions in the chromaticity diagram, MacAdam repeatedly performed color matching experiments by additive color mixing for a series of predetermined test colors (color centers). The experiment was conducted with a 2 visual field and constant luminance. The matches were constrained to a series of different directions in the chromaticity diagram. The matches did not repeat exactly, but fluctuated within the limit of visual perception. The standard deviations for the fluctuations in each direction around each color center were plotted by open circles as shown in Figure 4.4. MacAdam found that the standard deviations were distributed around the central color in an elliptical pattern. He performed the color matching experiments for 25 different test colors and obtained similar results for all the experiments. The ellipses thus obtained are called MacAdam ellipses. In Figure 4.5, the MacAdam ellipses are shown for the 25 color centers. Each ellipse is magnified 10 times to make it readily visible. The standard deviations expressed by the MacAdam ellipses do not themselves directly represent color differences. Separately, MacAdam performed another experiment to obtain the color

Figure 4.3 David Lewis MacAdam (1910–1998)

118


0.326

0.324 y0

y 0.322

0.320 x0 0.302

0.304

0.306

0.308

x

Figure 4.4 Standard deviation of color matching (open circles) with respect to center color (x0 y0 , and the best-fitting ellipse (MacAdam 1942). Reproduced by permission of Optical Society of America 0.8

520

530 540

510

550 560

0.6 570 500 580 590

y 0.4

600 610 650 620 630

490

0.2 480

470 460 450

0

380

0.2

0.4 x

0.6

Figure 4.5 MacAdam ellipses in the xy chromaticity diagram (10 times magnification) (MacAdam 1942). Reproduced by permission of Optical Society of America

difference as a jnd. He found that the standard deviation multiplied by three approximately corresponds to the jnd of color difference. Thus, the MacAdam ellipses illustrated in Figure 4.5 represent the fluctuation of just-noticeable color differences as a function


119

of position in the chromaticity diagram. Figure 4.5 shows that the MacAdam ellipses are far from being uniform. More specifically, although a very small difference of chromaticity coordinates is discernible for the blue–purple colors located in the lower left portion of the chromaticity diagram, a difference in chromaticity coordinates of about 10 times that is barely perceived for the green colors located in the upper portion. Since the chromaticity diagram is regarded as a ‘map of colors’, ideally the MacAdam ellipses should all be circles of the same radius. Consequently, various different chromaticity diagrams have been proposed that yield more closely the same perceived difference for equal distances in the diagram. These are called uniform-chromaticity-scale diagrams or UCS diagrams. The concept of a UCS diagram is important in practice; hence, many researchers have performed experiments and proposed theories for its development. Among them, a chromaticity diagram proposed by MacAdam was recommended by the CIE in 1960 as the CIE 1960 UCS diagram because of its improved uniformity relative to the xy chromaticity diagram, and because of the simplicity of the conversion equations (MacAdam 1937). In the MacAdam chromaticity diagram, chromaticity coordinates x and y or tristimulus values X Y and Z are converted into new chromaticity coordinates u and v by u = 4x/−2x + 12y + 3 = 4X/X + 15Y + 3Z

(4.1)

v = 6y/−2x + 12y + 3 = 6Y/X + 15Y + 3Z The diagram produced by plotting u as abscissa and v as ordinate is called the uv chromaticity diagram. In the X10 Y10 Z10 color specification system, uniform chromaticity coordinates u10 and v10 are obtained by the same equations with the subscript 10 added. Figure 4.6 shows the MacAdam ellipses converted into the uv chromaticity diagram. It can be seen that the uniformity of the ellipses is somewhat improved compared with that in the xy diagram. However, considering that circles of the same radius would indicate a completely uniform chromaticity diagram, the uv chromaticity diagram is seen to be still imperfect. The uv chromaticity diagram has been widely used since the 1960 recommendation. Examples are the definition of correlated

120

UNIFORM COLOR SPACES 0.4 520

540

580

560

600

620

500

640 660 770 nm

0.3

v 0.2 480

0.1 460 440 420

0

0.1

0.2

400 nm

0.3

0.4

0.5

0.6

u

Figure 4.6 MacAdam ellipses in the uv chromaticity diagram (10 times magnification) (Wyszecki and Stiles 1967). Reproduced by permission of Wiley

color temperature as described in Section 3.7 and the color rendering evaluation of light sources to be described in Section 7.1. However, Eastwood (1975) reported that a better uniformity is obtained by multiplying the v values by 1.5. In 1976, the CIE (2004a) modified the uv chromaticity diagram according to this report, and recommended a new u v chromaticity diagram defined by u = 4x/−2x + 12y + 3 = 4X/X + 15Y + 3Z

(4.2)

v = 9y/−2x + 12y + 3 = 9Y/X + 15Y + 3Z The chromaticity diagram thus obtained is called the CIE 1976 UCS chromaticity diagram or the u v chromaticity diagram. Figure 4.7 shows the MacAdam ellipses converted into this diagram, but the improvement in uniformity is not very distinct (Tominaga 1987). Chromaticity coordinates x y can be readily converted to u v using a nomogram given in Figure 4.8 instead of using Equation 4.2 (Hunt 2004). For example, values x = 0.2 and y = 0.3 can be converted into u and v by taking the intersections of the straight line connecting x = 0.2 and y = 0.3 with the u and v axes, leading to u and v values of 0.13 and 0.44, respectively.

121


0.6

550 600 650 500

0.4 v′

0.2

450

0

0.2

0.4

0.6

u′

Figure 4.7 MacAdam ellipses in the u v chromaticity diagram (10 times magnification) (Tominaga 1987). Reproduced by permission of Ohm Sha

0.9 0.8 0.7 y 0.6 0.5

0.7 0.6

x 0.6 0.5

u′ 0.5

0.4

0.4

0.3

0.3

0.3

0.5

v′

0.2

0.4 0.3

0.1

0.2 0.2

0.4

0.6

0.8 0.7

0.2

0.1 0.1

0.1

0

1.0

Figure 4.8 Nomogram for transforming CIE x y values to CIE u v values (Hunt 2004). Reproduced by permission of Wiley

122


4.2 UNIFORM LIGHTNESS SCALES (ULS) Subsequent to the uniformity of chromaticity coordinates, let us now consider the uniformity of lightness. As described in Section 2.3, lightness is a relative brightness, a perceptual attribute used mainly to describe object colors. Lightness approximately corresponds to a function of the tristimulus value Y . As described in Section 3.4, Y is equal to the luminous reflectance in the case of a reflecting object or the luminous transmittance in the case of a transmitting object. The description below mainly relates to reflecting objects. As with chromaticity coordinates, the problem is that Y does not have a uniform scale with respect to lightness. More specifically, lightness does not increase linearly with Y . Thus, the tristimulus value Y must be modified so that lightness increases linearly in a new scale that is called a uniform lightness scale (ULS). The Munsell color system provides an important clue to how to do this. Lightness in the Munsell system is a visually equi-spaced scale called Munsell value V . Figure 4.9 shows the relation between luminous reflectance Y and Munsell value V . There is a nonlinear relationship, given by (see Note 4.1) Y = 12219V − 023111V 2 + 023951V 3 −0021009V 4 + 00008404V 5

(4.3)

10 90.0

102.6

78.7 68.4

Munsell value (V )

8

59.1 43.1

6

30.1 19.8

4

12.0 6.6

2

3.1 1.2

0

20

40

60

80

100

Luminous reflectance (Y )

Figure 4.9 Relation between luminous reflectance Y and Munsell value V (points are labelled by values of Y

123

UNIFORM LIGHTNESS SCALES (ULS)

Referring to Figure 4.9, it can be seen that the nonlinearity is particularly distinct with a steep slope in the vicinity of black colors, and that the color quickly appears relatively light when only a small quantity of white (in terms of reflectance) is mixed with black. This is why black hair unexpectedly looks white when only a small amount of white hair is mixed with it. More specifically, black hair (with V = 0) changes into semi-white (V = 5) when only about 20% of white hair (V = 10) is mixed with the black. The uniformity of Munsell value can be illustrated by another example. The luminous reflectance Y of the white background (W) of this book is approximately 87%, and that of the black printed portion (K) is approximately 7%. An intermediate value for a gray color can be obtained as Gray 1 by taking the average value of Y , or as Gray 2 by taking the average of V after converting Y into V in accordance with Equation 4.3. The values thus obtained are: Y V

W 870 94

K 70 31

Gray 1 870 + 70/2 = 470

Gray 2 94 + 31/2 = 625

The upper level in Figure 4.10 shows W, Gray 1, and K, whereas the lower level in the same figure shows W, Gray 2, and K. It can be seen that the lightness of Gray 1 is not at the midpoint between W and K, but that it is considerably displaced towards W. On the other hand, Gray 2 is located approximately at the midpoint between W and K. In general, when a small change I in the intensity I of a stimulus is just noticeable, it is found experimentally that the following relation holds I/I = constant

(4.4)

Average in luminous reflectance

W

Gray 1

K

Average in Munsell value

W

Gray 2

K

Figure 4.10 Gray colors having the average value of white (W) and black (K) in terms of luminous reflectance (Gray 1), and in terms of Munsell value (Gray 2)

124


This is known as the Weber Law. If we assume that just-noticeable differences represent equal steps in sensation S, then the change in sensation S, that results from the change in I , is a constant and can be written as S = k I/I

(4.5)

where k is another constant. The minimum detectable stimulus value is called the absolute threshold, and, expressing it as I0 , the following equation can be obtained by integrating Equation 4.5 S = k logI/I0

(4.6)

This is known the Weber–Fechner Law. Figure 4.11 shows the relation between the luminance level L and L/L in the case of a white light stimulus. The inflection point at L = 5 × 10−3 cd/m2 shows the transition from rods to cones, and it can be seen that Equation 4.4 approximately holds true for the region corresponding to L greater than 100 cd/m2 . Although it depends on the experimental conditions, the ratio I /I is approximately equal to 0.01. Thus, as an alternative to the Munsell value, the logarithm of tristimulus value Y can be used as a uniform lightness scale.

+0.5 0.0

log (∆L/L)

–0.5 –1.0 –1.5 –2.0 –2.5 –6 –5 –4

–3 –2 –1 0 log L (cd/m2)

1

2

3

Figure 4.11 Relation between luminance L and L /L for a white stimulus (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

UNIFORM LIGHTNESS SCALES (ULS)

125

In fact, the value of (2 − log Y is widely used as a ‘visual density’ in the evaluation of photographs. Stevens proposed a power law expressed by S = k I/I0

(4.7)

for use in place of Equation 4.6. The power differs depending on the stimulus, and the values for different stimuli are listed in Table 4.1. Depending on the experimental conditions, is in the range 0.33–0.5 in the case of light. In general, the numerical equation to be used to express a nonlinear relationship may be selected arbitrarily. More specifically, for example, Figure 4.12 shows the similarity between a logarithmic function and a power law function. Considering the experimental uncertainties, there is no advantage or disadvantage of one function over the other. Tristimulus value Y is converted into a uniform lightness scale by a fifth-order polynomial in the case of the Munsell system, or by a cube-root function in a CIE formulation that is to be described below. The differences between equations with different functions are small, as shown in Figure 4.13. Table 4.1 Power for various types of stimuli (Nikkagiren Kannoukensa Iinkai 1973). Reproduced by permission of Wiley Senses

Stimulus

Vision

Brightness

Hearing

Lightness Sound intensity

Smell

Smell

Taste

Taste

Touch

Temperature Pressure Gravity Grip Vibration Roughness Hardness

Condition

5 field, dark adaptation Point source, dark adaptation Reflectance of gray color chip Both ears One ear Coffee Heptane (gas) Saccharin Sugar Salt Cold feeling on arm Warm feeling on arm Pressure on skin Lifting weight Grip meter Vibration of 60 Hz on a finger Vibration of 250 Hz on a finger Diameter of powdered emery Rubber squeezed between fingers

Power 033 05 12 06 054 055 06 08 13 13 10 15 11 145 17 095 06 15 08

126


2.0

1.5

F (R)

1.0 0.4156R0.3761 log10(R + 1)

F (R ) 0.5

0

10

20

30 R

40

50

60

Figure 4.12 Similarity between logarithmic function (solid line) and power law function (broken line). Reproduced by permission of Wiley

100

80

Lightness

60

L = 10 Y L* = 116(Y/100)1/3 – 16

40 10 V 20

Y

0 –20

0

2

4 6 Munsell value V

8

10

Figure 4.13 Uniform lightness scale expressed in various forms (Hunter and Harold 1987). Reproduced by permission of Wiley

Equation 4.3 shows the relation between Y and Munsell value V . In practice, the relation between Y and perceived lightness depends on the lightness of the background. Figure 4.14 shows the relation between Y and perceived lightness, V for several background

127

CIE UNIFORM COLOR SPACES

10 Yb = 1

8 Yb = 20

Yb = 50

6 Yb = 100

V 4

2

0

20

40

60

80

100

Y

Figure 4.14 Change in relation between luminous reflectance, Y , and lightness, V , with change in luminous reflectance Yb of background (Wyszecki and Stiles 1967). Reproduced by permission of Wiley

luminous reflectances Yb . Assuming that V = 10 when Y = 100 for any value of Yb , the relationship can be expressed approximately by V = YYb + 100/10Y + Yb

(4.8)

Comparing Figure 4.14 with Figure 4.9, it can be seen that the Munsell value function is close to the Yb =100 curve for high values of Y while it is close to the Yb = 20 curve for low values of Y .

4.3 CIE UNIFORM COLOR SPACES Although a perfect uniform chromaticity diagram is yet to be found, the uniformity of the u v diagram is much better than that of the xy diagram. However, in addition it is necessary to pay attention to attaining uniformity of lightness. In other words, although a more uniform relationship can be established in the chromaticity coordinates, if (u v Y ) is used in place of (x y Y ), there is no change in the uniformity of lightness. Thus, another colorimetric system, inclusive of lightness, is needed. This is called a uniform color space. The coordinate corresponding to lightness in a uniform color space has sometimes been called psychometric lightness.

128


Because of the practical importance, numerous uniform color spaces have been proposed. For example, Wyszecki (Figure 4.15) noted that Munsell value, V can be approximated by a cube-root function of Y , and that perceived chroma is higher for colors of higher lightness but the same chromaticity. Thus, he proposed the following uniform color space (Wyszecki 1963) based on the CIE 1960 uv chromaticity diagram (see Section 4.1) W ∗ = 25 Y 1/3 − 17 U ∗ = 13 W ∗ u − un

(4.9)

V ∗ = 13 W ∗ v − vn where, u and v are the chromaticity coordinates of the object, and un and vn are the chromaticity coordinates of a perfect reflecting diffuser. W ∗ represents the lightness, and is approximately equal to 10V , where V is the Munsell value. The total color difference E between two colors (U1∗ V1∗ W1∗ ) and (U1∗ V2∗ W2∗ ) in the uniform color space is defined by E = U ∗ 2 + V ∗ 2 + W ∗ 2 1/2

(4.10)

where U ∗ = U1 ∗ − U2 ∗ V ∗ = V1 ∗ − V2 ∗ ∗

∗

W = W1 − W2

¨ Figure 4.15 Gunter Wyszecki (1925–1985)

(4.11) ∗

129


This uniform color space was adopted by the CIE in 1964 as the CIE 1964 U ∗ V ∗ W ∗ color space, and was used in industry until two new color spaces were recommended in 1976. As is illustrated in Figure 4.16, there were very many uniform color spaces available at

All color stimuli

Object color

Surface color

CIE Y, x, y 1931

Dominant wavelength and purity 1931

Munsell 1929

Judd color triangle 1935

MacAdam’s ellipses 1942

Munsell renotation system 1943

MacAdam uv chromaticity diagram 1937

Friele 1961

Adam’s chromatic value 1942

FMC 1967

Breckenridge and Schaub chromaticity diagram 1939

Adams– Nickerson 1944, 1950 Hunter α, β 1942 Glasser cube root 1958

Hunter Rd, a, b 1948 Hunter LL, aL, bL 1958 Hunter Ln, an, bn 1966 CIE U* V* W* 1964 CIE L* u* v* (CIELUV) 1976 CIE L* a* b* (CIELAB) 1976

Figure 4.16 Genealogy of uniform color spaces (Hunter and Harold 1987). Reproduced by permission of Wiley

130


the time in addition to the CIE 1964 U ∗ V ∗ W ∗ space, and this was a cause of confusion. Accordingly, the CIE addressed the situation in 1976 and recommended the use of only two spaces, the L ∗ a ∗ b ∗ color space and the L ∗ u ∗ v ∗ color space (CIE 2004a). The color differences in these two color spaces are not the same, but as described in Section 4.6, approximate conversion is possible. The definitions of the color spaces are given below in the XYZ colorimetric system, but the X10 Y10 Z10 system should be used when the visual angle exceeds 4 . The equations are the same, but with a subscript 10 added to every symbol.

4.3.1 CIE 1976 L∗ a∗ b∗ Color Space A color space known as the CIE 1976 L ∗ a ∗ b ∗ Color Space (abbreviated CIELAB) is expressed by the following three-dimensional orthogonal coordinates L ∗ = 116Y/Yn 1/3 − 16 a ∗ = 500X/Xn 1/3 − Y/Yn 1/3 ∗

b = 200Y/Yn

1/3

− Z/Zn

1/3

(4.12)

where X Y and Z represent the tristimulus values of the object under consideration, and Xn , Yn and Zn represent the tristimulus values of a perfect reflecting diffuser under the same illuminant. The values are normalized so that Yn = 100. Lightness is represented by L ∗ , which is called CIE 1976 lightness. The same quantity is used in the CIELUV color space, which is described below. It should be noted that Equation 4.12 is limited to the conditions X/Xn > 0008856, Y/Yn > 0008856 and Z/Zn > 0008856. Under other conditions, a modified equation described in Section 4.3.3 should be employed.

4.3.2 CIE 1976 L∗ u ∗ v ∗ Color Space A color space known as the CIE 1976 L ∗ u ∗ v ∗ Color Space (abbreviated CIELUV ) is expressed by the following three-dimensional orthogonal coordinates L ∗ = 116Y/Yn 1/3 − 16 u ∗ = 13 L ∗ u − u n ∗

∗

v = 13 L v − vn

(4.13)

131


where Y , u and v represent the tristimulus value Y of the object under consideration and the chromaticity coordinates obtained according to Equation 4.2, and Yn , u n , and vn represent the tristimulus value Y and the chromaticity coordinates u and v of a perfect reflecting diffuser under the same illuminant. The values are normalized so that Yn = 100.

4.3.3 Modiﬁed Equation for Dark Colors Equations 4.12 and 4.13 are not applicable to very low values of X, Y and Z. To extend the applicability to all colors, the following modified equations are used (see Note 4.2). L ∗ = 116fY/Yn − 16 a ∗ = 500 fX/Xn − fY/Yn

(4.14)

b ∗ = 200 fY/Yn − fZ/Zn where fX/Xn = X/Xn 1/3

if

X/Xn > 24/1163

fX/Xn = 841/108X/Xn + 16/116

if

X/Xn 24/1163 (4.15)

The term 841/108 is approximately equal to 7.787, the term, (24/116)3 is approximately equal to 0.008856, and the term 16/116 is approximately equal to 0.1379. Functions fY/Yn and fZ/Zn are obtained in the same manner. Based on Equation 4.15, the CIE recommends the use of the following equation as an equivalent equation for L ∗ when Y/Yn is not greater than (24/116)3 . L ∗ = 116841/108Y/Yn + 16/116 − 16 9033Y/Yn

(4.16)

In the CIELAB and CIELUV color spaces, a ∗ and b ∗ , as well as u and v ∗ , are sometimes called color coordinates, and represent the combined attributes of hue and chroma. As defined by the CIE, the spaces were intended to apply to the case of adaptation to a visual field of chromaticity not too different from that of average daylight. In practice, they are applied to a wide range of adaptation conditions falling outside the range of average daylight, but ∗

132


further accumulation of experimental results is required to justify the validity of this extension. In the Japanese Industrial Standard (JIS), in addition to the two color spaces described above, the Adams–Nickerson color space and the Hunter color space are included as references for use in obtaining color difference values. Also included in the JIS as a reference is the CMC (l:c) color difference formula, which is used widely in the dye industry and has been adopted as British Standard BS 6923 and ISO Standard 105-J03 (see Note 4.3).

4.4 CORRELATES OF PERCEIVED ATTRIBUTES Approximate correlates of certain perceived attributes of color and color difference (lightness, chroma, hue angle, and hue difference) can be calculated in CIELAB or CIELUV color space in the following manner. The subscripts ab and uv indicate the CIELAB color space and CIELUV color space, respectively.

4.4.1 Color Difference The color difference Eab ∗ between two colors (L1 ∗ a1 ∗ b1 ∗ ) and (L2 ∗ a2 ∗ b2 ∗ ) is obtained by Eab ∗ = L ∗ 2 + a ∗ 2 + b ∗ 2 1/2

(4.17)

where L ∗ = L1 ∗ − L2 ∗ a ∗ = a1 ∗ − a2 ∗ ∗

∗

b = b1 − b2

(4.18)

∗

Similarly, the color difference Euv ∗ between two colors (L1 ∗ u1 ∗ v1 ∗ ) and (L2 ∗ u2 ∗ v2 ∗ ) is obtained in the CIELUV color space by Euv ∗ = L ∗ 2 + u ∗ 2 + v ∗ 2 1/2

(4.19)

where L ∗ = L1 ∗ − L2 ∗ u ∗ = u1 ∗ − u2 ∗ v ∗ = v1 ∗ − v2 ∗

(4.20)

CORRELATES OF PERCEIVED ATTRIBUTES

133

4.4.2 Lightness As described in Section 4.3.1, a correlate of lightness, called CIE 1976 lightness, can be calculated by the following equations L ∗ = 116fY/Yn − 16

if

Y/Yn > 24/1163

if

Y/Yn 24/1163

L ∗ = 116841/108Y/Yn 9033Y/Yn

(4.21)

4.4.3 Chroma and Saturation CIE 1976 chroma, C ∗ is obtained as a distance between the origin and the point expressed by the coordinates a ∗ and b ∗ or u ∗ and v ∗ , according to Cab ∗ = a ∗ 2 + b ∗ 2 1/2 Cuv ∗ = u ∗ 2 + v ∗ 2 1/2

(4.22)

In the case of CIELUV, a correlate of saturation suv is defined, by suv = 13u − un 2 + v − vn 2 1/2 = Cuv ∗ /L ∗

(4.23)

There is no easily defined correlate of saturation in CIELAB.

4.4.4 Hue Angle CIE 1976 hue angle h is obtained by the following equations hab = tan−1 b ∗ /a ∗ huv = tan−1 v ∗ /u ∗

(4.24)

where hab lies between 0 and 90 if a ∗ and b ∗ are both positive, between 90 and 180 if b ∗ is positive and a ∗ is negative, between 180 and 270 if b ∗ and a ∗ are both negative, and between 270 and 360 if b ∗ is negative and a ∗ is positive, with a similar convention for huv .

134


4.4.5 Hue Difference The hue difference H ∗ is obtained according to the following equations Hab ∗ = 2Cab1 ∗ Cab2 ∗ 1/2 sinhab /2 Huv ∗ = 2Cuv1 ∗ Cuv2 ∗ 1/2 sinhuv /2

(4.25)

These direct definitions of and Hab ∗ and Huv ∗ were suggested by Sève (see Note 4.4) and can be used to replace the original indirect definitions: Hab ∗ = Eab ∗ 2 − L ∗ 2 − Cab ∗ 2 1/2 Huv ∗ = Euv ∗ 2 − L ∗ 2 − Cuv ∗ 2 1/2

(4.26)

The color differences defined above in Equations 4.17 and 4.19 are simple distances in the CIELAB or CIELUV color space. However, it is has been found that this method of calculation is not optimal for obtaining color difference values that correlate with visual perception. According to Equation 4.26, the color difference E ∗ is obtained from lightness, chroma, hue differences L ∗ , C ∗ and H ∗ by E ∗ = L ∗ 2 + C ∗ 2 + H ∗ 2 1/2

(4.27)

It can be seen that L ∗ , C ∗ and H ∗ are included with the same weight. However, there are cases in which correlation with the perceived magnitude of color differences is improved by using different weights, l c and h, for L ∗ , C ∗ and H ∗ , respectively as follows E ∗ = L ∗ /l2 + C ∗ /c2 + H ∗ /h2 1/2

(4.28)

This point is discussed further in Note 4.3 in the context of the CMC (l:c) color difference formula and in Section 4.7.

4.5 COMPARING CIELAB AND CIELUV COLOR SPACES As shown in Figure 4.16, the CIELAB color space was obtained by modifying Adams’ color space, a color space widely used in industry prior to 1976 (see Note 4.3). It is based on the color vision model of Adams fitted to the Munsell color system. The following relative

135

COMPARING CIELAB AND CIELUV COLOR SPACES

tristimulus values Xc , Yc , and Zc are used in place of the tristimulus values X Y and Z R P x d/ P x d = X/Xn Xc = vis vis Yc = R P y d/ P y d = Y/Yn (4.29) vis vis R P z d/ P z d = Z/Zn Zc = vis

vis

where Xn Yn , and Zn are the tristimulus values of a perfect reflecting diffuser. CIELAB color space is based on the cube root of these relative tristimulus values. By normalizing the color matching functions x y, and z according to x d = y d = z d = 1 (4.30) vis

vis

vis

Borges (1991) has shown that the following approximation holds R P x d ∼ R x d P x d (4.31) vis

Accordingly, the approximated as

vis

vis

relative Xc ∼

tristimulus

vis

R x d

value

Xc

can

be

(4.32)

Thus, Xc is approximately independent of the spectral distribution P of the illuminant. The same can be said of the other relative tristimulus values, Yc and Zc . The fact that the relative tristimulus values are independent of the illuminating light reflects color constancy, a known characteristic of color vision. This feature is an advantageous characteristic of Adams’ color space. CIELAB color space inherits this characteristic from Adams’ color space. Note however, that Equation 4.31 is only an approximation and that this ‘color constancy’ feature of CIELAB does not match real color vision exactly. On the other hand, CIELUV color space was obtained by modifying the CIE 1964 U ∗ V ∗ W ∗ uniform color space by incorporating the u v chromaticity diagram. Thus, the inherited advantage of this color space is that straight lines in the chromaticity diagram readily express additive color mixture. CIELUV color space is widely used in industrial fields that depend on the additive mixing of light, such as in color TV sets, video monitors, illuminations, etc.

136


To further study the relationship between the characteristics of CIELAB color space and CIELUV color space, representative red (R), green (G), blue (B), cyan (C), magenta (M), and yellow (Y) color chips were measured. Figure 4.17 shows the coordinates a ∗ and b ∗ , as well as u ∗ and v ∗ , of the six color chips, obtained for standard illuminant A (open circles) and D65 (filled circles). As predicted above, it can be seen that CIELAB color space is hardly influenced by the change of illuminant. Because the conversion equations are different, the coordinates a ∗ b ∗ differ from u ∗ v ∗ even when the same tristimulus values are used originally. For example, Figure 4.18 shows how the color coordinates a ∗ b ∗ (filled circles) correspond to u ∗ v ∗ (open circles) for the six color chips under standard illuminant D65 . It can be seen that there is a considerable difference between the two sets of coordinates. Figure 4.19 shows the results obtained by plotting a ∗ b ∗ and u ∗ v ∗ for a larger series of colors with an L ∗ value of 50 under standard illuminants A and D65 . It can be seen that the coordinates correspond to each other in a complex manner and that there is no particular rule to relate them. To illustrate the uniformity of the CIELAB and CIELUV color spaces, MacAdam ellipses and the equal hue and equal chroma curves of the Munsell color system are shown for the two color spaces in Figures 4.20 and 4.21 (Robertson 1977). It can be seen from Figure 4.20 that the CIELUV color space is superior for MacAdam ellipses. This is quite natural considering that the CIELUV color space was developed based on the MacAdam ellipses. However, if complete uniformity had been achieved, the v* b* Y 75 50

50 R

25

G

Y

75

G

Y

25

R –50

–25 C

25

–50

–25

M

–25 –50

50 a*

50 75

–25

C B

25

B

u*

100 R125 M

M –50 B –75

(a) CIELAB

(b) CIELUV

Figure 4.17 Chromaticity coordinates of 6 color chips under standard illuminant A (open circles) and D65 (filled circles)

137


b*, v* 75

Y

50 G 25 –50

R

–25

25

50

75

100 a*, u*

M

–25 C –50 B –75

Figure 4.18 Correspondence between color coordinates a ∗ b ∗ (filled circles) and u ∗ v ∗ (open circles) for six color chips under standard illuminant D65 b*, υ* 100 50 a*, u* 50

–200 –150 –100 –50

200

100

–50 –100 –150

(a) Standard illuminant A

b*, υ* 100 50 150 50

–200 –150 –100 –50

100

–50

200

250 a*, u*

–100

–150

(b) Standard illuminant D65

Figure 4.19 Correspondence of color coordinates a ∗ b ∗ and u ∗ v ∗ (arrowhead) of colors of L ∗ = 50

138


150

100

L* = 50

100 50 b*

0 v*

0 –50

L* = 50

50

–50

–100

Optimal colors

–100 –200 –150 –100 –50 0 a*

50

100 150

Optimal –150 colors –150 –100 –50

0

50

100

150 200

u*

Figure 4.20 MacAdam ellipses for CIELAB color space (left) and CIELUV color space (right) (Robertson 1977). Reproduced by permission of Wiley 150

V=5 GY

100

Y

100 50

YR

GY

Y YR

G

V=5 R

50 b*

0

R G

0

RP

BG –50

B

PB –100 –200 –150 –100 –50 0 50 a*

v* –50 –100

RP B PB

P 100

BG

150

–150 –150 –100 –50

P 0 u*

50

100

150 200

Figure 4.21 Equi-hue and -chroma curves of the Munsell color system in CIELAB color space (left) and CIELUV color space (right) (Robertson 1977). Reproduced by permission of Wiley

ellipses would all be circles of the same radius. Thus, the CIELUV color space needs more improvement. On the other hand, from Figure 4.21, it can be seen that the CIELAB color space is superior when evaluated in terms of the Munsell system. Considering that the CIELAB color space evolved from Adams’ color space, which in turn was based on the Munsell system, this fact again is quite natural. However, if complete uniformity had been achieved, the equal hue and equal chroma curves would be concentric circles. In this case again, there is much to be improved. It can be seen from this discussion that both color spaces are far from being completely uniform, and that they require improvement. A reason why complete uniformity is not achieved may be that although a simple numerical equation can sometimes express a single aspect of color perception under controlled conditions, in the real world color perception has many attributes and is subject


139

Psychologically evlauated values

4.0

2.0

0.0

–2.0

–4.0 1.40

1.60

1.80

20.0

Color difference

Figure 4.22 Relation between color difference and psychologically evaluated values under the simple experimental condition of chroma difference only. (Billmeyer and Saltzman 1981). Reproduced by permission of Wiley

to the influence of many different parameters so that much more complicated equations are needed. For example, Figure 4.22 shows how psychologically evaluated values correspond to color difference under a limited simple experimental condition. It can be seen that a clear correlation is obtained. However, under a wider range of experimental conditions in which the conditions vary considerably, the agreement of calculated values with psychologically evaluated values becomes extremely poor as is shown in Figure 4.23 (Billmeyer and

Psychologically evaluated values

6 5

4 3 2 1

0 5 10 15 Color difference (ANLAB40, see Note 4.3)

Figure 4.23 Relation between color difference (ANL AB40) and psychologically evaluated values under a wider range of experimental conditions (Billmeyer and Saltzman 1981). Reproduced by permission of Wiley

140


Saltzman 1981). The reason for the scattered results is presumably the missing parameters (e.g., the illumination level, the color of the background, the size of the specimen, the distance between specimens, etc.) so that a simplified equation is inadequate. Accordingly, numerous studies have been and continue to be undertaken to find better equations.

4.6 CONVERSION OF COLOR DIFFERENCE Because the CIELAB and CIELUV color spaces are both derived from the tristimulus values X Y and Z the colorimetric values, L ∗ a ∗ b ∗ and L ∗ u ∗ v ∗ can be converted to each other via X Y Z . However, the color difference, E ∗ indicates only the scalar difference between two colors, and does not include information on the direction. Thus, it is impossible to convert a difference in one space to a difference in the other space. However, because CIELAB and CIELUV are both used widely, it would be useful, even though impossible in a strict sense, to have a rough idea how a color difference expressed in one space relates to the same color difference expressed in the other. More specifically, for example, the question is whether a difference of Eab ∗ = 1.0 obtained in the vicinity of yellow with Y = 50 corresponds approximately to a Euv ∗ difference of 1.0 or 5.0, or even 10.0. A nonlinear optimization method can solve this problem, and the color difference can be converted within a predetermined range (Ohta 1977). Let us consider all the colors in CIELAB space around a given color when Eab ∗ = 10.0. In CIELAB space, these colors fall on a spherical surface of radius 10.0 around the given color. When the sphere is converted to CIELUV space, the spherical shape is not retained because the conversion from one space to the other is not simple but rather is highly nonlinear. For example, a sphere with its center at a yellow color of Y =50 (L0 ∗ = 76.1) and a radius of 10.0 in CIELAB space transforms to a pseudo ellipsoid as shown in Figure 4.24 when converted into CIELUV space. In this computation, standard illuminant D65 and the color matching functions of the X10 Y10 Z10 colorimetric system are used. The range of color difference (maximum value and minimum value) in CIELUV can be obtained as the lengths of the major and minor axes of the pseudo ellipsoid. More specifically, let

= Euv ∗ /Eab ∗

(4.33)

141

CONVERSION OF COLOR DIFFERENCE

v*

80 +5.0 +

+7.5

L*= L0*

70 40

30

20 –5.0

–7.5

0

50

60

u*

+ 60

Figure 4.24 A CIELAB sphere with Eab ∗ = 10.0 and centered at a yellow color of Y = 50, converted into CIELUV space (Ohta 1977). Reproduced by permission of Wiley

be the ratio of the color differences in the two color spaces. It can be seen that is not a constant value, but by denoting the maximum and minimum values by max and min , respectively, the following relation holds

min max

(4.34)

By applying a nonlinear optimization method, the values of min and max were calculated for a series of L ∗ u ∗ v ∗ and are given in Figure 4.25. In the figure, the outermost boundary shows the range of optimal colors. Figure 4.25 can be used in the following manner. For example, let one of a pair of color chips have the colorimetric values of L ∗ = 50, u ∗ = 50, and v ∗ = 30. Referring to the figure, the following values can be read from (c) and (C)

min = 070

max = 176

(4.35)

If the color difference between the pair of color chips is given by Eab ∗ =1.0, then the color difference Euv ∗ is in the range 070 176

(4.36)

If the color difference between the pair of color chips is given by Euv ∗ , the bounds for the color difference Eab ∗ can be found in a similar manner.

142


(b) ρmin

100

(B) ρmax

L* = 30

100

50

50 1.6

0.9 0.8

0.2

100

2.4

2.6 2.8 3.0

100

(C) ρmax

L* = 50

L* = 50

1.6

1.8

50 0.2 0.4 0.6 0.8

0 0

50

100 150 0.95

1.3 1.4

–150 –100

–50

L* = 70

0 –150 –100

–50

(D) ρmax

100

0.2 0.4 0.6 0.8 0.95

50

0

50

100

–50

150

1.3 1.4

–150 –100

–50

L* = 90

(E) ρmax 1.3 1.4

0

50

50 1.6

0

50

100

150

u*

100

150

L* = 90

50 0

0 –50

L* = 70 2.0

100

1.6

–150 –100

1.8

–100 2.2

0.2 0.4 0.6 0.8 0.95

50

u*

–50

0.95

100

100 150 2.4

0

–100

(e) ρmin

50

–100

0.6

100

2.2

–50

0.8

–100

2.0

0 0

–50

(d) ρmin

u*

–50

0.2

100

–150 –100 –50

150

0.4

–100

(c) ρmin

2.2

50

–50 0

0.6

–50

0.4

–150 –100

150

1.8 2.0

1.4

1.3

0.2

–150 –100 –50

L* = 30

–150 –100

–50

50

100

150

u*

1.8

–50

–50

–100

–100

Figure 4.25 The values of min and max for conversion of color difference (Ohta 1977). Reproduced by permission of Wiley

143

COLOR DIFFERENCE EQUATIONS BASED ON CIELAB

4.7 COLOR DIFFERENCE EQUATIONS BASED ON CIELAB After the CIELAB and CIELUV color difference formulas were introduced by the CIE 1976, it quickly became apparent that, despite being as good or better than any other formulas then available, they could be further improved by adjusting the L ∗ C ∗ and H ∗ components, depending on their position in color space. The first major attempt to do this resulted in the CMC (l:c) formula (Clarke et al. 1984). This is based on the CIELAB formula and was adopted first as a standard in the UK textile industry and later as an ISO Standard (ISO 1995). The formula includes rather complicated functions, SL SC and SH which are used to adjust L ∗ Cab ∗ and Hab ∗ respectively to obtain better agreement with visual judgments. Details are given in Note 4.3. Subsequent work indicated that similar performance could be obtained with much simpler versions of SL SC and SH and, in 1994, the CIE recommended the trial of a formula, known as CIE94. The formula is as follows: E94 ∗ = L ∗ /kL SL 2 + Cab ∗ /kC SC 2 + Hab ∗ /kH SH 2 1/2 where SL = 1 SC = 1 + 0045Cab ∗ and SH = 1 + 0015Cab ∗

(4.37)

The functions SL SC and SH are weighting functions that, in effect, set different tolerances for L ∗ Cab ∗ and Hab ∗ as a function of position in color space. The factors kL , kC and kH are corrections, known as parametric factors, that allow for different sample and viewing parameters. In the textile industry, for example, it is common practice to set kL =2 and kC = kH =1. Meanwhile, a formula known as BFD (l:c) had been developed (Luo and Rigg 1987) that, although similar to the CMC (l c) formula in many ways, contained an additional cross-product term, involving both Cab ∗ and Hab ∗ . This was found to be necessary because in the blue region, the axes of color difference ellipses in the a ∗ b ∗ diagram are not aligned with the Cab ∗ and Hab ∗ directions, as is required by a formula without this extra term. The CIE continued its work and, in 2000, proposed a new formula, known as CIEDE2000 (CIE 2001, Cui et al, 2001). This built on the ideas of CMC (l:c), BFD (l:c) and CIE94 and used additional new data. The formula retains the basic structure of CIE94, but has several new features, including a cross-product term as in BFD (l:c).

144


It is expected that this formula will be adopted as a CIE standard. Details are as follows: E00 = L /kL SL 2 + C /kC SC 2 +H /kH SH 2 + RT C /kC SC H /kH SH 1/2 where 0015L − 502 SL = 1 + 20 + L − 502 SC = 1 + 0045C SH = 1 + 0015C T T = 1 − 017 cosh − 30 + 024 cos2h +032 cos3h + 6 − 020 cos4h − 63 RT = − sin2 RC = 30 exp−h − 275 /25 2 7 C RC = 2 7 C + 257 L = L∗ a = 1 + Ga ∗ b = b∗ C = a 2 + b 2 1/2 h = tan−1 b /a   ∗ 7 C ab  G = 05 1 − ∗ 7 Cab + 257

(4.38)

NOTE 4.1 CALCULATION OF MUNSELL VALUE V FROM LUMINOUS REFLECTANCE Y As shown in Equation 4.3, Y is given by a fifth-order polynomial Y = fV. Thus, it is not an easy task to obtain V directly from Y and, in general, V is obtained by a successive approximation technique using the Newton–Raphson method (see Section 7.7).

145

CALCULATION OF MUNSELL VALUE

That is, Vg corresponding to a given Yg is obtained by first defining a function V as V = fV − Yg

(4.39)

The required value Vg is obtained when V = 0. Thus, if an initial value (first guess) of the value of Vg is V0 , a closer approximation V1 is obtained as V1 = V0 − V0 / V0

(4.40)

where V is the first derivative of V, The same procedure is then repeated except for substituting the first approximation V1 into Equation 4.40 as a new initial value. By repeating the procedure several times, Vg can be obtained with any desired precision. In general, the value converges within a precision of 0.001 by repeating the above procedure three or four times. An approximation for obtaining V directly has been proposed by McCamy (1992b) as follows: 1. When Y 0.9 V = UY W

(4.41)

2. When Y > 0.9 V = AY 1/3 − B − C/DY − E2 + F + G/Y H + J sinKY 1/3 + 1 +M/Y sin NY − 2 − P/QY sin SY − T

(4.42)

where A = 249268

G = 00133

P = 00037

B = 15614

H = 23

Q = 044

C = 0985

J = 00084

S = 128

D = 01073

K = 41

T = 053

E = 3084

M = 00221

U = 087445

F = 754

N = 039

W = 09967

(4.43)

The trigonometric angle functions above are in units of degree. The maximum error for the approximation of V is reported to be 0.0035.

146


NOTE 4.2 MODIFIED CIELAB AND CIELUV EQUATIONS FOR DARK COLORS CIE 1976 lightness L ∗ is defined by L ∗ = 116p 1/3 − 16

(4.44)

where p = Y/Yn . However, when p < (16/116)3 ( 0.0026241), it can be seen that this equation is not realistic because L ∗ becomes negative. Thus, as shown in Figure 4.26, a tangent with a gradient of m is drawn from the origin so that it contacts the curve of Equation 4.44 (Pauli 1976). This line is defined by L ∗ = mp

(4.45)

CIE 1976 lightness for dark colors is then obtained in accordance with Equation 4.45. From Equations 4.44 and 4.45, the following can be derived: dL ∗ /dp = 116/3p −2/3

(4.46)

dL ∗ /dp = m

(4.47)

10 8

p = 0.008856

6 4

L* = mp

L*

L* = 116p1/3 – 16

2 0

p = 0.0026241

–2 –4 0

0.004 0.008 p = Y/Yn

0.012

Figure 4.26 Derivation of modified equation for L ∗

147

OTHER COLOR DIFFERENCE FORMULAS

Thus, by taking the tangential contact point to be at coordinate pt , the following equations can be derived: 116pt1/3 − 16 = mpt

(4.48)

116/3pt−2/3

(4.49)

=m

From these relations, it follows that pt = 24/1163 0008856 m = 116/3/116/242 9033 Thus, for pt 0008856 L ∗ = 9033p

(4.50)

A similar modified equation is used in the rare cases where X/Xn or Z/Zn is less than 0.008856, as shown in Equation 4.15.

NOTE 4.3 OTHER COLOR DIFFERENCE FORMULAS In addition to the CIELAB and CIELUV color difference formulas, many other formulas have been developed and used over the years. For example, the Japanese Industrial Standard (JIS) includes the following three alternative color difference formulas as references. 1. Adams–Nickerson color difference formula. This formula utilizes another color space that is approximately uniform in perception. Adams proposed this formula in 1942 based on a coordinate system derived from the Munsell value function. Later, Nickerson (Figure 4.27) made improvements to the original function. Accordingly, the formula is known as the Adams–Nickerson color difference formula. The color difference EAN is computed as follows: EAN = 40023VY 2 + VX − VY 2 + 04VZ − VY 2 1/2 (4.51) where VY is the difference in lightness coordinates between two surface colors, and VX − VY VZ − VY are the differences in color coordinates. The values for VX VY , and VZ are obtained by substituting the relative tristimulus values Xc Yc , and Zc into the Munsell value function. Because standard illuminant C is used Xc = 101998 X

Yc = Y

Zc = 084672 Z

(4.52)

148


Figure 4.27 Dorothy Nickerson (1900–1985)

Thus, the following equations can be obtained for the tristimulus values X Y , and Z . 101998 X =12219VX − 023111VX 2 + 023951VX 3 − 0021009VX 4 + 00008404VX 5 Y =12219VY − 023111VY 2 + 023951VY 3 − 0021009VY 4 + 00008404VY 5

(4.53)

084672 Z =12219VZ − 023111VZ 2 + 023951VZ 3 − 0021009VZ 4 + 00008404VZ 5 Originally, these equations were obtained for 100X/XMgO , 100Y/YMgO , and 100Z/ZMgO where XMgO , YMgO , ZMgO are the tristimulus values of a surface of smoked magnesium oxide, the old reference standard, which is now replaced by a perfect reflecting diffuser. However, the values of X Y and Z can be used directly. Because the coefficient in the right-hand side of Equation 4.51 is 40, the formula is also known as the ANLAB40 color difference formula. By multiplying EAN by 1.1, one can obtain a value corresponding approximately to color difference in CIELAB color space. This is because the cube-root function used in CIELAB is quite close in shape to the Munsell value function. 2. Hunter color difference formula. This color difference formula was proposed by Hunter (Figure 4.28) in 1948. It is convenient when values are directly read on a photoelectric colorimeter, and

OTHER COLOR DIFFERENCE FORMULAS

149

Figure 4.28 Richard Sewall Hunter (1909–1991)

utilizes another color space that is approximately uniform in perception. The color difference EH is calculated according to EH = L2 + a2 + b2 1/2

(4.54)

where L is the difference in lightness L between two surface colors, and a b are differences in color coordinates a b. The values of L a and b for standard illuminant C are obtained by L = 10 Y 1/2 a = 175102 X − Y/Y 1/2

(4.55)

b = 70Y − 0847 Z/Y 1/2 where X Y and Z are the tristimulus values in the XYZ colorimetric system. 3. CMC (l:c) color difference formula. This color difference formula, which was adopted as a British Standard (BSI 1988) and later as an International Standard (ISO 1995), is widely used in the textile industry. The color difference ECMC is calculated according to the following equation: ECMC = L ∗ /lSL 2 + Cab ∗ /cSC 2 + Hab ∗ /SH 2 1/2 (4.56) where l and c are coefficients that depend on the type of object. For most dyed products, the preferred values are l = 2 and c = 1. SL SC , and SH are correction factors for lightness difference,

150


chroma difference, and hue difference, respectively, which are obtained by SL = 0511

for L ∗ 16

SL = 0040975L ∗ /1 + 001765L ∗

for L ∗ > 16

SC = 00638Cab ∗ /1 + 00131Cab ∗ + 0638 SH = FT + 1 − FSC

(4.57)

F = Cab ∗ 4 /Cab ∗ 4 + 1900 1/2 T = 056 + 02 coshab + 168 T = 036 + 04 coshab + 35

for 164 < hab < 345 for hab 164 or hab 345

where L ∗ Cab ∗ , and hab are the lightness, chroma, and hue angle of the reference specimen in CIELAB color space. If there is no reference specimen, the average of all the specimens is taken for these values.

NOTE 4.4 DIRECT CALCULATION OF HUE DIFFERENCE H* The hue difference in CIELAB and CIELUV was originally determined indirectly from the color difference E by H ∗ 2 = E ∗ 2 − L ∗ 2 − C ∗ 2

(4.58)

but it can also be obtained directly (Seve 1991). For example, in the case of CIELAB, the hue difference H ∗ between the colorimetric values (L1 ∗ a1 ∗ b1 ∗ ) and (L2 ∗ a2 ∗ b2 ∗ ) can be obtained as H ∗ 2 = E ∗ 2 − L ∗ 2 − C ∗ 2 = L ∗ 2 + a ∗ 2 + b ∗ 2 − L ∗ 2 − C ∗ 2 = a ∗ 2 + b ∗ 2 − C ∗ 2

(4.59)

The subscript ab is omitted in this and the following equations for clarity. In view of the following relations a ∗ = a1 ∗ − a2 ∗ b ∗ = b1 ∗ − b2 ∗ C1 ∗ = a1 ∗ 2 + b1 ∗ 2 1/2 C2 ∗ = a2 ∗ 2 + b2 ∗ 2 1/2 C ∗ = C1 ∗ − C2 ∗

(4.60)

DIRECT CALCULATION OF HUE DIFFERENCE

151

the following can be obtained H ∗ 2 = a1 ∗ − a2 ∗ 2 + b1 ∗ − b2 ∗ 2 − a1 ∗ 2 + b1 ∗ 2 1/2 − a2 ∗ 2 + b2 ∗ 2 1/2 2 = 2a1 ∗ 2 + b1 ∗ 2 1/2 a2 ∗ 2 + b2 ∗ 2 1/2 − 2a1 ∗ a2 ∗ − 2b1 ∗ b2 ∗ (4.61) By introducing hue angles h1 and h2 , sin h1 = b1 ∗ /a1 ∗ 2 + b1 ∗ 2 1/2 = b1 ∗ /C1 ∗ sin h2 = b2 ∗ /C2 ∗ cos h1 = a1 ∗ /a1 ∗ 2 + b1 ∗ 2 1/2 =

(4.62)

a1∗ /C1∗

cos h2 = a2∗ /C2∗ the following can be obtained H ∗ 2 = 2C1 ∗ C2 ∗ − 2C1 ∗ C2 ∗ cos h1 cos h2 − 2C1 ∗ · C2 ∗ sin h1 · sin h2 = 2C1 ∗ C2 ∗ 1 − cos h1 · cos h2 − sin h1 · sin h2 = 2C1 ∗ C2 ∗ 1 − cos h

(4.63)

where h = h1 − h2

(4.64)

Since cos h = 1 − 2 sin 2 (h/2), H ∗ 2 = 4C1 ∗ C2 ∗ sin2 h/2

(4.65)

H ∗ = 2C1 ∗ C2 ∗ 1/2 sinh/2

(4.66)

and

If h is sufficiently small and C1 ∗ is nearly equal to C2 ∗ , they can be written as C ∗ to obtain the simple relation H ∗ = C ∗ h/180

(4.67)

5 Measurement and Calculation of Colorimetric Values The measurement of color according to the CIE colorimetric system is known as color measurement. The instruments used are called colorimeters, spectrophotometers and spectroradiometers. The values obtained (tristimulus values, chromaticity coordinates, etc.) are known as colorimetric values. The methods used can be classified into either direct measurement of tristimulus values or spectral measurement followed by calculation. A third class, which involves visual color matching in a visual colorimeter, is rarely used these days because the results depend considerably on the observers, and because the low visual field luminance leads to large uncertainties.

5.1 DIRECT MEASUREMENT OF TRISTIMULUS VALUES Direct measurement of tristimulus values comprises reading the colorimetric values directly from the output of photoelectric photoreceptors that satisfy a condition known as the Luther condition.


154

MEASUREMENT AND CALCULATION

This refers to a situation in which the product of the spectral responsivity of photoreceptor and the spectral transmittance of a correction filter is proportional to one of the CIE color matching functions or to a linear combination of them. An instrument with three such channels (one for each color matching function) is known as a photoelectric colorimeter. The Luther condition can be realized by either of the following two methods.

5.1.1 Template Method In this method the Luther condition is realized by using a template placed in a spectrum. Figure 5.1 shows an optical system for applying this method to the colorimetry of a reflecting object. The radiation reflected by the test object passes through a prism and is dispersed into a spectrum. Three templates, an X template, a Y template, and a Z template, are placed one at a time in the spectrum. The X template is designed so that the spectral response of the entire system is proportional to the x color matching function. Thus, the X tristimulus value of the object can be measured directly once the system is calibrated using a standard of known tristimulus value. Similarly, the Y and Z tristimulus values can be measured by using the Y and Z templates respectively. Despite their theoretical simplicity, template colorimeters are not widely used in practice because they are complicated and expensive to construct.

5.1.2 Optical Filter Method Figure 5.2 illustrates the realization of the Luther condition by using a combination of optical filters in place of a prism and templates. Lens

Light source

Lens

Photoreceptor

Spectrum

Prism

Lens

Template

Slit

X, Y, Z output

Lens Test object

X template

Y template

Z template

Figure 5.1 Schematic diagram of a template photoelectric colorimeter (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

155

DIRECT MEASUREMENT OF TRISTIMULUS VALUES

Test object Light source

Photoreceptor (X )

(Y )

X, Y, Z output

Combination of filters

(Z ) Tristimulus filters

(a) Serial arrangement

(b) Parallel arrangement

Figure 5.2 Schematic diagram of a filter photoelectric colorimeter (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

The optical filters can be combined either (a) in a serial arrangement, or (b) in a parallel arrangement. The degree of approximation of the color matching functions that can be achieved by each of the arrangements is shown in Figure 5.3, and it can be seen that a higher accuracy can be obtained with the parallel arrangement. Filter colorimeters are widely used because of their simple structure and because they are relatively inexpensive. Often, an additional calculation function is added to convert the tristimulus values into other colorimetric values such as L ∗ a ∗ b ∗ u ∗ and v ∗ . 2.0 z (λ) Spectral response

1.5 y (λ)

1.0

x (λ) 0.5

0

x (λ)

400

500 600 Wavelength (nm) (a) Serial arrangement

700

Figure 5.3 CIE 1931 color matching functions (solid line) and spectral responses of filter photoelectric colorimeter (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

156


z (λ)

Spectral response

1.6

1.2 y (λ)

x (λ)

0.8 x (λ)

0.4 0

400

500 600 Wavelength (nm) (b) Parallel arrangement

700

Figure 5.3 (Continued)

5.2 SPECTRAL COLORIMETRY Spectral colorimetry is a method by which colorimetric values are obtained by calculation from the measured spectral distribution of a source or the measured spectral reflectance (or transmittance) of an object. The measurements are made by a spectroradiometer (for sources) or a spectrophotometer (for reflecting or transmitting objects). A spectroradiometer or spectrophotometer uses a monochromator to disperse polychromatic radiation (‘white light’) into monochromatic components and a photoreceptor to measure the amount of radiation at each wavelength. Figure 5.4 shows the principle of a monochromator using a prism as the dispersing element. The incident radiation passes through an entrance slit and is converted into a parallel beam. A prism then disperses the beam to produce a spectrum of radiation of different wavelengths. A lens focuses the dispersed radiation onto an exit slit. Thus, monochromatic radiation of any desired wavelength can be extracted by changing the position of the exit slit. This type of monochromator is referred to as a single monochromator because it uses a single dispersing system. A diffraction grating is frequently used as the dispersing element instead of a prism. A diffraction grating comprises numerous equi-distant parallel lines ruled on a plane surface. A spectrum is produced by interference between the beams of radiation transmitted by the numerous slits so formed. An example of a practical spectrophotometer using a diffraction grating is shown

157

SPECTRAL COLORIMETRY

Aperture

Lens Lens

Exit slit

Prism

Entrance slit

Figure 5.4 Principle of a prism monochromator (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

in Figure 5.5. The actual construction is more complicated than the one shown, but the principle is the same. The radiation emitted from the source passes through an optical system comprising a slit, a collimating mirror, and a diffraction grating. It then passes through a second, similar system and is directed by a rotary sector mirror alternately onto a reference object and the test object, which are placed over apertures in an integrating sphere (also called Ulbricht sphere), a sphere whose inner surface is coated with a diffusely reflecting white paint having little or no wavelength selectivity. At least two windows, known as ports, are provided in the sphere. Light enters through one port (the entrance port), is repeatedly diffused

Light source Lens

Slit Diffraction grating

Diffraction grating

Rotating sector mirror

Light trap

Collimating mirror

Photoreceptor

Integrating sphere

Test object (transmittance)

Collimating mirror Reference object Test object (reflectance) (reflectance)

Figure 5.5 Optical system of a practical spectrophotometer

158


and reflected inside the sphere, and is emitted from the sphere through the exit port. The reflected radiation is measured by a photoreceptor that is directed at a portion of the integrating sphere wall. A spectrophotometer such as this, with two dispersing elements, is called a double monochromator spectrophotometer. Use of a double monochromator reduces the stray radiation (radiation of unwanted wavelengths). With a single monochromator, especially one using a diffraction grating, stray radiation filters must be inserted in the beam at appropriate times. For example a yellow filter avoids the false detection of blue or purple wavelengths when the monochromator is set to ‘yellow’, but must be removed when the monochromator is set to ‘blue’ or ‘purple’ The source and the photoreceptor positions in Figure 5.5 can be interchanged. By putting the source at the position of the photoreceptor, the test object is irradiated by polychromatic radiation. This is necessary for the measurement of fluorescent materials (see Note 5.1). To exclude the specular component of the reflected flux, a trap, such as a black coating, can be placed at the appropriate part of integrating sphere wall. This is discussed in more detail in the Section 5.3.

5.3 GEOMETRICAL CONDITIONS FOR MEASUREMENT When one observes a colored paper or other object that has some gloss, the color of the object changes, depending on the viewing angle. This indicates that the reflection from a glossy object depends greatly on the angles at which the light is incident and viewed. Although usually less dramatically, the transmission from an object can also vary with incident and viewing angles. To ease comparisons when reflection or transmission is measured, the CIE has defined some standard terms and geometrical conditions. The geometrical conditions are denoted by symbols such as 450 or d:8 . The numbers before and after the colon indicate the angles of incidence and viewing, respectively, measured from the normal to the surface of the object being measured. The symbol d indicates the use of an integrating sphere. Table 5.1 shows that different terms are used according to the solid angle of the measured beam. If the solid angle approaches zero, the measured quantity is called radiance factor; if the solid angle approaches 2 sr corresponding to the entire hemisphere,

159

GEOMETRICAL CONDITIONS Table 5.1 Classification of reflectance Solid angle of viewing system

Radiometric quantity

Photometric quantity

Infinitesimal Finite but limited

Radiance factor Reflectance factor

2 sr

Reflectance

Luminance factor Luminous reflectance factor Luminous reflectance

Test object

Test object

Perfect reflecting diffuser (a) Radiance factor

Photoreceptor

Photoreceptor

Optical fiber

Photoreceptor

Optical fiber

Photoreceptor

Photoreceptor

When the values are considered as a function of wavelength, a prefix ‘spectral’ is added, for example, ‘spectral reflectance’.

Perfect reflecting diffuser (b) Reflectance factor

Integrating sphere

Photoreceptor

Perfect reflecting diffuser

Test object (c) Reflectance

Figure 5.6 Method for measuring radiance factor, reflectance factor, and reflectance (Kawakami 1986). Reproduced by permission of Asuara Shoten

the measured quantity is called reflectance; and, for all other conditions, the measured quantity is called reflectance factor. Photometric quantities are defined similarly. Figure 5.6 illustrates these geometrical conditions. The curves on the test object and the perfect reflecting diffuser indicate the intensity of the reflected radiant flux as a function of viewing angle. In this book, radiance factor,

160


reflectance factor, and reflectance are collectively referred to as reflectance unless it is particularly necessary to distinguish one from the others. The geometrical conditions described above in a general way have been defined precisely by the CIE (CIE 2004a). Some examples are shown in Figure 5.7. The reflectance factor is measured under any of the conditions (1), (2), and (3), while reflectance is measured under condition (4). The CIE defines an ideal perfect reflecting diffuser with a reflectance of 1.0 as the reference standard for the measurement of radiance factor, reflectance and reflectance factor (see Note 5.2). When the test object has a specular reflection component (i.e., it is glossy), a black optical trap may be placed in the integrating sphere in conditions (3) and (4), to exclude the specular component. In this case, if it is desired to include the specular component, a white cap with the same reflecting properties as the integrating sphere must replace the trap. The quantity measured with the trap in place is called diffuse reflectance whereas, with the white cap in place, the quantity measured is called total reflectance. The geometries are distinguished by adding the letter e (for excluded) or i (for included) after the symbol d. To obtain proper integration within an integrating sphere, a baffle must be included to block radiation that would pass directly

Photoreceptor

Photoreceptor

Test object (1) (45°: 0°)

Test object (2) (0°: 45°)

Photoreceptor Sphere cap (white or black) Integrating sphere Integrating sphere

Sphere cap (white or black)

Light baffle

Photoreceptor Light baffle Test object (3) (d: 0°)

Test object (4) (0°: d)

Figure 5.7 Standard CIE geometrical conditions for measuring a reflecting object (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

161

CALCULATION OF COLORIMETRIC VALUES

Photoreceptor Integrating sphere

Integrating sphere

Photoreceptor

Test object

Test object

Test object

Photoreceptor (1) (0°: 0°)

Baffle

Black trap (2) (0°: d)

Baffle (3) (d: d)

Integrating sphere

Figure 5.8 Geometrical conditions for measuring a transmitting object (Baba 1987). Reproduced by permission of Asuara Shoten

from the irradiated point on the sphere wall to the test object, or from the test object to the point on the sphere wall viewed by the photoreceptor, as is shown in Figure 5.7. For 45 geometries, the CIE distinguishes between an annular arrangement (including all azimuthal directions) and a directional arrangement (at one azimuth angle only) by adding the letters a or x, respectively. The current CIE recommendations for reflection measurements are di:8 , de:8 , 8 :di, 8 :de, d:d, d:0 , 45 a:0 , 0 :45 a, 45 x:0 and 0 :45 x with tolerances being specified for each of the angles (CIE 2004a). It is recognized by the CIE that some of the definitions need to be more precise so it is likely that future CIE publications will introduce small changes in the tolerances although the basic angles will probably remain unchanged. Figure 5.8 shows some of the CIE recommended geometrical conditions for measuring transmission. The full list currently recommended by the CIE is 0 :0 , di:0 , de:0 , 0 :di, 0 :de and d:d (CIE, 2004a). In this case, the symbols di and de represent inclusion and exclusion, respectively, of the regular component. The total of the regular transmittance measured with 0 :0 geometry and the diffuse transmittance measured with de:0 (or 0 :de) geometry yields the total transmittance measured with di:0 (or 0 :di) geometry.

5.4 CALCULATION OF COLORIMETRIC VALUES As described in Section 3.4, the tristimulus values X Y and Z can be obtained from the measured spectral reflectance (transmittance) by using Equations 3.12 and 3.13. The calculation involves obtaining the area defined by the product of three functions, i.e., PRx PRy or PRz, as shown in Figure 5.9.

162


P

R

λ Spectral distribution P(λ) of irradiating illumination

λ Spectral reflectance R(λ) of object

P (λ) × R (λ)

PR

λ

x

y

λ Color matching function x(λ)

z

λ Color matching function y(λ)

PRy

PRx

λ

Color matching function z(λ)

PRz

X λ

Y λ

Z λ

Figure 5.9 Computation procedure of tristimulus values X Y and Z (Billmeyer and Saltzman 1981). Reproduced by permission of Wiley

In direct colorimetry (Section 5.1) the area is obtained by analog integration, whereas in spectral colorimetry (Section 5.2), the integration is approximated by summation. As an example of spectral colorimetry we take the measurement of the red color of the national flag of Japan (the flag of the Rising Sun). The dimensional specifications of the flag are well defined. The ratio of the height to the width must be 2:3. The red sun must be located with its center at the center of the flag and its diameter must be 3/5 of the height of the flag (Figure 5.10). The color is not so rigorously defined, being described merely as red on a white background. Let us consider two versions of the flag that are

163


3a Red color 3 b 5

b = 2a

White background

Figure 5.10 Flag of the Rising Sun

readily available. We take P to be the CIE standard illuminant D65 and use the XYZ (two-degree) colorimetric system. We approximate Equations 3.12 and 3.13 by summation over a wavelength range of 380–780 nm with the interval of wavelength set at 5 nm. These selections of wavelength range and interval are generally sufficient when the functions P and R are smooth, as they are here. However, if the functions P and R change rapidly with wavelength or, if a higher accuracy is required, a broader wavelength range and/or a narrower wavelength interval must be selected.

5.4.1 Tristimulus Values and Chromaticity Coordinates Let the spectral reflectances of the red color of the two flags be R1 and R2 , respectively. The results obtained by measuring R1 and R2 under the 0 :d geometrical condition are given in Figure 5.11 and Table 5.2. As stated above, the tristimulus values can be computed by summation. Because the wavelength interval for the summation is common for both numerator and denominator, it can be omitted from the computation. Thus Equations 3.12 and 3.13 can be replaced by X =k

n

RPx

i =1

Y =k

n

RPy

(5.1)

i =1

Z =k

n

RPz

i =1

k = 100/

n i =1

Py

(5.2)

164


100

Reflectance

75

50

2

1 25

0

400


700

Figure 5.11 Spectral reflectance for flag 1 (solid line) and flag 2 (broken line)

Table 5.2 Spectral reflectance of flags 1 and 2 Wave length (nm) 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510

Reflectance flag 1

flag 2

Wave length (nm)

00895 00833 00768 00684 00594 00517 00441 00377 00323 00278 00243 00215 00193 00178 00165 00155 00148 00141 00137 00135 00131 00130 00129 00128 00128 00128 00129

00500 00446 00401 00357 00320 00293 00268 00250 00236 00222 00213 00205 00199 00194 00190 00187 00185 00183 00182 00182 00181 00180 00180 00181 00182 00183 00185

515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645

Reflectance flag 1 00129 00129 00130 00133 00138 00144 00153 00163 00180 00208 00251 00330 00443 00637 00904 01326 01834 02461 03255 04093 04822 05449 05934 06273 06495 06637 06717

Reflectance

flag 2

Wave length (nm)

flag 1

flag 2

00189 00194 00200 00208 00214 00219 00223 00226 00235 00251 00276 00323 00391 00506 00665 00923 01256 01700 02324 03075 03827 04580 05260 05819 06240 06529 06705

650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730 735 740 745 750 755 760 765 770 775 780

06757 06770 06753 06713 06648 06601 06559 06526 06518 06509 06480 06473 06474 06464 06459 06442 06428 06422 06428 06432 06412 06394 06390 06396 06401 06404 06409

06811 06867 06875 06854 06802 06745 06690 06643 06621 06601 06559 06533 06521 06516 06512 06491 06474 06466 06461 06472 06473 06461 06466 06473 06486 06488 06493


165

Using the data from Table 5.2 and Appendices 1 and 3, the X tristimulus value of flag 1 can be calculated as X1 = k R1 380P380x 380 + R1 385P385x 385 +

+ R1 780P780x 780 = k 00895 × 4998 × 00014 + 00833 × 6338 × 0022 +

+ 06409 × 6338 × 0001 = k × 43893 Similarly Y1 = k × 22654 Z1 = k × 4163 k = 100/211328 X1 , Y1 , Z1 , x1 , y1 , u1 and v1 can then be obtained as X1 = 100 × 43893/211328 = 2077 Y1 = 100 × 22654/211328 = 1072 Z1 = 100 × 4163/211328 = 197 x1 = X1 /X1 + Y1 + Z1 = 2077/2077 + 1072 + 197 = 06207 y1 = Y1 /X1 + Y1 + Z1 = 1072/2077 + 1072 + 197 = 03204 u1 = 4X1 /X1 + 15Y1 + 3Z1 = 4 × 2077/2077 + 15 × 1072 + 3 × 197 = 04431 v1

= 9Y1 /X1 + 15Y1 + 3Z1 = 9 × 1072/2077 + 15 × 1072 + 3 × 197 = 05146

166


520

0.8 540

560

0.6 500 580

y 0.4

D65

2 Red color

0.2

600

Vermilion

1

620 650 770 nm

480 470

0

450 380

0.2

0.4 x

0.6

0.8

Figure 5.12 Chromaticity coordinates of flags 1 and 2

Similarly, the values for flag 2 are X2 = 1802 Y2 = 950 Z2 = 213 x2 = 06078 y2 = 03204 u2 = 04318 v2 = 05123 Figure 5.12 shows the coordinates of the two flags in the xy chromaticity diagram. It can be seen that both are located in the vermilion region, close to the red region, but that there is a small difference in chromaticity between the two colors.

5.4.2 Dominant Wavelength and Excitation Purity The dominant wavelength and excitation purity of each color can be read from Figure 5.12 by drawing a straight line from the point for the standard illuminant D65 (0.313, 0.324) through the point representing the color. For flag 1, the intersection of the straight line with

COLORIMETRIC VALUES IN CIELAB AND CIELUV

167

the spectrum locus is at x = 0681. From this, dominant wavelength d1 and excitation purity pe1 for flag 1 can be obtained as d1 = 615 nm pe1 = 100 × 0621 − 0313/0681 − 0313 = 837 Similarly, the dominant wavelength d2 and the excitation purity pe2 for the flag 2 are obtained as d2 = 615 nm pe2 = 100 × 0608 − 0313/0681 − 0313 = 802 It can be seen that the dominant wavelengths of the two flags are the same, but that the excitation purity (correlating with saturation) is higher for flag 1 than for flag 2.

5.5 COLORIMETRIC VALUES IN CIELAB AND CIELUV UNIFORM COLOR SPACES The tristimulus values Xn , Yn and Zn and the chromaticity coordinates un and vn of standard illuminant D65 are Xn = 9504

Yn = 10000

un = 01978

vn = 04683

Zn = 10889

Thus, the colorimetric values in CIELAB and CIELUV can be obtained as follows.

5.5.1 Colorimetric Values and Color Difference The colorimetric values L1∗ a1∗ b1∗ u1∗ and v1∗ for flag 1 are L1 ∗ = 116Y1 /Yn 1/3 − 16 = 1161072/10001/3 − 16 = 3910 ∗

a1 = 500 X1 /Xn 1/3 − Y1 /Yn 1/3 = 500 2077/95041/3 − 1072/10001/3 = 6365

168


b1 ∗ = 200 Y1 /Yn 1/3 − Z1 /Zn 1/3 = 200 1072/10001/3 − 197/108891/3 = 4250 u1 = 13L1 ∗ u1 − un ∗

= 13 × 391004431 − 01978 = 12469 v1 ∗ = 13L1 ∗ v1 − vn = 13 × 391005146 − 04683 = 2353 Similarly, the colorimetric values L2 ∗ a2 ∗ b2 ∗ u2 ∗ and v2 ∗ for flag 2 are L2 ∗ = 116Y2 /Yn 1/3 − 16 = 3693 ∗

a2 = 500 X2 /Xn 1/3 − Y2 /Yn 1/3 = 5910 ∗

b2 = 200 Y2 /Yn 1/3 − Z2 /Zn 1/3 = 3737 ∗

u2 = 13L2 ∗ u2 − un = 11234 v2 = 13L2 ∗ v2 − vn ∗

= 2112 Thus, the color differences Eab ∗ and Euv ∗ between flag 1 and flag 2 are Eab ∗ = L1 ∗ − L2 ∗ 2 + a1 ∗ − a2 ∗ 2 + b1 ∗ − b2 ∗ 2 1/2 = 3910 − 36932 + 6365 − 59102 + 4250 − 37372 1/2 = 719 Euv ∗ = L1 ∗ − L2 ∗ 2 + u1 ∗ − u2 ∗ 2 + v1 ∗ − v2∗ 2 1/2 = 3910 − 36932 + 12469 − 112342 + 2353 − 21122 1/2 = 1277


169

These color differences are relatively high values, considering that a color difference E ∼ 1.0 is usually discriminable. It can be deduced therefore that the colors of the two flags differ significantly from another, and that this could lead to a problem if the simple description ‘flag of the Rising Sun’ were to be specified for a display of multiple flags. To prevent such nonuniformity, color measurement and color science can be used effectively. For example, in case of the national flag of the United States of America (the StarSpangled Banner), colors are specified by using color chip numbers in a color appearance system. In the case of the national flag of Algeria, which has a green area on the left, a white area on the right, and the Islamic symbol (crescent and star) on a central red area, the green and red colors are specified quantitatively by dominant wavelengths (541.1 and 656.2 nm). Similarly, the red color of the Canadian flag is specified by CIE coordinates (x = 0576 y = 0315 Y = 95) calculated with respect to illuminant D65 and the 1931 Standard Observer. As discussed in Section 4.6, color differences in CIELUV color space are different from those in CIELAB. It can be seen from Figure 4.25 that sometimes, in the red region, Euv ∗ is larger than Eab ∗ by about two times or more. In fact, in the above example of the Japanese flags, Euv ∗ /Eab ∗ = 1277/719 = 178.

5.5.2 Correlates of Visual Attributes As described in Section 4.4, correlates of various visual attributes of color can be derived in the CIELAB and CIELUV systems. For lightness, L ∗ is used directly, but other correlates require some further calculation. Continuing the example of the two versions of the Japanese flag, the chroma correlates Cab1 ∗ Cuv1 ∗ Cab2 ∗ and Cuv2 ∗ are obtained from Equations 4.22 as Cab1 ∗ = a1 ∗ 2 + b1 ∗ 2 1/2 = 63652 + 42502 1/2 = 7653 ∗

Cuv1 = u1 ∗ 2 + v1 ∗ 2 1/2 = 124692 + 23532 1/2 = 12689 Cab2 ∗ = a2 ∗ 2 + b2 ∗ 2 1/2 = 59102 + 37372 1/2 = 6992

170


Cuv2 ∗ = u2 ∗ 2 + v2 ∗ 2 1/2 = 112342 + 21122 1/2 = 11431 The chroma differences between the two flags are thus Cab ∗ = 661 Cuv ∗ = 1258 Similarly, the hue angles of the flags are obtained from Equations 4.24 as hab1 = tan−1 4250/6365 = 3373 huv1 = tan−1 2353/12469 = 1069 hab2 = tan−1 3737/5910 = 3231 huv2 = tan−1 2112/11234 = 1065 and the differences in hue angle are hab = 142 huv = 004 From Equations 4.25, the hue differences between the two flags are Hab ∗ = 27653 × 69921/2 sin142/2 = 181 Huv ∗ = 212689 × 114311/2 sin004/2 = 008 Alternatively, the hue differences can be calculated from Equations 4.26: Hab ∗ = Eab ∗ 2 − L ∗ 2 − Cab ∗ 2 1/2 = 7192 − 3910 − 36932 − 7653 − 69922 1/2 = 182


171

Huv ∗ = Euv ∗ 2 − L ∗ 2 − Cuv ∗ 2 1/2 = 12772 − 3910 − 36932 − 12689 − 114312 1/2 = 033 The results are the same, within rounding errors. A correlate of saturation is defined only in the CIELUV color space. From Equation 4.23, the values for the two flags are suv1 = 13 u1 − un 2 + v1 − vn 2 1/2 = 13 04431 − 019782 + 05146 − 046832 1/2 = 32452 suv2 = 13 u2 − un 2 + v2 − vn 2 1/2 = 13 04318 − 019782 + 05123 − 046832 1/2 = 30953 CIELUV saturation can also be obtained from L ∗ and Cuv ∗ as suv1 = Cuv1 ∗ /L1 ∗ = 12689/3910 = 32453 suv2 = Cuv2 ∗ /L2 ∗ = 11431/3693 = 30953 Again, the results are the same, apart from rounding errors. These calculations enable us to say that flag 1 has a higher chroma (and saturation) than flag 2, but that it has almost the same hue. Values of L ∗ , C ∗ , h and s are easier to correlate with the visual appearance of a color than are tristimulus values and chromaticity coordinates. The correlation with visual appearance is better than that of simpler derived quantities such as purity and dominant wavelength, Furthermore, the differences, L ∗ C ∗ and H ∗ correlate approximately with perceived differences of lightness, chroma and hue, respectively. Despite the usefulness of the correlations, they are not perfect and efforts continue in CIE Committees and elsewhere to find better ones. This is discussed in Section 6.8.

172


NOTE 5.1 SPECTRAL COLORIMETRY OF FLUORESCENT MATERIALS Fluorescent materials are widely used in fluorescent brighteners, traffic signs, advertisements, etc., because of their strong colors. A fluorescent material absorbs incident radiation, and, within a small fraction of a second, when the excited fluorescent molecule recovers its steady state, emits radiation of a longer wavelength. Thus, an erroneous result is obtained if spectral colorimetry is performed using an optical system with the monochromator in the incident beam. For example, if a fluorescent material having a bright green color is measured by using the optical system shown in Figure 5.13(b), a spectral radiance factor shown by the broken line in Figure 5.14 is obtained, and is completely different from visual evaluation. This is because, when radiation in the excitation wavelength range, for example 400 nm, is incident on the test object, a fluorescent component of higher wavelength enters the photoreceptor along with the reflected 400 nm radiation and increases the apparent reflectance. Accordingly, for spectral colorimetry of a fluorescent material, an optical system such as that shown in Figure 5.13(a) must be employed. In this system, the test object is irradiated with polychromatic radiation, and the reflected and fluorescent radiation is dispersed and measured. A spectral radiance factor as shown by the solid line in Figure 5.14 is obtained, which is in agreement with visual evaluation. However, the fluorescent component in the spectral radiance factor thus measured is influenced by the relative spectral distribution of the irradiation, so a defined standard source must be used in the spectrophotometer. This is different from the Light source

Photoreceptor Light Spectrometer source Photoreceptor

Spectrometer

Standard

Test object

Exchange (a) Polychromatic illumination

Standard

Test object

Exchange (b) Monochromatic illumination

Figure 5.13 Optical system for spectral colorimetry (Nayatani 1980). Reproduced by permission of Asuara Shoten

REFERENCE STANDARD FOR REFLECTION MEASUREMENTS

173

Excitation wavelength region

120

Radiance factor

(a)

80

(b) Reflected component

40 Emission wavelength region 0

300

400

500 Wavelength (nm)

600

700

Figure 5.14 Spectral radiance factor obtained by measuring conditions (a) and (b) for green-emitting fluorescent material (Grum and Bartleson 1979). Reprinted from Vol. 2 Color Measurement, F. Grum and C. J. Bartleson, Optical Radiation Measurements, Copyright (1979), with permission of Elsevier

situation with non-fluorescent materials where any source can be used in the measurement, with the standard source/illuminant being introduced only in the calculation. The reflectance of the test object free of the fluorescent component can be obtained by connecting the lower values of the two curves in Figure 5.14 (solid line up to 480 nm, dotted line for 480–510 nm, and broken line above 510 nm).

NOTE 5.2 REFERENCE STANDARD FOR REFLECTION MEASUREMENTS At one time, magnesium oxide (MgO) obtained by depositing the smoke from burning magnesium was used as the standard for measuring reflectance, reflectance factor and radiance factor. This is because such a standard had a very high reflectance, which was assumed to be unity. However, since 1969, a perfect reflecting diffuser having a true reflectance of unity has been used in place of magnesium oxide. The absolute reflectance R of MgO relative to this ideal standard is shown in Figure 5.15. Values reported before 1969 must be corrected by multiplying the values by this R. Because a perfect reflecting diffuser does not exist in practice, durable white reference standards such as those shown in

174


1.00

Absolute reflectance

0.98 0.96 0.94 0.92 0.90 0.2

0.4

0.6

0.8 1.0 1.0 Wavelength (µm)

1.5

2.0

2.5

Figure 5.15 Absolute reflectance of deposited magnesium oxide (MgO) (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

BaSO4 (pressed powder)

1.00 0.98 Absolute ref lectance

0.96

1.00 Halon

Halon MgO (freshly smoked)

0.98

BaSO4 (paint)

0.96

0.94

MgO M 14 Glass

0.92

0.92

BaSO4 (pressed powder)

0.90

0.90

Vitrolite Glass

0.88

0.88 BaSO4 (paint)

0.86

0.94

0.86

0.84

0.84

0.82

0.82

0.80 300

400

500 600 Wavelength λ (nm)

700

0.80 800 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Wavelength λ (µm)

Figure 5.16 Absolute spectral reflectance curves of different materials used as transfer or working standards of reflectance (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

Figure 5.16 must be calibrated by comparing the reflected radiation with the incident radiation directly. Such calibrations are difficult to do accurately and, consequently, are usually performed only in National Metrology Institutes.

6 Evolution of CIE Standard Colorimetric System The establishment of the CIE standard colorimetric system was described in Chapter 3. It has remained the basic system for the specification of color stimuli for over 70 years. Nevertheless, the tristimulus values and chromaticity coordinates themselves have various insufficiencies in the practical application of some industrial problems. For example, the nonuniformity of XYZ and Yxy spaces has already been pointed out in Chapter 4 where more uniform color spaces were described. Thus, to make it more widely applicable to practical problems, the CIE system has evolved beyond the principles established in 1931. In the present chapter, descriptions are given of various expansions of the system, i.e., color mixing, maximum values of luminous efficacy, optimal colors, chromatic adaptation, color appearance models and metamerism. The evolution is still in progress, and work on color vision models is an example. The methods described here are not designed for a specific problem alone, and are applicable to problems widely encountered in various fields.


176

EVOLUTION OF CIE STANDARD COLORIMETRIC SYSTEM

6.1 ADDITIVE MIXING The phenomenon of generating a new color stimulus through the interaction of multiple other color stimuli is called color mixing. Color mixing can be either additive or subtractive. Additive mixing occurs when the component color stimuli are simultaneously incident on the eye. The stimuli may be superimposed optically, they may occur in rapid temporal alternation (flicker), or they may be interlaced in a spatial pattern not visible to the eye. Because the physical intensity of the color stimulus that results from the mixture is obtained as the sum of the components, the mixture is called ‘additive’ color mixture. Let us consider two color stimuli, [R] and [G]. The tristimulus values for unit amount (for example, 1 W) of the color stimuli are given by (XR YR ZR and (XG YG ZG respectively, and the chromaticity coordinates are given by (xR yR and (xG yG respectively. The tristimulus values (XF YF ZF of the color stimulus [F] obtained by additive mixing of amounts R and G of these two color stimuli are expressed by the following equations according to Grassmann’s Laws. XF = RXR + GXG YF = RYR + GYG

(6.1)

ZF = RZR + GZG These equations can be expressed in matrix form:     XF XR XG R  YF  =  YR YG    G ZF ZR ZG 

(6.2)

Thus, the color stimulus [F] is located in the composite line of vectors [R] and [G] in XY Z space, and, in the xy chromaticity diagram, as shown in Figure 6.1, it is on the straight line connecting the points R (xR , yR and G (xG , yG . Although they do not all have the same luminance, a variety of colors can be obtained as mixtures of [R] and [G] on the straight line RG. Their position along the line depends on the amounts R and G. For example, let the tristimulus values and chromaticity coordinates of color stimuli [R] and [G] be XR = 20 XG = 2

YR = 10 YG = 10

ZR = 3 ZG = 3

xR = 06061 xG = 01333

yR = 03030 yG = 06667

(6.3)

177

ADDITIVE MIXING

520

0.8 540

[G] 560

0.6 500

[Y] y

580

[F] 0.4

600

D65

[C]

[R] 0.2

620 650 770 nm

[M] 480

0

470 450

[B] 380

0.2

0.4 x

0.6

0.8

Figure 6.1 Additive mixing of red [R], green [G], and blue [B]

By mixing 0.5 units of each (R = G = 05), the following result can be obtained from Equation 6.2.        XF 20 2 05 11  YF  =  10 10    =  3 (6.4) 3 3 05 0 ZF Thus, the chromaticity coordinates xF and yF for the color stimulus [F] are xF = XF /XF + YF + ZF = 11/11 + 10 + 3 = 04583

(6.5)

yF = YF /XF + YF + ZF = 10/11 + 10 + 3 = 04167 The chromaticity coordinates xm and ym of the midpoint of the straight line RG are xm = xR + xG /2 = 06061 + 01333/2 = 03697

(6.6)

178


ym = yR + yG /2 = 03030 + 06667/2 = 04849 It can be seen that the midpoint coordinates differ from those of Equation 6.5. This shows that, when two color stimuli are mixed in equal amounts, i.e., R = G = 05, the chromaticity of the mixture is not necessarily at the midpoint in the chromaticity diagram. Then, let us consider the case of three-stimulus mixture by adding a third stimulus [B]. For example, after producing a stimulus [Y] by mixing stimuli [G] and [R], all colors on the straight line YB can be obtained as mixtures by adding a third stimulus [B], although the luminance is not always the same. Thus, although not at constant luminance, all the points inside the triangle RGB can be obtained by changing the mixing ratio of [R], [G] and [B]. The range of the colors thus obtained is called a color gamut. The three component stimuli that used are for the additive mixing are called additive primaries, and, in order to achieve a large gamut, are most often red [R], green [G] and blue [B]. An achromatic color stimulus [W] can be obtained, for example, by properly mixing [R] and [C], with [C] having been obtained by mixing [G] and [B]. Two color stimuli (for example [R] and [C]) capable of producing a particular achromatic stimulus by mixing are called complementary colors. An infinite number of complementary pairs can be found that produce any particular achromatic stimulus [W]. This is illustrated in Figure 6.1 where the stimulus D65 is shown as a mixture of [R] and [C], [G] and [M], or [B] and [Y]. Now, let R G and B be the amounts of [R], [G] and [B] respectively in a mixture. Then, the tristimulus values (XF YF ZF of the stimulus [F] obtained by mixing them can be obtained by an extension of Equation 6.2 as 

  XF XR  YF  =  YR ZF ZR

XG YG ZG

  R XB YB   G  B ZB

(6.7)

Let the tristimulus values and the chromaticity coordinates for stimulus [B] be XB = 20

YB = 10

ZB = 70

xB = 02000

yB = 01000 (6.8)

and let the values for [R] and [G] be as given in Equations 6.3. Then, a stimulus having the same chromaticity coordinates as those of

179

ADDITIVE MIXING

standard illuminant D65 with a luminance of YD = 10 can be calculated as an additive mixture of [R], [G], and [B]. Such a stimulus will be metameric to D65 . Because D65 has chromaticity coordinates xD and yD given by xD = 03127

yD = 03290

(6.9)

XD and ZD are obtained by XD = YD xD /yD = 10 × 03127/03290 = 95046

(6.10)

ZD = YD 1 − xD − yD /yD = 101 − 03127 − 03290/03290 = 10890 From Equation 6.7, the following is obtained 

  XF 20  YF  =  10 3 ZF

    2 20 R 95046 10 10   G  =  100000  3 70 B 108906

(6.11)

By multiplying both sides of the equation by the inverse of the first matrix of the left-hand side, the following can be obtained    −1  95046 R 20 2 20  G  =  10 10 10   100000  108906 B 3 3 70    1/18 −4/603 −1/67 95046 1/9 0   100000  =  −1/18 0 −3/670 1/67 108906   02991 =  05831  01178 

(6.12)

Thus, 0.2991, 0.5831 and 0.1178 are the desired amounts of [R], [G] and [B] respectively. In the unit plane (R + G + B = 1.0), the luminance of the color obtained by additive mixing is intermediate between the luminances

180


0.8 [G]

0.6 0.6 0.7 0.8 0.9

y 0.4

0.2

[R] 0.6 0.5 0.4 0.3

0.4

[B]

0

0.2

0.4

0.6

x

Figure 6.2 Color gamut for each luminance value obtained by additive mixing (numerals indicate luminance ratio with respect to white) (MacAdam, J. SMPTE, 56, 285–512 (1951))

of the primaries. The actual luminance of any color within the RGB triangle depends on its position in the triangle or, in other words, on its chromaticity coordinates. In the example above, the three primaries all had the same Y value of 10.0, and the calculated mixture also had Y = 100. However, it is more common to normalize the units of the [R], [G] and [B] primaries so that a given white color is obtained by adding unit amounts of the three. This is the maximum luminance that can be obtained within the RGB triangle. For other positions within the triangle (i.e., for other chromaticities), the luminance is always less. The color gamut for a given luminance level changes as illustrated in Figure 6.2 (MacAdam 1951). In Figure 6.2, the luminances of the colors are expressed by the ratio with respect to the luminance of the white color.

6.2 SUBTRACTIVE MIXING In contrast to additive mixing, which is a process of mixing colored lights, a process in which a new color results from the superposition of absorbing media, such as color filters or dyes, is known

181

SUBTRACTIVE MIXING

as subtractive mixing. The intensity of the color stimulus that is obtained by this mixing decreases as if each component is subtracted. Mathematically speaking, the intensity does not decrease in a subtractive manner, but in a multiplicative manner. For example, when gray filters having transmittances of 40 and 60% are superposed, the actual transmittance is 24% 06 × 04 = 024 instead of 0% which is calculated by subtraction as 1 − 06 − 04 = 00. For colored filters, the calculation is more complicated but the same principle applies. The three types of superposed absorbing media used in subtractive mixing are called subtractive primaries. Typically, cyan [C], magenta [M] and yellow [Y] dyes, as shown in Figure 6.3, corresponding to the complementary colors of [R], [G] and [B] are used. The tristimulus values of colors obtained by subtractive mixing cannot be calculated as easily as in additive mixing. Let filters [C] and [M] be two primaries for subtractive mixing. Then, let the spectral transmittance for unit quantity (for example, 1.0 at peak density) of each of the filters be TC and TM respectively, and consider superposing c and m units of them. Assuming that the Lambert–Beer law (see Note 6.2) holds in such a case, the spectral transmittance TF of the color [F] obtained by subtractive mixing can be expressed as TF = TC c TM m

[Y]

[M]

(6.13)

[C]

Density

0.8

0.4

0

400


700

Figure 6.3 Spectral density of three primaries dyes, cyan [C], magenta [M] and yellow [Y] for subtractive mixing (Ohta 1971)

182


Thus, the tristimulus values XF YF and ZF of subtractive mixture can be obtained from Equations 3.12 and 3.13 as integrals over the visible spectrum: XF = k YF = k ZF = k

vis

TF Px d

vis

TF Pyd

(6.14)

vis

TF Pzd

where, the constant k is k = 100/

vis

TF Pyd

(6.15)

Equation 6.14 cannot be further simplified and therefore cannot be related to the tristimulus values [C] and [M] directly. Accordingly, the results for subtractive mixing cannot be easily computed, and, although the subtractive complementary colors exist, they cannot be obtained as simply as in the case of additive complementary colors. For example, color photographs are obtained by subtractive mixing using [C], [M] and [Y] dyes having spectral densities such as are shown in Figure 6.3. Because the three dyes exist in three superimposed layers in color photographs, they can be treated in the same manner as superimposed color filters. The situation is more complicated than for additive mixing because, even for a single dye, when the peak density is changed from 0.5 to 1.0, 1.5, 2.0, 2.5 and 3.0, and the chromaticity coordinates are computed for standard illuminant D65 by Equations 6.13 to 6.15, we obtain the results shown in Figure 6.4 (Ohta 1971). The chromaticity coordinates of the primaries in additive mixing remain the same, independent of the intensity. However, it can be seen from Figure 6.4 that the chromaticity coordinates of the primaries in subtractive mixing change greatly with changing density. In particular, the [Y] dye becomes reddish with increasing density eventually approaching a brown color. Let us consider the case of mixing three dyes [C], [M], and [Y]. First, the tristimulus values obtained by mixing [C] and [M] alone can be calculated by Equations 6.13–6.15. By varying the amounts (densities) of the dyes, i.e., c and m , numerous colors can be

183

SUBTRACTIVE MIXING

520

0.8 540

[G] 560

0.6 500

[Y] + [C] 580

y

[Y]

0.4

600

D65

[R]

[M] + [Y]

620 650 770 nm

[C]

0.2

480

[C] + [M] [B] [M] 470 450 380

0

0.2

0.4 x

0.6

0.8

Figure 6.4 Subtractive mixing of cyan [C], magenta [M] and yellow [Y] dyes (filled circles correspond to peak densities 0.5–3.0 in steps of 0.5) (Ohta 1971)

obtained at a constant luminous transmittance Y , for example Y = 10. The chromaticity coordinates of these colors fall on the locus [C] + [M] shown by a broken line in Figure 6.4. Accordingly, the total color gamut for colors with Y = 10 or higher obtained by subtractive mixing of the three dyes in Figure 6.3 falls in the range defined by the broken lines [C] + [M], [M] + [Y] and [Y] + [C] in Figure 6.4 (Ohta 1971). In the case of additive mixing, as is shown in Figure 6.1, the color gamut corresponds to a triangular region surrounded by straight lines. Thus, it is difficult to predict the results of color mixing and the complementary colors in the case of subtractive mixing, because the color gamut corresponds to a region surrounded by curved lines. For an approximate prediction, the rule developed in Note 6.1 can be used, but this is not exact. Although the Lambert–Beer law, as described in Note 6.2, holds true at high precision for filters and transparent dyes such as are used in color photography, more complicated models such as the Kubelka–Munk model must be used for dyes that have high scattering (Judd and Wyszecki 1975).

184


6.3 MAXIMUM VALUE OF LUMINOUS EFFICACY AND OPTIMAL COLORS Using the CIE colorimetric system, a theoretical limit for improving the spectral properties of illuminants and dyes can be obtained. The value obtained by dividing the luminous flux by the corresponding radiant flux is called luminous efficacy. First, let us obtain the maximum value of the luminous flux for a radiation (illuminant) at given chromaticity coordinates. As is explained in Chapter 1, the radiometric quantities of an illuminant are related to the photometric quantities by the spectral luminous efficiency V. Spectral luminous efficiency V is a function of wavelength, and is bell-shaped with the peak value of 1.0 at a wavelength of 555 nm. Accordingly, the maximum value of luminous efficacy is obtained with a monochromatic light of wavelength 555 nm. The value is 683 lm/W. The luminous efficacy for other illuminants is always lower than 683 lm/W. On the other hand, because numerous metamers exist in general for an illuminant at any given chromaticity, it can be understood that different spectral efficacies can be obtained and that a maximum value exists among them. The maximum value of luminous efficacy as a function of chromaticity can be calculated with the results illustrated in Figure 6.5

0.8 683 lm . w–1

0.6

650 600 550

y

500

0.4

450 400 350 300 250 200 150

0.2 100 50

0

0.2

0.4 x

0.6

0.8

Figure 6.5 Maximum values of luminous efficacy (MacAdam 1950). Reproduced by permission of Optical Society of America

LUMINOUS EFFICACY AND OPTIMAL COLORS

185

(see Note 6.3 and MacAdam 1950). Through computation, the illuminant that provides the maximum value of luminous efficacy at a given set of chromaticity coordinates is shown to be obtained by the additive mixing of two monochromatic lights. For example, since the chromaticity coordinates of standard illuminant D65 are x = 0313 and y = 0329, the maximum value of luminous efficacy can be read from Figure 6.5 to be about 425 lm/W. This is obtained by mixing monochromatic lights of wavelengths of about 450 nm and about 570 nm. Such illuminants would be economically advantageous from the point of view of providing maximum illuminance for a given chromaticity, but have little practical value from the viewpoint of color rendering properties (See Section 7.1). Nevertheless, they may still provide target values in developing artificial daylights. In fact, considering that the luminous efficacy of standard illuminant D65 is about 203.5 lm/W, there is room for improvement in luminous efficacy while still taking other practical considerations such as color rendering into consideration. The designers of modern fluorescent lamps attempt to take advantage of this consideration by concentrating the radiant power in three bands of wavelength. This can give higher luminous efficacy while still giving excellent color rendering properties. With respect to object colors, let us consider an object having a constant lightness and a dominant wavelength of d . The excitation purity pe depends on the spectral properties, but a maximum value can be found. The color at the maximum value is called an optimal color. Referring to Figure 6.6, two types of spectral reflectance (transmittance) are found for optimal colors. In case of a color belonging to Type 1, 100% of the light is reflected (transmitted) in the wavelength region from 1 to 2 , and no light is reflected (transmitted) in the other regions. In contrast to Type 1, optimal colors of Type 2 do not reflect (transmit) light in the wavelength region from 1 to 2 , but reflect (transmit) 100% of the light in the other regions. The color gamut obtained for optimal colors for standard illuminants A and D65 are shown in Figures 6.7 and 6.8 respectively in the u v chromaticity diagram (see Note 6.4 and MacAdam 1935). In Figure 6.9, the optimal color gamut is compared with the gamut that can be produced with a typical color film (reproducible color gamut). The theoretical limit of the effect that can be obtained by improving the spectral properties of photographic dyes can be seen easily. However, it is also true that dyes of the type illustrated in Figure 6.6 show unrealistic square spectral reflectances (transmittances). By restricting dyes to smooth spectral curves with an excitation purity as high as possible, so-called pseudo optimal colors can be calculated. The resulting color gamut is shown in the uv chromaticity


1.0

Reflectance (transmittance)

186

Type 1 0.5


0

λ1

400

λ2 600 500 Wavelength (nm)

700

1.0

Type 2 0.5

0

500 λ1 600 Wavelength (nm)

400

λ2 700

Figure 6.6 Spectral reflectance (transmittance) of optimal colors

0.6

520

540

560

580 A 70 80 90 60 50 40 30

500

600

620

640 780 nm

20

0.4 Y = 10

v' 480

0.2 470 460 450 380

0

0.2

0.4 u'

0.6

Figure 6.7 Optimal color gamut corresponding to Y = 10 to 90 for standard illuminant A

187

LUMINOUS EFFICACY AND OPTIMAL COLORS

0.6

520 540

560

500 80 70 60 50 40 30

0.4 v'

580

600

90 D65

620

780 640 nm

20 480

0.2

10 470 460 450 380

0

0.2

0.4

0.6

u'

Figure 6.8 Optimal color gamut corresponding to Y = 10 to 90 for standard illuminant D65

520 540

0.6

560

500

580

600

620 640 780 nm

D65

0.4 v' 480

Y = 10 0.2 470 460 450 380

0

0.4

0.2

0.6

u'

Figure 6.9 Optimal color gamut (solid line) and reproducible color gamut of color film (broken line)

diagram in Figure 6.10. The spectral reflectance of these colors exhibit realistic distributions as is shown in Figure 6.11 (Ohta 1982). It can be seen from Figure 6.10 that the color gamut of true optimal colors is not much bigger than that of pseudo-optimal colors. Thus, the color gamut of optimal colors can be set as the theoretical limit for improving the spectral properties of photographic dyes.

188


520

0.4

540

560

500

580

M

600

620

D65

640 660 770

90 m

Y = 50

v 0.2 480

10 470 460 450 440 420

0

400

0.2

0.4

0.6

u

1.0

M

0.5

0 i=1 5 λ = 400

10 15 20 25 500 600 Wavelength (nm)

30 700



Figure 6.10 Optimal color gamut (broken lines) and pseudo optimal color gamut (solid line) (Ohta 1982). Reproduced by permission of Wiley

1.0

m

0.5

0 i=1 5 λ = 400

10 15 20 25 500 600 Wavelength (nm)

30 700

Figure 6.11 Optimal color (broken lines) giving chromaticity points M and m in Figure 6.10 and spectral reflectance (transmittance) of pseudo optimal colors (solid line) (Ohta 1982). Reproduced by permission of Wiley

6.4 CHROMATIC ADAPTATION PROCESS When the intensity and spectral distribution of the ambient illumination vary, our visual system continues to function without being greatly influenced by the variation. This is realized by adaptation, either luminance adaptation (the adaptation of vision to the ambient

189

CHROMATIC ADAPTATION PROCESS

luminance level), or chromatic adaptation (the adaptation of vision to the color of the ambient illumination). For example, although the luminance of the skin of black people observed under sunlight is higher than that of the skin of white people observed under overcast conditions, the skin of black people is still perceived as black. This is because of luminance adaptation. Similarly, because of chromatic adaptation, the perceived color of objects does not change much, even when the spectral distribution of the illuminant changes by a large amount, such as from daylight to incandescent light. We often experience this phenomenon without being conscious of it, perceiving object colors as unchanged even in different illuminants. This is known as color constancy. As an example of chromatic adaptation, let us consider the case of moving from outdoors under daylight illumination into a room illuminated with an incandescent lamp. Chromatic adaptation does not occur instantaneously, but proceeds in two steps as is shown in Figure 6.12.

6.4.1 Illuminant Color Shift When we enter a room illuminated with an incandescent lamp after being outdoors under daylight, all the objects illuminated by the incandescent lamp appear yellowish. This can be interpreted as a change of the chromaticity point SD of an object under daylight

SA′ 0.55 SA v′

A

SD 0.50 SD′ D 0.45

0.15

0.2 u′

0.25

Figure 6.12 Illuminant color shift (from SD to SA , adaptive color shift (from SA to SD ), resultant color shift (from SD to SD , and corresponding color SD for SD , when illuminant changes from D to A

190


(D) to another chromaticity point SA under the incandescent lamp (A). This change, from SD to SA , is called the illuminant color shift. Because the illuminant shift corresponds to the change in chromaticity coordinates obtained by colorimetric calculations, it is also called the colorimetric shift.

6.4.2 Adaptive Color Shift When our eye adapts to the incandescent lamp (A), the sensitivity balance for the three types of cones in the visual system changes. Thus, the original daylight color appearance is largely recovered and the yellowish appearance disappears. This recovery takes place quite quickly and often we are unaware of any major change. However, in general, the appearance of colors observed under incandescent light, even after full adaptation, does not completely agree with that observed under daylight. Referring to Figure 6.12, the appearance of the object under the incandescent lamp (A) matches the appearance of an object that has a chromaticity point SD under daylight (D). This change from SA to SD is known as the adaptive color shift.

6.4.3 Resultant Color Shift When our eye sufficiently adapts to the incandescent lamp (A), we see the total color change from SD to SD , combining the illuminant color shift and the adaptive color shift. This is called the resultant color shift and is generally much smaller than either the illuminant or the adaptive color shift. Thus, the color under daylight (D) that matches the appearance of the object SD viewed under (A) is SD . This color is called the corresponding color for SD viewed under (A). Conversely, if (A) is regarded as the reference illuminant, SA represents the corresponding color for SA viewed under (D). The illuminant and adaptive color shifts occur in opposite directions to each other and to about the same extent. Color constancy holds when the degree of shift is completely the same for both. In practice, however, color constancy is rarely perfect, and this imperfection is observed as the resultant color shift. Figure 6.13 shows a series of corresponding colors for the case of a change from daylight to an incandescent lamp (MacAdam 1956). When a color photograph or color television image is taken, since there is no mechanism for color constancy, the change in illuminant color becomes a problem. In the case of a color photograph, if, for example, a daylight color negative film is used under an

191

VON KRIES’ PREDICTIVE EQUATION

520

0.8

540

560

0.6 500 580

y 0.4 Daylight

600

Incandescent light

620 650 700 nm

0.2 480 470

380

450

0

0.2

0.4 x

0.6

0.8

Figure 6.13 Corresponding colors when daylight is changed to incandescent light (colors change in the direction of arrowhead) (MacAdam 1956). Reproduced by permission of Optical Society of America

incandescent lamp, a yellowish color image will be obtained corresponding to the illuminant color shift. Therefore, a color correction corresponding to the illuminant color shift must be made in the printing process, or a color temperature conversion filter must be used at the time of photographing to modify the relative spectral distribution of the illuminant. In the case of color television, an adjustment of the responsivity of the red, green and blue receptors of the camera (called white balance) is made to compensate for the illuminant color shift in advance.

6.5 VON KRIES’ PREDICTIVE EQUATION FOR CHROMATIC ADAPTATION As described in Section 6.4, color constancy is not perfect. Thus, even after chromatic adaptation is complete, our skin color, for example, illuminated by daylight appears slightly different from that when it is illuminated by an incandescent lamp. That is, the skin color under daylight which looks equivalent to the skin color under the incandescent lamp (the corresponding color) is slightly different

192


Figure 6.14 Johannes von Kries (1853–1928)

from the actual skin color perceived under daylight. It is extremely important in practice to be able to calculate corresponding colors by predicting the effect of chromatic adaptation. Numerous predictive equations have been proposed. The classic equation is that proposed by von Kries (Figure 6.14) and it provides the basis for a number of more modern equations. To obtain corresponding colors, the von Kries hypothesis (von Kries 1902) assumes three types of receptors to be present in our eye, each receptor sensing nominally red, green and blue colors, and each differing in spectral responsivity (see Note 6.5 for a method of calculating such fundamental spectral responsivities). In the hypothesis, when the spectral distribution of the illuminant is changed, chromatic adaptation occurs by a change of responsivity balance between the three receptors in such a manner that a white color is always sensed as white, while the shape of the spectral responsivity of each photoreceptor remains unchanged. For instance, let us consider the case shown in Figure 6.15, in which daylight is changed to an incandescent lamp. The responsivity for red, green and blue colors is balanced under daylight because the spectral distribution of daylight is approximately flat. In the case of the incandescent lamp, however, the red component is stronger and the blue component weaker. Thus, the responsivity of the red receptor is lowered and that of the blue receptor is increased to obtain equal responses, and, as a result, the perceived color remains approximately unchanged. According to the von Kries hypothesis, the tristimulus values (X Y Z ) of the corresponding color under a reference illuminant for an object that has tristimulus values (X Y Z ) under a test illuminant can be expressed (see Note 6.6) as

193

VON KRIES’ PREDICTIVE EQUATION



  X a11  Y  =  a21 Z a31

a12 a22 a32

  X a13 a23   Y  a33 Z

(6.16)

where the element aij of the matrix is obtained from the tristimulus values of the test light and the reference light. For instance, in the case of standard illuminant A and standard illuminant D65 , as the test light and the reference light respectively, Equation 6.16 can be evaluated (see Note 6.7) as      1127 −0438 0427 X X  Y  =  −0011 1011 0002   Y  (6.17) 0 0 3068 Z Z Then, the corresponding color can be obtained from Equation 6.17. For example, the corresponding color under standard illuminant D65 for an object color having the following tristimulus values under standard illuminant A X = 2800

Y = 2126

Z = 527

(6.18)

can be obtained by substituting the values of Equations 6.18 into Equation 6.17:      1127 −0438 0427 2800 X  Y  =  −0011 1011 0002   2126  0 0 3068 527 Z   2449 =  2120  (6.19) 1617

Energy

Blue Green Red

Wavelength Energy

Blue Green Red

Eye

Illumination Wavelength

Balance of eye responsivity

Spectral distribution

Figure 6.15 Change in balance of eye responsivity with change of illumination

194


6.6 CIE PREDICTIVE EQUATIONS FOR CHROMATIC ADAPTATION Chromatic adaptation can be mostly explained by the von Kries predictive equation, but the equation is still insufficient to explain some of the details including the following phenomena: 1. When a gray scale is illuminated by a colored light, light grays take on the hue of the illuminant whereas dark grays take on that of the complementary color. This is known as the Helson– Judd effect. 2. As the illuminance on a group of achromatic colors increases, light gray colors appear more whitish, while dark gray colors appear more blackish. This is known as the Stevens effect. 3. When a chromatic color is illuminated at different levels of illuminance, the chroma (colorfulness) appears to increase with increasing illuminance. This is known as the Hunt effect. Many different equations (known as chromatic adaptation transforms or CATs) have been proposed that are improvements over the von Kries transform. One of these was recommended by the CIE for field trials in 1994. Later, in 1998, the CIE recommended a different transform as part of a color appearance model known as CIECAM97s and, still later, in 2004, a modified version in the CIECAM02 color appearance model. This model, including the chromatic adaptation transform, is described in Section 6.8. The CIE 1994 transform (CIE 1994) was originally proposed by Nayatani (Figure 6.16). He combined a linear process, as in the von Kries transform, and a non-linear process in which the exponent

Figure 6.16 Yoshinobu Nayatani (1927–)

CIE PREDICTIVE EQUATIONS FOR CHROMATIC ADAPTATION

Test light

195

Reference light

Test color X, Y, Z

Corresponding color X′, Y′, Z′

ρ0

ρ ′0

Visual field

Reference visual field

Figure 6.17 Test color and corresponding color

changes with the adaptation level (Nayatani et al. 1981, Takahama et al. 1984). The 1994 version is slightly modified from the earlier versions and consists of the following three steps. The experimental set-up is shown in Figure 6.17 and relates a test color under one adaptation condition and the corresponding color under a reference condition. 1. The CIE tristimulus values, X Y and Z of the test color are converted by the following equation into tristimulus values, R G and B based on fundamental spectral responsivities.      X R 040024 070760 −008081  G  =  −022630 116532 004570   Y  (6.20) B 0 0 091822 Z where the chromaticity coordinates of the fundamental primaries, on which the spectral responsivities are based, are not those of Pitt (Note 6.5), but rather a slightly different set based on those proposed by Hunt and Pointer (1985) which, in turn, were based on a set proposed by Estévez (1979). 2. The tristimulus values, R G and B obtained above for the test color are converted into the tristimulus values, R G and B for the corresponding color by a nonlinear conversion according to

R = 1000 + nK 1/f1 R0 R + n/

1000 + nf1 R0 /f1 R0 − n

G = 1000 + nK 1/f1 G0 G + n/

1000 + nf1 G0 /f1 G0 − n

B = 1000 + nK 1/f2 B0 B + n/

1000 + nf2 B0 /f2 B0 − n

(6.21)

196


where n is a noise factor that is set equal to 1.0. The various other symbols in Equation 6.21 are explained below. 3. The tristimulus values X’, Y’, Z’ for the corresponding color are obtained from the tristimulus values R G B obtained above according to 

  X 185995  Y  =  036119 0 Z

−112939 063881 0

  021990 R  G  0 108906 B

(6.22)

which is the inverse of Equation 6.20. In Equation 6.21, 0 and 0 are the reflectances of a gray color in the test field and in the reference field, respectively and are restricted to a range of 0.18 to 1.0. The symbols , and are transformed relative chromaticity coordinates obtained by = 048105 x0 + 078841 y0 − 008081/y0

= −027200 x0 + 111962 y0 + 004570/y0

(6.23)

= 0918221 − x0 − y0 /y0 where x0 and y0 are the chromaticity coordinates of the test light. The coordinates , and are obtained from the chromaticity coordinates x0 and y0 of the reference light by the same equations. The functions f1 x and f2 x are expressed by f1 x = 6469 + 6362 x 04495 /6469 + x 04495 f2 x = 078448414 + 8091x 05128 /8414 + x 05128 (6.24) The symbols R0 , G0 , B0 and R0 , G0 , B0 represent the effective adapting responses in the test field and reference fields, respectively, and are given by 

   R0  G0  = 0 E0 /   B0     R0  G0  = 0 E0 /   B0

(6.25)

where E0 and E0 are the illuminances (in lux) of the test and reference fields, respectively. The coefficient K in Equation 6.21 corrects

197

COLOR VISION MODELS

for the any difference between the test and reference illuminances and is given by K=

1000 + n/20 + n2/3f1 R0

1000 + n/20 + n2/3f1 R0 × 1000 + n/20 + n1/3f1 G0 × 1000 + n/20 + n1/3f1 G0

(6.26)

where n, the noise factor, again equals 1.0. For example the corresponding color under D65 , for the object color defined in Equation 6.18, is calculated by the CIE 1994 transform to be X = 2417

Y = 2107

Z = 1764

(6.27)

for E0 = E0 = 1000 lx and 0 = 0 = 02 (see Note 6.8 for details of the calculation). It can be seen that these values are in fair agreement with those of Equation 6.19 obtained by the von Kries transform, although not exactly the same. The main advantage of the CIE 1994 transform and other more modern variations is that they are still effective when the illuminance changes or the saturation of the illumination is high, because they include a prediction of the Helson–Judd, Stevens, and Hunt effects. More recently, Nayatani et al. (1999) have proposed a modified version of the CIE 1994 transform. This includes the introduction of a factor to take account of incomplete adaptation and also proposes a reduction of the noise factor, n from 1.0 to 0.1. Furthermore, the CIE has introduced new transforms in the color appearance models described in Section 6.8. Other authors, such as Luo and Hunt (1998) have also published chromatic adaptation transforms which give results as least as good as those of the CIE transforms. It is clear that studies of the validity and practicability of all these transforms must be performed in industrial applications and these may lead to new CIE recommendations in the future. All these new transforms involve much more complicated calculations than the original von Kries transform. However, this is not an obstacle in these days where advanced computers are readily available.

6.7 COLOR VISION MODELS Many attempts have been made to explain particular color vision phenomena by constructing models (known as color vision models) based on known or postulated mechanisms of the human eye. If these color vision models are sufficiently rational, it should also

198


Third stage (sensation)

Observer

Chromatic signal

Brightness signal

Blue– yellow

Red– green

Black and white response

D =B–R

T = R–G

A = R+G

B

G

R

Second stage (optic nerve)

First stage (cones)

Figure 6.18 Guth’s color vision model (Guth et al. 1980). Reproduced by permission of Optical Society of America

be possible to explain other color vision phenomena by the same model. Most of the color vision models proposed to date are based on a stage theory (also known as a zone theory), which assumes a three-color response in the cones as the first stage, and a conversion of color signals in the optic nerve layer as the second stage. Color vision models based on the zone theory have been proposed by many researchers such as Muller and Judd, Adams, Hurvich and Jameson, Guth, and Ingling. Figure 6.18 is a schematic diagram of the model due to Guth et al. (1980). Other models are similar in general outline but differ in details. The CIE has been particularly concerned with models that predict color appearance. Such models are needed in color image reproduction applications. In 1997, the CIE proposed a model known as CIECAM97s that could be used to promote uniformity of practice and compatibility between various components in modern imaging systems. Based on experience gained, the CIE later proposed some small modifications in a model known as CIECAM02. This model is described in Section 6.8.

6.8 COLOR APPEARANCE MODELS Color appearance models attempt to bridge the gap between psychophysical specifications of color stimuli (usually in the form of CIE tristimulus values) and color appearance. A series of

199

COLOR APPEARANCE MODELS

mathematical formulae is used to calculate quantities that correlate with appearance attributes such as hue, saturation, lightness, brightness and many others. The CIE system of colorimetry embodied in the 1931 and 1964 color matching functions does a good job of predicting metameric matches. A few breakdowns have been found (Robertson 1993), but their effect is minor in most circumstances. However, the system, in its basic form, does not do a good job of predicting color appearance and, indeed, it was never intended to do so. To determine color appearance, it is necessary to take into account other factors such as the observer’s state of adaptation, the nature of the surround, and ambient light. It is also necessary to have some knowledge of the human color vision system. A number of ‘color appearance models’ have been proposed that calculate correlates of perceived attributes, such as hue, etc., from the CIE tristimulus values. All of these models are based to some extent on the functioning of the visual system, but all include some empirical adjustments to improve the accuracy with which visual observations are predicted. Most models follow the same general outline. A flowchart of a typical model is shown in Figure 6.19. A transformation of the CIE tristimulus values is used to approximate the responses of

Spectral Power Distribution

Surround Conditions

CIE Tristimulus Values

Cone Responses

Adaptation

Non-linear Compression

Achromatic Signal

Opponent Processes

Appearance Attributes

Figure 6.19 Flowchart of a typical color appearance model

200


the three types of cone in the human retina. These responses are then adjusted to take into account the state of adaptation of the observer. Following this, there is usually some sort of compression of each of the three signals to model the fact that the visual system is nonlinear and limited in its range of response. At some stage, either before or after the compression, an achromatic response is generated. Such a response is generally believed to exist and to control functions such as visual acuity. The compressed signals are used as input to opponent processes that produce red–green and blue–yellow responses. The two opponent responses and the achromatic response are then combined in various ways, largely empirical, to produce indices that correlate with visual judgements of a wide variety of attributes of perceived color. The CIELAB and CIELUV uniform color spaces were introduced by the CIE to provide a basis for predicting the perceptibility of color differences. They were not intended to be full color appearance models although they have sometimes been used that way. Neither uses a transformation to cone responses because, at the time of their introduction, the added complexity did not seem to be justified. The CIELAB model uses an adaptation transform of the von Kries type in which the three responses (CIE XYZ tristimulus values in this case) are divided by responses representing the state of adaptation of the observer. The adapted responses are then compressed by a cube-root function and subtracted from each other to give two opponent quantities, a* and b*. The achromatic quantity is CIELAB lightness, L *, a cube-root function of the Y tristimulus value. Plotting these three quantities as rectangular coordinates produces a uniform color space. Embedding a cylindrical coordinate system produces simple correlates of lightness, chroma and hue. On the other hand, the CIELUV space is based on a projective transformation of the CIE chromaticity diagram. Adaptation is predicted by a translation in the projected diagram. Compression is again handled by a cube-root function but compression in all three ‘channels’ is determined by the signal in the Y channel. One of the major drawbacks of both CIELAB and CIELUV as color appearance models is that the chromatic axes do not correspond well to the four unique hues, red, yellow, green and blue. Examples of color appearance models proposed over the years include those of Richter (1980), Nayatani (1990), Hunt (1991), Guth (1991), Fairchild (1996) and Luo (1996). The existence of so many different models produced confusion and, with the advent of digital imaging and the ability to transmit images and reproduce them with different media and in different viewing conditions, it became urgent that the CIE standardize one color appearance model for


201

interim use in practice, while researchers continued to develop even better models for the future. As was the case with color difference formulae in 1976, current evidence was insufficient to show which model was best. Nevertheless, in view of the pressing industrial need for standard practice, the CIE decided to choose and recommend one model to be used until a demonstrably better model could be formulated. The model builds on the work of many researchers and was thought to encompass the best of what was currently available. The original version, known as CIECAM97s was published (CIE 1998b) and initial results indicated that it performed as well, if not better than any previously published model for a wide range of experimental data. Nevertheless, a few minor improvements were suggested and a slightly modified version, known as CIECAM02 was published (CIE 2004b). The first step in the CIECAM02 model is a transformation to a set of ‘fundamental’ responses, R, G and B by      R 07328 04296 −01624 X/Y  G  =  −07306 16975 00061   Y/Y  (6.28) B 00030 00136 09834 Z/Y The spectral responsivities associated with this transformation are somewhat sharper in shape than those normally attributed to the cone responses, but they give a better prediction of chromatic adaptation. The next step is a chromatic adaptation transform that includes a factor, D, to specify the degree of adaptation. For full adaptation, D should be set equal to unity and for no adaptation it should be set equal to zero. The CIE gives guidance on specifying D in intermediate conditions and suggests D = 0.6 as a realistic minimum value. The transformation equations are RC = Yw D/Rw + 1 − DR

(6.29)

GC = Yw D/Gw + 1 − DG

(6.30)

BC = Yw D/Bw + 1 − DB

(6.31)

where RC GC and BC are chromatically adapted values and Xw Yw and Zw are the tristimulus values of the white point that controls the adaptation. Determining this point in real conditions is an area of active research, but, for example, when viewing hardcopy material it could be a perfect reflecting diffuser or the media white in the same conditions. For a computer or TV monitor, the white point is often taken to be the ‘device white’ at 20% luminance. The quantities Rw , Gw and Bw are calculated from Xw , Yw and Zw by Equation 6.28.

202


The values of RC , GC and Pointer–Estévez space by    038971 R  G  =  −022981 00000 B  1096124  0454369 −0009628

BC are then transformed to the Hunt–  068898 −007868 118340 004641  00000 10000

  RC −0278869 0182745 0473533 0072098   GC  (6.32) −0005698 1015326 BC

where the second 3×3 matrix is the inverse of the 3×3 matrix in Equation 6.28. A post-adaptation nonlinear compression is then applied in this space by R a =

400FL R /100042 2713 + FL R /100042

(6.33)

G a =

400FL G /100042 2713 + FL G /100042

(6.34)

B a =

400FL B /100042 2713 + FL B /100042

(6.35)

In these equations, Ra Ga , and Ba are post-adaptation compressed values. FL is a factor that depends on the luminance, LA of the adopted white point and is calculated by FL = 02k 4 5LA + 011 − k 4 2 5LA 1/3

(6.36)

k = 1/5LA + 1

(6.37)

where

In Equations 6.33–6.35, if any of the values of R , G or B are negative, its absolute value must be used and the corresponding value of Ra , Ga or Ba must be made negative. Having calculated the coordinates of the color in Hunt–Pointer– Estévez space, the next step, prior to calculating indices of color appearance, is to calculate an intermediate Cartesian representation (a,b) and a hue angle (h). This is done by a = R a − 12G a /11 + B a /11

(6.38)

b = 1/9R a + G a − 2B a

(6.39)

−1

h = tan b/a

(6.40)


203

To denote hue appearance, CIECAM02 uses a system similar to that of the Natural Color System (Hård and Sivik 1981). The hue circle is divided into four quadrants separated by the unique hues, red, yellow, green and blue. Hues are then denoted by a quantity called ‘hue quadrature’, H where H =0 (or 400) for unique red , 100 for unique yellow, 200 for unique green, and 300 for unique blue, with intermediate hues spaced evenly within the four quadrants. The hue angle, h (Equation 6.40) is defined to be 20.14 (or 380.14 for unique red, 90.00 for unique yellow, 164.25 for unique green, and 237.53 for unique blue. The hue quadrature, H for any intermediate hue is then derived from the hue angle, h by interpolation between the two nearest unique hues (denoted by subscripts 1 and 2) according to the following formula. H = H1 +

100h − h1 /e1 h − h1 /e1 + h2 − h/e2

(6.41)

Equation 6.41 includes an eccentricity factor, e to account for the empirical fact that achromatic colors are not at the center of contours of low saturation. This factor is calculated by e=

1

cosh + 360/ + 38 4

(6.42)

Indices of lightness and brightness are based on an achromatic response, A calculated by A = 2Ra + Ga + 1/20Ba − 0305Nbb

(6.43)

where Nbb is a background brightness induction factor, calculated by Nbb = 0725Yw /Yb 02

(6.44)

and Yw and Yb are the Y tristimulus values of the white point and the background respectively, background being defined as the area immediately adjacent to the stimulus of interest. An index of lightness J is calculated from the achromatic responses of the stimulus (A and the white point (Aw by J = 100A/Aw cz

(6.45)

where c is a constant representing the impact of the surround (the field outside the background) and is assigned the values 0.525 for

204


a dark surround, 0.59 for a dim surround, and 0.69 for an average surround. The base exponential nonlinearity z is defined by

(6.46) z = 148 + Yb /Yw CIECAM02 brightness, Q is computed from J by

Q = 4/c J/100Aw + 4FL025

(6.47)

where FL is defined in Equation 6.36. Lastly, in order to define indices of chroma (C), colorfulness (M), and saturation (s, it is first useful to define an intermediate quantity, t by t=

50000/13Nc Ncb ea 2 + b 2 Ra + Ga + 21/20Ba

(6.48)

where Nc is a chromatic induction factor that is assigned the values 0.8 for a dark surround, 0.9 for a dim surround, and 1.0 for an average surround. The chromatic brightness induction factor, Ncb is set equal to the background brightness induction factor, Nbb defined in Equation 6.44. The three indices are then defined by

(6.49) C = t 09 J/100164 − 029Yb /Yw 073 M = CFL 025

s = 100 M/Q

(6.50) (6.51)

The CIECAM02 color appearance model, like its predecessor, CIECAM97s is very complicated and largely empirical. Hopefully, future models will follow the workings of the human visual system more closely. Nevertheless, CIECAM02 is the result of intense study by many researchers and not only provides a standard model that can unify practice, but is also as good as or better than any model proposed to date.

6.9 ANALYSIS OF METAMERISM As described in Section 3.5, metamerism is the name given to the phenomenon in which a pair of color stimuli, differing in spectral distribution, match in color under certain conditions. Such a pair of color stimuli is called a metameric pair. In general, the color match is no longer valid if any of the conditions, such as the spectral distribution of the light illuminating an object color, changes.

205

ANALYSIS OF METAMERISM

This susceptibility to breakdown is why this type of color match is distinguished from matches between stimuli with the same spectral distribution by giving it the special name, ‘metamerism’. The degree to which a match is susceptible to change is called the degree of metamerism, and an index that quantifies this degree is called a metamerism index. The breakdown of a metameric match is common when the conditions change but there are some special situations where the match does not break down. For example, Figure 6.20 shows a calculated example of a metameric pair which exhibits metamerism (i.e., matches) under standard illuminant C as well as under standard illuminant A. Figure 6.21 shows an example in which metamerism holds under three different illuminants, i.e., daylight, an incandescent lamp, and a fluorescent lamp. It can be seen however that the difference in spectral reflectance (transmittance) is narrowed with the increasing number of illuminants under which metamerism holds true. In developing indices of metamerism, the deviation from matching is often classified by the conditions that are changed. That is, the terms object-color metamerism, illuminant metamerism, or observer metamerism are used depending on the condition that is changed to expose the metamerism. In addition to these terms, the term geometrical metamerism is sometimes used to describe

Reflectance

0.8

0.6

0.4

0.2

0

400


700

Figure 6.20 Object colors showing metamerism under standard illuminants A and C (XYZ colorimetric system) (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

206


60

Reflectance

40

20

20

0

400


700

Figure 6.21 Object colors showing metamerism under daylight, incandescent light, and cool white fluorescent light (Berns 2000). Reproduced by permission of Wiley

matches that depend on the geometrical conditions under which object colors are viewed. However, this is really a misnomer because the phenomenon does not involve stimuli that match while differing in spectral distribution. Table 6.1 shows how the spectral properties of object colors, illuminants or observers can be evaluated by varying the conditions to establish the degree of metamerism. Studies have been conducted that suggest methods for evaluating the degree of metamerism and the spectral properties of illuminants Table 6.1 Evaluation methods for degree of metamerism Object of evaluation (metameric pair)

Means of evaluation (variable)

Examples of application

Object colors (spectral reflectances) Illuminants (spectral distributions) Observers (color matching functions)

Illuminants Observers

Illuminant metamerism Observer metamerism∗

Object colors Observers Illuminants Object colors

Evaluation of illuminants Observer metamerism∗ Evaluation of color separation systems and colorimeters

* In the evaluation method of JIS and CIE, the standard deviate observer is used instead of using a number of individual observers.

207


using various mathematical techniques (Ohta and Wyszecki 1975, Stiles et al. 1977). For instance, let us consider a pair of metameric object colors. The colors of the objects match only under certain conditions, and, in general, will not match if the conditions (in this case, the spectral distribution of the illuminant or the color matching functions of the observer) are changed. Accordingly, the degree of deviation from color matching (the stability of color matching) can be obtained by evaluating the metameric pair for differing illuminants or observers, and thus the degree of illuminant metamerism or the degree of observer metamerism can be obtained (see Sections 7.4 and 7.5). The deviation from matching is illustrated below for metameric object colors. As a case in which the observer is changed, let us consider changing the XYZ colorimetric system to the X10 Y10 Z10 system. The twelve object colors shown in Figure 6.22 are used. Although these colors are not actual object colors, having been produced by mathematical operation, they are fairly smooth and possess realistic spectral reflectances. All the objects have exactly the same gray

0.8 5 8

Reflectance

11 10

0.6

10

8

9

5

10 11

0.4

11 6

6

9

6

9

5

8

8

0.2 5

9

6 10

0

10

11

6

12 2

1

Reflectance

11

6 3

0.8 7

4 1

4

4

12 7

3

0.4

2

3 2

0.2

12 3

4

12 1

400

7 3

4

4

0

8

10

9 5

0.6

9

6

500

1

2 1

600 Wavelength (nm)

2

700

Figure 6.22 Twelve gray object colors metameric under standard illuminant C in the XYZ colorimetric system (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

208


color in the XYZ colorimetric system under standard illuminant C. Their tristimulus values and chromaticity coordinates are X = 2941

Y = 3000

x = 0310

y = 0316

Z = 3543 (6.52)

However, the colors no longer match if they are evaluated in the X10 Y10 Z10 colorimetric system, and the chromaticity coordinates become distributed, as shown in Figure 6.23. The range of distribution shows the degree of difference between the two sets of color matching functions, and, at the same time, the degree of difference between the spectral reflectances of the 12 objects. The deviation from matching can also be shown for the case in which the illuminant is changed. Figure 6.24 shows the chromaticity coordinates, evaluated in the XYZ colorimetric system with respect to standard illuminant A, for 100 object colors that are metameric (xD = 0.313, yD = 0.329, Y = 50) under standard illuminant D65 . The spectral reflectance of the 100 object colors were generated by a Monte Carlo method in which random numbers, uniformly distributed between 0 and 1.0, were assigned to the spectral reflectances, under the constraint that the generated colors satisfy the given condition of matching under illuminant D65 . Figure 6.24

0.340

12

0.330

8

5 4

y2 and y10

3

0.320

2 7 9

0.310

10

6 11 1

0.300

0.290 0.300

0.310 0.320 x2 and x10

Figure 6.23 Chromaticity coordinates for twelve gray colors in the X10 Y10 Z10 colorimetric system that are metameric in the XYZ system with the coordinates indicated by the cross (Wyszecki and Stiles 1967). Reproduced by permission of Wiley

209


0.430

0.420

y 0.410

0.400

0.390 0.430

0.440

0.450 x

0.460

Figure 6.24 Deviation of gray metamers (Y = 50) from color matching with change of standard illuminant from D65 to A (by Monte Carlo method). Open circle shows chromaticity coordinates of standard illuminant A. Cross shows center of ellipse

also shows a probability ellipse containing 95% of the distributed chromaticity coordinates. The size of the ellipse colorimetrically indicates the difference in spectral distribution between standard illuminants A and D65 . In contrast to Figure 6.24, in which 100 object colors are used, Figure 6.25 shows the theoretical limits for the deviation from

0.460 0.440

y

0.420

51.0 Å

0.400

Y = 50.0

0.380 49.0 0.360 0.400 0.420 0.440 0.460 0.480 x

Figure 6.25 Theoretical limits for deviation from matching for metameric gray colors (Y = 50) when illuminant changes from D65 to A

210


0.460 0.440 P

0.420 y

Å 0.400 0.380 0.360 0.400 0.420 0.440 0.460 0.480 x

Figure 6.26 Theoretical limit (envelope of pattern shown in Figure 6.25) for the deviation from metamerism and that calculated by the Monte Carlo method (hatched portion)

matching of an infinite number of metamers. These limits were obtained by linear programming (see Note 6.9). Figure 6.26 compares the probability ellipse of Figure 6.24 with the envelope of the pattern shown in Figure 6.25. It can be seen that there is a great difference between them. This can be understood by comparing the shape of spectral reflectances R included in the calculation. For example, Figure 6.27 shows the spectral reflectance of the color that provides point P on the envelope shown in Figure 6.26. The probability of obtaining this distribution of spectral reflectance in

Reflectance

1.0

0.5

0

i=1 400

5

10 15 20 500 600 Wavelength (nm)

25

30 700

Figure 6.27 Spectral reflectance that provides point P in Figure 6.26

211

COLOR MIXING RULE

practice is extremely low. Thus, the region obtained by the Monte Carlo method is far smaller than that obtained by the linear programming method.

NOTE 6.1 COLOR MIXING RULE By dividing the spectrum of a white color [W] (400–700 nm) into three regions of 400–500 nm, 500–600 nm, and 600–700 nm, regions approximately corresponding to blue [B], green [G], and red [R] can be obtained. Thus, the following equation holds

R + G + B = W

(6.53)

By additively mixing each pair of two colors from the above three ‘primaries’, as understood from Figure 6.1, three primaries for subtractive mixing can be obtained as

R + G = Y

G + B = C

(6.54)

B + R = M In the case of subtractive mixing, on the other hand, the tristimulus values resulting from color mixing can be accurately obtained only by assuming the Lambert–Beer law. But for a rough prediction, the same operation as in Equation 6.54 can be performed. More specifically, the spectral transmittance of a color resulting from the subtractive mixing of two colors is obtained as a product of the spectral transmittances of the two components, so that, if an [R] filter is superimposed on another [R] filter, [R] alone is transmitted. If an [R] filter is superimposed on a [G] filter, no light is transmitted commonly. Thus, if subtractive mixing is expressed by the symbol *, we can write

R∗ R = R

G∗ G = G

B∗ B = B

R∗ G = G∗ B = B∗ R = K

(6.55)

212


where [K] represents black, and indicates that no light is transmitted. Of course, these equations are idealized and hold true exactly only if each filter has 100% transmittance in its own region of the spectrum and zero transmittance outside this region, with no overlap of transmitting regions. By combining the three primaries of subtractive mixing, we obtain, from Equations 6.54 and 6.55

C∗ M = G + B∗ B + R = G∗ B + R∗ G + B∗ B + B∗ R = B

M∗ Y = B + R ∗ R + G = B∗ R + G∗ B + R∗ R + R∗ G = R

Y∗ C = R + G∗ G + B

(6.56)

= R∗ G + B∗ R + G∗ G + G∗ B = G

C∗ M∗ Y = G + B∗ B + R∗ R + G = B∗ R + G = B∗ R + G∗ B = K

In these equations, except for the last one, [K] is omitted because it has no influence on the resulting color. Figure 6.28 shows graphically a simplified model of spectral transmittance for real cyan and magenta dyes. It can be clearly seen, both from Figure 6.28 and from the preceding derivation, that [B] is generated by subtractive mixing of [C] and [M]. The rule of color mixing introduced here shows the case in which the primaries are mixed at equivalent quantities, but the same treatment can be applied when the mixing ratio is changed. However, this prediction method is only valid for recognizing the rough color, and, if an accurate result is necessary, Equations 6.13, 6.14 and 6.15 must be used.

213

LAMBERT–BEER LAW

[C]

1.0

0.8 0.7 0.5

0.1 0 1.0

[M]

0.9

Reflectance

0.8

0.5

0.1 0 1.0

[C] * [M] = [B]

0.7 × 0.8 = 0.56 0.5 0.8 × 0.1 = 0.08 0.1 × 0.9 = 0.09 0

400


700

Figure 6.28 Spectral transmittance produced by mixing cyan [C] and magenta [M] dyes

NOTE 6.2 LAMBERT–BEER LAW The logarithm of the ratio of the incident light intensity, I0 to the transmitted light intensity, I is linearly proportional to the thickness, d of the absorbing medium. This is known as Lambert’s law and is expressed by logI0 /I = d

(6.57)

where is a constant. This logarithm is also proportional to the concentration c of the absorbing substance contained in the absorbing medium. This is known as Beer’s Law and is expressed by logI0 /I = c

(6.58)

214


where is a constant. By combining Lambert’s law and Beer’s law, the Lambert–Beer law is obtained, expressed by logI0 /I = kcd

(6.59)

where k is a constant.

NOTE 6.3 METHOD FOR CALCULATING THE MAXIMUM VALUE OF THE LUMINOUS EFFICACY OF RADIATION As an alternative to the calculation method of MacAdam (1950), the maximum value of the luminous efficacy of radiation can be obtained in the following manner. From Equation 1.20, luminous flux (lm) can be obtained as PVd (6.60) v = Km vis

where P (W/nm) is the spectral concentration of radiant flux, and Km = 683 lm/W. This indicates that a radiant flux of 1 W at = 555 nm corresponds to a luminous flux of 683 lm. By approximating Equation 6.60 by summation, and omitting the wavelength interval, for simplification, the following equation can be obtained: v = Km

n

Pi Vi

(6.61)

i =1

where i indicates discrete wavelength points, and n = 31 (assuming = 10 nm in the wavelength region 400–700 nm). By properly normalizing the summation for spectral distribution P, the following equation can be obtained n

Pi = 1

(6.62)

i =1

The chromaticity coordinates xs and ys of P are related to the tristimulus values X Y and Z by xs = X/X + Y + Z =

n

Pi · x i /

i =1

n

Pi x i + y i + z i

i =1

ys = Y/X + Y + Z =

n i =1

Pi · y i /

n i =1

Pi x i + y i + z i

(6.63)

CALCULATING OPTIMAL COLORS

215

By rewriting Equation 6.63, we obtain n

Pi 1 − 1/xs x i + y i + z i = 0

i =1 n

Pi x i + 1 − 1/ys y i + z i = 0

(6.64)

i =1

Then, the problem is reduced to obtaining the maximum value for v (the object function) in Equation 6.61, under the constraints of Equations 6.62 and 6.64, for the unknown Pi ( i = 1 n ) in view of Vi = y i . Because the constraint equations and the object function are both given as linear functions of the unknown, Pi , the problem can be solved easily by linear programming. Since programs for the numerical solution of linear programming are provided as built-ins in most computers, the solution can be found readily. The method of linear programming can be used not only for the computation of the maximum value of the luminous efficacy of radiation, but also widely in the solution of many problems of color engineering such as the computation of optimal colors, theoretical limits for the deviation from metamerism, and the spectral distribution of an illuminant having prescribed properties. (See Notes 6.4, 6.9, 7.1)

NOTE 6.4 METHOD FOR CALCULATING OPTIMAL COLORS As an alternative to the method of MacAdam (1935), the problem of obtaining the region of optimal colors can be solved by a method similar to that used in solving the maximum value of luminous efficacy. More specifically, the locus of optimal colors at a certain value of luminance factor Ys is obtained by first fixing the chromaticity coordinate x to a certain value xs . Then, the maximum and minimum values of the chromaticity coordinate y are obtained at that point. Let the spectral reflectance be Ri (i = 1 n), and, by normalizing the spectral distribution Pi of the illuminant by n i =1

Pi y i = 100

(6.65)

216


the conditions for Y = Ys and x = xs are obtained as n

Ri P i y i = Ys

(6.66)

i =1 n i =1

Ri P x i /

n

Ri Pi x i + y i + z i = xs

(6.67)

i =1

Similarly to Equation 6.64, Equation 6.67 can be rewritten as n

Ri Pi 1 − 1/xs x i + y i + z i = 0

(6.68)

i =1

By using Equations 6.66 and 6.67, y is obtained as: y = Y/X + Y + Z = Ys /

n

Ri Pi x i + y i + z i

(6.69)

i =1

= xs Ys /

n

Ri · Pi · x i

i =1

Thus, by obtaining the minimum and maximum values for the following object function X=

n

Ri P i x i

(6.70)

i =1

for the unknown Ri (i = 1 n) under the conditions of 0 Ri 1 and the restriction of Equations 6.66 and 6.68, and substituting the resulting values of Ri into Equation 6.69, the maximum and minimum values for y can be obtained. The solution for this problem can be found easily again by linear programming, because the constraints and the object function are all linear. By sequentially changing the value of xs , the total locus of optimal colors at Y = Ys can be obtained.

NOTE 6.5 METHOD FOR OBTAINING FUNDAMENTAL SPECTRAL RESPONSIVITIES The color matching functions obtained by additive color mixing experiments include negative portions, and their shape depends on the basic stimuli that are used in the experiments. Thus, these color

217

FUNDAMENTAL SPECTRAL RESPONSIVITIES

matching functions cannot be regarded as the spectral responsivities of the receptors (cones). The spectral responsivities of the three types of cones are called fundamental spectral responsivities (or ‘fundamentals’), and it is assumed that they can be related to the CIE color matching functions by some kind of conversion equation. The primary colors (fundamental primary colors) corresponding to these fundamental spectral responsivities can match all real colors by the mixing of positive amounts, and are therefore located outside the xy chromaticity diagram. The chromaticity coordinates of these fundamental primary colors cannot be obtained directly, but they can be postulated from the color matching results of people having abnormal color vision (color blindness). As illustrated in Figure 6.29, a color-blind person having ‘red’ and ‘green’ cones (more correctly called long- and medium-wavelength cones respectively), but lacking ‘blue’ cones can distinguish all the colors such as [C1 ] and [C2 ] located on the straight line RG in which red light [R] and green light [G] are mixed. However, this person cannot distinguish, for example, the colors falling on the straight line in which [C1 ] and [B1 ] are mixed or the straight line in which [C2 ] and [B2 ] are mixed. All the colors located on straight lines B1 C1 , B2 C2 , etc., appear the same to such a colorblind person. Thus, those colors are called confusion colors, and the straight lines are called confusion lines. By obtaining the confusion lines for a plurality of colors [C1 ], [C2 ], etc., the chromaticity coordinates for the fundamental primary color [B] can be obtained as the cross point. Color matching experiments have been performed, based on this theory of color blindness, on people having abnormal color vision. More specifically, confusion lines have been obtained for people

[R]

[C2] [C1] [G] [B2]

[B1]

[B]

Figure 6.29 Confusion lines C1 B and C2 B

218


with three different types of color blindness, labelled P, D and T. Typical results are shown in Figure 6.30. The three types of color blindness P, D and T are assumed to be due to a lack of long- (‘red’), medium- (‘green’) and short- (‘blue’) wavelength photoreceptors, respectively. As an example, Pitt obtained the following chromaticity coordinates for the fundamental primaries. xR = 0747

yR = 0253

zR = 0000

xG = 1081

yG = −0081

zG = 0000

xB = 0180

yB = 0000

zB = 0820

(6.71)

Conversion from the XYZ colorimetric system to the RGB system can be performed in accordance with an equation analogous to Equation 3.47 in Note 3.2 as −1  −1     X xR xG xB R SR 0 0  G  =  0 SG 0  s yR yG yB   Y  Z B zR zG zB 0 0 SB 

(6.72)

520

0.8

540

560

0.6 500

y

580

0.4

600 620 650 nm (P)

0.2 480 470

380

450

0

0.2

0.4 x

0.6

0.8

Figure 6.30 Confusion lines for three types of abnormal color vision (P, D and T respectively) (Wyszecki and Stiles 1982). Reproduced by permission of Wiley

219

FUNDAMENTAL SPECTRAL RESPONSIVITIES

520

0.8

540

560

0.6 500

y

580

0.4

600 620 650 770 nm

0.2

470 450

0

(D)

380

0.2

0.4

0.6

0.8

x

520

0.8

540

560

0.6 500 580

y

0.4

600 620 650 770 nm

0.2 480 470 450

0

(T) 0.2

Figure 6.30 (Continued)

0.4 x

0.6

0.8

220


Thus, by substituting the values of Equation 6.71, we obtain 

  R SR G  =  0 0 B  SR = 0 0

−1  −1   0747 1081 0180 X 0 0 SG 0   0253 −0081 0   Y  0 0 0820 Z 0 SB −1    02425 32365 −00532 X 0 0 SG 0   07575 −22365 −01663   Y  (6.73) 0 0 012195 X 0 SB

Normalizing by setting (R G B) = (X Y Z ) = (1,1,1), we obtain 1 = 02425 + 32365 − 005321/SR 1 = 07575 − 22365 − 016331/SG

(6.74)

1 = 0 + 0 + 121951/SB and SR = 02919

SG = −06089

SB = 08200

(6.75)

By substituting these values into Equation 6.73, we obtain the following result 

   R 02919 0 0 G  =   0 −06089 0 B 0 0 08200    02425 32365 −00532 X  −07575 −22365 −0163   Y  0 0 12195 Z    00708 09447 −00155 X =  −04612 13618 01013   Y  0 0 10000 Z

(6.76)

Since Equation 6.76 also holds for color matching functions, the fundamental spectral responsivities, PR (), PG () and PB (), are obtained as PR = 00708 x + 09447 y − 00155 z PG = −04612 x + 13618 y + 01013 z PB = 10000 z

(6.77)

221

DEDUCING VON KRIES’ EQUATION

2.0 PB(λ)

Spectral responsivity

1.5 PG(λ) 1.0 PR(λ) 0.5

0

400


700

Figure 6.31 Fundamental spectral responsivities, PR (), PG () and PB ()

Figure 6.31 shows these responsivities, PR (), PG (), and PB (). It can be seen that PR () overlaps with PG () over quite a large area, emphasizing that the notations R, G and B, implying ‘red’, ‘green’ and ‘blue’, respectively are not very accurate. This is why many people now prefer the notations L, M and S, implying ‘long’, ‘medium’ and ‘short’ wavelengths, respectively. The spectral responsivities shown in Figure 2.3 were obtained by the method described in this Note. Several other methods have been used to derive sets of fundamental responsivities, all of them giving broadly similar results, but differing in detail. One of the most recent and reliable sets is that derived by Stockman and Sharpe (1998, 2000).

NOTE 6.6 DEDUCING VON KRIES’ PREDICTIVE EQUATION FOR CHROMATIC ADAPTATION Instead of CIE tristimulus values X Y and Z consider the responses of the three photoreceptors (cones) as tristimulus values. Let R0 , G0 and B0 be the tristimulus values of a test illuminant in this system, and R G and B be those of a test object viewed under the test illuminant. Similarly, let R0 , G0 and B0 be the tristimulus values of a reference illuminant, and R G and B be those of the same test

222


object viewed under the reference illuminant. According to the von Kries hypothesis, the ratios R/R0 etc will remain constant. Thus R/R0 = R /R0 G/G0 = G /G0

(6.78)

B/B0 = B /B0 By expressing Equation 6.78 in matrix form, the following is obtained     R R (6.79) K  G  = K  G  B B where



 1/R0 0 0 K =  0 1/G0 0  0 0 1/B0



 1/R0 0 0 K =  0 1/G0 0  0 0 1/B0

(6.80)

The tristimulus values R G and B can be obtained by a linear conversion of the tristimulus values X Y and Z and are expressed by         X R X R  G  = M  Y   G  = M  Y  (6.81) B Z B Z where M is a conversion matrix, and the elements depend on the tristimulus values of the three fundamental primaries. By combining Equations 6.79, 6.80, and 6.81, the following can be obtained     X X (6.82) KM  Y  = K M  Y  Z Z By multiplying both sides of Equation 6.82 by the inverse of (K M, i.e., (M−1 K −1 , the following relation can be obtained     X X  Y  = M−1 K −1 KM  Y  Z Z   X = M−1 DM  Y  (6.83) Z

223

APPLICATION OF VON KRIES’ EQUATION

where, matrix (D) is D = K −1 K   0 0 R0 /R0   G0 /G0 0  = 0 0 0 B0 /B0

(6.84)

NOTE 6.7 APPLICATION OF VON KRIES’ EQUATION FOR CHROMATIC ADAPTATION Let us obtain a corresponding color by using standard illuminant A as the test light and standard illuminant D65 as the reference light. By using the fundamental primaries obtained by Pitt, the matrices (M and (M−1 of Note 6.7 are obtained from Equation 6.76 as 

0071 0945  M =  −0461 1362 0000 0000  2558 −1775  0133 M−1 =  0866 0000 0000

 −0016  0101  1000  0220  0000  0000

(6.85)

The tristimulus values, X0 , Y0 and Z0 for the perfect reflecting diffuser under the test illuminant, and X0 , Y0 and Z0 for the perfect reflecting diffuser under the reference illuminant are X0 = 1098

X0 = 950

Y0 = 1000

Y0 = 1000

Z0 = 350

Z0 = 1089

(6.86)

By substituting these values into Equation 6.81, the following values are obtained R0 = 101682

R0 = 95500

G0 = 88976

G0 = 103190

B0 = 35500

B0

= 108900

(6.87)

224


By substituting these into Equation 6.84, the following relation is obtained   99500/101682 0 0  0 103190/88976 0 D =  0 0 108900/35500   0979 0 0  0 1160 0 = (6.88) 0 0 3068 Thus, the tristimulus values for the corresponding color, X Y and Z can be obtained from Equation 6.83 as     X X  Y  = M−1 DM  Y  X Z    2558 −1775 0220 0979 0 0  0133 0000   0 1160 0 =  0866 0 0 1000 0 0 3068    0071 0945 −0016 X ×  0071 0945 −0016   Y  Z 0 0 1000    1127 −0438 0427 X 1011 0002   Y  =  −0011 (6.89) 0 0 3068 Z 0.6 560 580

590

Y

v' 0.5

G R A D65

B 0.4 0.1

P 0.2

0.3 u'

Figure 6.32 Corresponding colors between test illuminant A and standard illuminant D65 (Nayatani 1980). Reproduced by permission of Asuara Shoten

225

CIE 1994 CHROMATIC ADAPTATION TRANSFORM

Figure 6.32 shows, in the u v chromaticity diagram, the relationship of various test colors and the corresponding colors computed by Equation 6.89.

NOTE 6.8 APPLICATION OF CIE 1994 CHROMATIC ADAPTATION TRANSFORM Let us obtain a corresponding color by using standard illuminant A as the test illuminant and standard illuminant D65 as the reference, with E0 = E0 = 1000 lx

0 = 0 = 02

(6.90)

The chromaticity coordinates x y for the test light and x y for the reference light are x = 04476

x = 03127

y = 04074

y = 03290

(6.91)

Thus, from Equation 6.23, the following can be obtained = 11186

= 10000

= 09330

= 10000

= 03268 = 10000

(6.92)

By substituting these values into Equation 6.25, the following results can be obtained R0 = 7121

G0 = 5939

B0 = 2081

R0 = 6366

G0 = 6366

B0 = 6366

(6.93)

When 0 = 0.2, Equation 6.26 reduces to K = 1, and Equation 6.21 becomes R = 08481 R + 110183 − 10000 G = 11050 G + 109866 − 10000

B = 43256 B + 1

07823

(6.94)

− 10000

As an example, let us consider again, as in Section 6.5, the case of the corresponding color (under D65 of an object color having

226


tristimulus values of X = 2800, Y = 2126 and Z = 527 under illuminant A. From Equation 6.20, we obtain      2800 040024 070760 −008081 R  G  =  −022630 116532 004570   2126  527 0 0 091822 B   258244 =  186791  (6.95) 48390 Then, by substituting this result into Equation 6.94, the following results can be obtained: R = 08481 258244 + 110183 − 10000 = 231612 G = 11050 186791 + 109866 − 10000

(6.96)

= 198943 B = 43256 48390 + 107823 − 10000 = 162010 By substituting these values into Equation 6.22, the correspondisng color is      X 185995 −112939 021990 231612  Y  =  036119   198943  063881 0 0 0 108906 162010 Z   241729 =  210743  (6.97) 176439 These are the values given in Equations 6.27 of the main text.

NOTE 6.9 THEORETICAL LIMITS FOR DEVIATION FROM METAMERISM Let us consider the deviation from metamerism when an illuminant P () is changed to a second illuminant P = P, using a method proposed by Ohta and Wyszecki (1975). Suppose the spectral reflectance of an object color is R and the chromaticity coordinates and luminance factor under illuminant P are

THEORETICAL LIMITS OF METAMERISM

227

x = xg y = yg , and Y = Yg , respectively. There are many possible values of R that satisfy this condition of metamerism. However, under P , the colors will no longer match. To make the omputation simple, the spectral distributions, are normalized n

Pi y i =

i =1

n i =1

Pi y i = 100

(6.98)

These constrains of metamerism (i.e., equality of xg , yg and Yg can be written, as in Notes 6.3 and 6.4, as n

Ri Pi 1 − 1/xg x i + y i + z i = 0

i =1 n

Ri Pi x i + 1 − 1/yg y i + z i = 0

(6.99)

i =1 n

Ri Pi y i = Yg

i =1

If Y = Ys and x = xs under illuminant P (), the constraints can be written as n i =1 n i =1

Ri Pi y i = Ys Ri Pi 1 − 1/xs x i + y i + z i = 0

(6.100)

Then, since x = X/(X + Y + Z), y can be obtained as y = Y/X + Y + Z = xs · Ys /X

(6.101)

Thus, by selecting the following object function with unknown Ri (i = 1, n X=

n i =1

Ri Pi x i

(6.102)

and obtaining the minimum and the maximum values under the restriction of Equations 6.99 and 6.100, the maximum and the minimum values for y can be obtained from Equation 6.101. Because all the constraints and the object function are linear

228


with respect to the unknown Ri (i = 1 n), this problem also can be solved by applying linear programming. The limits of deviation from metamerism can thus be obtained by computing the maximum and minimum values of y while sequentially changing the values of xs and Ys . The results shown in Figure 6.25 were calculated by this method.

7 Application of CIE Standard Colorimetric System The basis of the CIE standard colorimetric system was established in 1931, and as described in Chapters 4 and 6, further evolution concerning uniform color spaces and the like occurred, based on the 1931 system and its 1964 counterpart. The present chapter describes, as examples of applications, the methods widely applied in the industrial field for evaluating the color rendering of light sources, artificial daylight sources, metamerism, and whiteness. The present chapter also describes a method of designing the spectral distribution of illumination, and the principles of computer color matching of object colors.

7.1 EVALUATION OF THE COLOR RENDERING PROPERTIES OF LIGHT SOURCES When an object is viewed under illuminants of different spectral distributions, as described in Section 6.4, the perceived color remains approximately constant because of the phenomenon of color


230

APPLICATION OF CIE STANDARD COLORIMETRIC SYSTEM

constancy. However, because color constancy is not perfect, the color appearance differs slightly from one illuminant to another. This effect of different illuminants (test lights) on the color appearance of object colors is called color rendering. Daylight or the light emitted from an incandescent lamp, which are common illuminants in daily life, are usually used as standards, with which the color rendering properties of other lights are compared. The better the color appearance of objects match, the higher the color rendering properties of an illuminant are judged to be. However, as in all industrial applications, it is helpful to establish a quantitative evaluation method. Methods for evaluating color rendering properties can be roughly classified into methods based on the difference in spectral distribution and methods based on the difference in color appearance of a series of object colors (test colors) used as representative colors. At present, the second method is generally used. In particular, the CIE evaluation method for color rendering (CIE 1995) is normally used. The CIE method, which is described below, uses a color rendering index that expresses numerically the degree of matching of color appearance under a test light source with the color appearance under a reference illuminant.

7.1.1 Reference Illuminant For most purposes, a black-body radiator or one of the series of CIE daylight illuminants defined in Section 3.8 is used as the reference illuminant. As a rule, when the correlated color temperature of the test source is lower than 5000 K, a black-body radiator is used, and, when the correlated color temperature of the test source is 5000 K or higher, a CIE daylight illuminant is used. In general, in either case, the reference illuminant used in the evaluation is one whose correlated color temperature is equal to that of the test source. For special purposes, however, a CIE standard illuminant or any other illuminant can be used. Although the color temperature of the test source can be computed to acceptable accuracy from its chromaticity coordinates (see Note 3.7), a graphical method based on Figure 7.1 can also be used.

7.1.2 Test Colors In the computation of the CIE color rendering index, 15 colors (numbered from 1 to 15) are used as the test colors. The Munsell

231

COLOR RENDERING PROPERTIES OF LIGHT SOURCES

0.36

P

50

0.34

00

0

0 28056 28

0

320

300

0

00

36

00

34

0.28

50

00 45

0.30

00 40

v 0.32

60 62 00 65 50 00 70 00 75 0 80 0 85 00 00 90 00 100 00

260

00

K 2300 2400

55

00

110 120 00 00

150 0 170 0 00 200 00 2500 0

D 0.18

0.20

0.22

0.24

0.26

0.28

u

Figure 7.1 Black-body locus (P) and daylight locus (D) in uv chromaticity diagram

notations, the color names, and the spectral reflectances are given in Table A5 in the Appendix. The spectral reflectances are also shown in Figure 7.2 and the colors are illustrated in Color Plate 4. Test colors 1–8, which have Munsell notations, V/C in the range from 6/4 to 6/8, are colors with medium lightness and medium saturation, and represent average object colors. They are used to obtain an average color rendering index, known as the CIE General Color Rendering Index. Test colors 9–14 are used to obtain special color rendering indices and represent examples of important object colors. Test colors 9–12 are representative high saturation red, yellow, green, and blue colors respectively. Test colors 13 and 14 represent the skin color of ‘white’ Caucasian people and the color of a typical green leaf, respectively. Test color 15 is standardized only in the Japanese version of the method and represents the average face color of a Japanese woman. For special purposes, other test colors, described by their spectral reflectance, can be used.

232


0.8

8

0.6

Reflectance

1 0.4

2

0.2

3 4

7 6 5

0 9

0.8

13

10

15

0.6 0.4 0.2 0

14

11 12 400

500

600


600

700

Figure 7.2 Fifteen test colors for use in computation of color rendering index

7.1.3 Color Space and Chromatic Adaptation Correction For the computation of color difference, the following colorimetric values U ∗ V ∗ and W ∗ according to the CIE 1964 uniform color space (see Section 4.3) are used W ∗ = 25 Y 1/3 − 17 U ∗ = 13 W ∗ u − un ∗

(7.1)

∗

V = 13 W v − vn where, u and v represent the chromaticity coordinates in the CIE 1960 UCS chromaticity diagram, and are obtained from the tristimulus values X Y and Z or from the chromaticity coordinates x and y according to u = 4X/X + 15Y + 3Z = 4x/−2x + 12y + 3

(7.2)


233

v = 6Y/X + 15Y + 3Z = 6y/−2x + 12y + 3 In Equation 7.1, un and vn are the u,v chromaticity coordinates of the illuminant. Use of the CIE 1964 uniform color space is discontinued for most purposes now. Accordingly, in theory, its use in the evaluation of color rendering could be replaced by the CIE 1976 L ∗ a ∗ b ∗ color space or the CIE 1976 L ∗ u ∗ v ∗ color space. Similarly, the CIE 1960 UCS chromaticity diagram could be replaced by the CIE 1976 UCS chromaticity diagram. However, little change is likely to occur in the color rendering index by this change so it has not been adopted by the CIE. In general, the chromaticity coordinates of the test source (indicated below by the subscript k) do not match the chromaticity coordinates of the reference illuminant (indicated below by the subscript r). However, as discussed in Chapter 6, this deviation is compensated by chromatic adaptation. To account for this adaptive color shift, the chromaticity coordinates uk and vk of the test source and the chromaticity coordinates uki and vki for each test color (indicated by subscript i ) under the test source are corrected by the following equations to obtain uk and vk , and uki and vki , respectively uk = ur vk = vr uki = 10872 + 0404cr cki /ck − 4dr dki /dk / 16518 + 1481cr cki /ck − dr dki /dk vki = 5520/16518 + 1481cr cki /ck − dr dki /dk

(7.3)

where: uk and vk are the chromaticity coordinates of the test source after applying the chromatic adaptation correction; ur and vr are the chromaticity coordinates of the reference illuminant; uki and vki are the chromaticity coordinates of the test colors after applying the chromatic adaptation correction; cr and dr are coefficients computed from the chromaticity coordinates ur and vr according to Equation 7.4; ck and dk are coefficients computed from the chromaticity coordinates uk and vk according to Equation 7.4; cki and dki are coefficients computed from the chromaticity coordinates uki and vki according to Equation 7.4

234


Coefficients c and d are computed by c = 4 − u − 10v/v d = 1708v + 0404 − 1481u/v

(7.4)

These equations represent a von Kries type of transformation (see Section 6.5).

7.1.4 Computation of Color Rendering Index Color rendering indices include a general color rendering index, Ra and special color rendering indices, Ri (i =1–15) for each of the test colors. Among the special color rendering indices, R9 –R15 are often given priority over the others in practical applications. To compute the indices, the tristimulus values X ri Yri and Zri , as well as Xki Yki and Zki are obtained from the spectral distribution Sr () and Sk (), respectively, where i = 1, 2, … , 15. Then, the chromaticity coordinates uri and vri as well as uki and vki are obtained by Equation 7.2. Then, the chromaticity coordinates uki and vki are converted into uki and vki to account for the adaptive color shift. By substitution into Equation 7.1, the following results are obtained Wri ∗ = 25Yri 1/3 − 17 Uri ∗ = 13Wri ∗ uri − ur Vri ∗ = 13Wri ∗ vri − vr

(7.5)

Wki ∗ = 25Yki 1/3 − 17 Uki ∗ = 13Wki ∗ uki − uk Vki ∗ = 13Wki ∗ vki − vk The color differences Ei (i = 1, 2, … , 15) according to the CIE 1964 uniform color space are obtained from these values in accordance with Ei = Uri ∗ − Uki ∗ 2 + Vri ∗ − Vki ∗ 2 + Wri ∗ − Wki ∗ 2 1/2

(7.6)

The special color rendering indices Ri for each of the test colors are obtained by Ri = 100 − 46Ei

(7.7)

235


The value 4.6 used as the coefficient in Equation 7.7 was introduced by the CIE so that the general color rendering index of a warm white fluorescent lamp would be Ra = 50. The general color rendering index Ra is an average value of the special color rendering indices for test colors 1–8, and is obtained by Ra =

8

i =1

Ri /8

(7.8)

Figure 7.3 shows the spectral distributions of three fluorescent lamps. The color rendering index for each of the lamps has been computed and is given in Table 7.1. From a theoretical point of view, there are several improvements that could be made to the procedure for calculating the CIE color rendering index. For example, a better chromatic adaptation transform could be incorporated. However, such improvements would cause changes in ratings that have become well established in practice so, to date, the CIE has not recommended any changes.

15

30

(a)

Spectral radiant power

10

(c)

25

5

20

0

15 (b)

10

10

5

5

0

400

500

600


500

600

700

Figure 7.3 Spectral distribution of representative fluorescent lamps (Mori 1987). (a) Daylight type used in 1940s (b) Daylight type (solid line) and white color type (broken line) (c) High-color-rendering daylight white color type (solid line) and three-band daylight white type (broken line). Reproduced by permission of Asuara Shoten

Correlated color temperature 5690 K

6350 K 4150 K 5010 K

5000 K

Chromaticity x y

0.328 0.335

0.314 0.345 0.378 0.388

0.345 0.351

0.346 0.359

Type of lamp

Daylight type in 1940s (measured on a preserved product) Daylight type White color type High color rendering daylight white type Three wavelength daylight white type P-5000

P-5000

D-6250 P-4100

D-5700

Reference

81.0

98.1

71.3 58.6

95.0

Ra

27

42

97

54 35

−70 −107 98

88

R10

79

R9

67

98

60 37

96

R11

51

96

69 43

91

R12

93

97

67 53

98

R13

69

99

94 93

94

R14

94

97

52 38

96

R15

Table 7.1 Color rendering indices of representative fluorescent lamps (Mori 1987). Reproduced by permission of Asuara Shoten

EVALUATION OF DAYLIGHT SIMULATORS

237

7.2 EVALUATION OF THE SPECTRAL DISTRIBUTION OF DAYLIGHT SIMULATORS The color rendering properties of general-purpose light sources are evaluated by comparing the color appearance of objects viewed under the test light with the color appearance of the same objects viewed under a reference light. However, the matching of the spectral distribution of a light source to a standard is important in other specialized applications, such as, for example, the evaluation of color matching, metamerism, and fluorescent materials. In particular, the CIE daylight illuminants, D50 , D55 , D65 and D75 cannot be reproduced exactly in practice and so-called daylight simulators are used as an alternative. Accordingly, the degree of matching of the spectral distribution of daylight simulators must be evaluated relative to the standard. The CIE has established a method (CIE 1999) for doing this based on the CIE metamerism index for change of illuminant (see Section 7.4). Two different indices are used, one to evaluate the visible region of the spectrum, the other to evaluate the ultraviolet region. The color matching functions of the X10 Y10 Z10 system are used, and the spectral distribution of the daylight simulator must be measured with a wavelength interval of 5 nm in a wavelength region 300–700 nm.

7.2.1 Evaluation of Spectral Distributions in the Visible Region The spectral distribution of a daylight simulator is evaluated in the visible region by its chromaticity coordinates and its metamerism index for five pairs of object colors that are metameric under the standard illuminant. First, the chromaticity coordinates u10 and v10 , obtained from the spectral distribution S() of the daylight simulator are required to be within a circle having a radius of 0.015 drawn around the chromaticity coordinates of the standard illuminant. This range, transformed to the x10 y10 chromaticity diagram is given in Figure 7.4. Second, the visible-range metamerism index is obtained from the color differences that the five sets of metameric pairs show under the test simulator. Table A6 in the Appendix gives the spectral reflectances of the five pairs. One member of each pair (the standard color for the metamerism index calculation) is the same for all four standard illuminants (D50 , D55 , D65 and D75 , whereas the other member (the test color) is different for each illuminant. Figure 7.5 gives, as an example, the spectral reflectances of the five metameric

238


0.400

0.350

D55 D65

y10 D75

0.300

0.250

0.300

0.350 x10

Figure 7.4 Allowable range of chromaticity coordinates of daylight simulators

Spectral reflectance

1.0

Set 1

Set 2

Set 3

0.5

0 400

500

600

700


500

600

700

1.0 Spectral reflectance

Set 4

Set 5

0.5

0 400

500


600

700

Figure 7.5 Spectral reflectances of five metameric pairs used in the evaluation of D65 simulators in the visible range


239

pairs for D65 . Solid and broken lines represent the standard and test colors, respectively. The metamerism index Mv is obtained from the following equation as the average of the five color differences Ei (i = 1, 2, , 5) Mv =

5

i =1

Ei /5

(7.9)

As a rule, the L ∗ a ∗ b ∗ color system is used for the computation of the color difference.

7.2.2 Evaluation of Spectral Distributions in the Ultraviolet Region The spectral distribution of a daylight simulator in the ultraviolet region is evaluated by using a metamerism index obtained from the color difference exhibited by the three sample pairs given in Tables A7 and A8 in the Appendix. The samples shown in Figure 7.6 (from Table A7) fluoresce when excited by radiation in the ultraviolet region, and the total spectral radiance factors under standard illuminant D65 , for example, match the spectral radiance factors of the non-fluorescent samples given in Table A8 and Figure 7.7. To obtain the total spectral radiance factors under the simulator, the values of the spectral distribution of the simulator S() are normalized according to Sn = 100S/

700

Sy 10

(7.10)

=400

where Sn () is the normalized relative spectral distribution of the simulator; y 10 () is the color matching function for the tristimulus value, Y10 in the X10 Y10 Z10 system, and is the wavelength interval for the summation ( = 5 nm). The total spectral radiance factor T () for the fluorescent sample under the simulator can be obtained according to

T = S + NF/Sn

(7.11)

where S () is the reflected spectral radiance factor, F () is the spectral distribution of the fluorescent component, normalized to

240



0.10 0.08

F1(λ) = F2(λ) = F3(λ)

0.06 0.04 0.02

Spectral radiance factor

0 400

Excitation efficiency

1.0

1.0

500 Wavelength λ (nm)

β1S(λ) β2S(λ)

0.5

β3S(λ)

0 400

0.5

Q1(λ′) Q2(λ′) Q (λ′) 3

0 300

600

400 Wavelength λ′ (nm)


700

500

Figure 7.6 Spectral properties of fluorescent samples used in the evaluation of daylight simulators in the ultraviolet range

Spectral radiance factor

1.0

0.5

0 400

β1(λ) β2(λ) β3(λ)


700

Figure 7.7 Spectral radiance factors of non-fluorescent samples used in the evaluation of daylight simulators in the ultraviolet range

241


yield F () = 1.0, is the wavelength of the reflected light and the emitted fluorescent light, and N is obtained according to N=

460 =300

Sn Q

(7.12)

where Q( ) is the excitation efficiency of the fluorescence; is the wavelength interval for the summation ( = 5 nm); and is the wavelength of the excitation light. Then, the color differences Ej ( j = 1, 2, 3) between the three fluorescent samples (Table A7) and the three non-fluorescent samples (Table A8) are computed under the daylight simulator. The ultraviolet-range metamerism index Mu is obtained by the following equation 3 Mu = Ej /3

(7.13)

j=1

As a rule, the L ∗ a ∗ b ∗ color system is used for the computation of color difference.

7.2.3 Grades A daylight simulator can be graded, as is shown in Table 7.2, by using the visible-range metamerism index Mv and the ultravioletrange metamerism index Mu . The grades are expressed by writing the grade of the visible-range index and then that of the ultravioletrange index, for example, as AB. A daylight simulator graded BC or better is sufficiently useful for most practical purposes. As examples, the evaluated results for a fluorescent lamp, CIE illuminant C, and the xenon arc lamp shown in Figure 7.8 are given in Table 7.3. The L ∗ a ∗ b ∗ color system was used to obtain these results. Table 7.2 Evaluation grades of metamerism indices for daylight simulator (CIE 1999) CIELAB

CIELUV

<0.25 0.25–0.5 0.5–1.0 1.0–2.0 >2.0

<0.32 0.32–0.65 0.65–1.3 1.3–2.6 >2.6

Grade A B C D E

242



20

10

0

300

400


700

Figure 7.8 Spectral distribution of fluorescent lamp (solid line), CIE illuminant C (broken line), and xenon lamp (dotted line) used as daylight (D65) simulators (Mori et al. 1983)

Table 7.3 Metamerism indices for representative daylight (D65 ) simulators (Mori 1983) Illuminant

MIv

MIu

Grade

Fluorescent lamp CIE illuminant C Xenon arc

0.34 0.27 0.40

0.32 3.28 1.18

BB BE BD

7.3 EVALUATION OF WHITENESS Whiteness is an important factor in the evaluation of certain papers and textiles. Thus, it is useful to develop an index of whiteness. It is generally assumed, in the development of such indices, that degree of whiteness approximately follows a yellow–blue line in the x ,y chromaticity diagram. In other words, increasing whiteness corresponds to decreasing yellowness. In addition, except for purely white colors, such materials often have a greenish or purplish tint that can be measured by the degree of departure from an ‘untinted’ yellow–blue line. Accordingly, a whiteness index is sometimes supplemented by a tint index. Methods for evaluating whiteness and tint can be roughly classified as:

243

EVALUATION OF WHITENESS

1. Methods based on the spectral reflectance of the test object at one specified wavelength (e.g., 457 nm) or two specified wavelengths (e.g., 450 and 700 nm) 2. Methods based on colorimetric values, such as the tristimulus values X Y and Z ; or the luminous reflectance, Y and chromaticity coordinates, x , y. The evaluation method recommended by the CIE falls into the second category. The CIE whiteness index W or W10 is computed (CIE 2004a) according to W = Y + 800xn − x + 1700yn − y W10 = Y + 800xn10 − x10 + 1700yn10 − y10

(7.14)

where Y is the tristimulus value Y of the test object in the XYZ colorimetric system; x y are the chromaticity coordinates of the test object in the XYZ colorimetric system; xn , yn are the chromaticity coordinates of the perfect reflecting diffuser in the XYZ colorimetric system; Y10 is the Y tristimulus value of the test object in the X10 Y10 Z10 colorimetric system; x10 , y10 are the chromaticity coordinates of the test object in the X10 Y10 Z10 colorimetric system; and xn10 , yn10 are the chromaticity coordinates of the perfect reflecting diffuser in the X10 Y10 Z10 colorimetric system. The CIE tint index TW or TW10 is obtained (CIE 2004a) by TW = 1000xn − x − 650yn − y TW10 = 900xn10 − x10 − 650yn10 − y10

(7.15)

These indices, defined by Equations 7.14 and 7.15 are applicable only to test objects whose whiteness index W or W10 , and tint index TW or TW10 are in the ranges of 40 < W < 5Y − 280 −30 < TW < 30

40 < W10 < 5Y10 − 280 − 30 < TW10 < 30

(7.16)

In other words, they apply only to ‘near-white’ objects. These equations are only approximations, and the values thus obtained do not match perfectly with practical psychological evaluation results. Hence, this is a subject for further study.

244


7.4 EVALUATION OF DEGREE OF METAMERISM FOR CHANGE OF ILLUMINANT As described in Section 6.9, a color match between two objects of different spectral reflectance breaks down when the viewing conditions change. The degree of deviation from matching is known as the degree of metamerism, and can be expressed by a metamerism index. The deviation from matching can occur either when the spectral distribution of the illuminant changes or when the color matching functions of the observer change. In either case, the degree of metamerism can be evaluated by applying the CIE colorimetric system. The metamerism index in the former case, in which the illuminant is changed, is called the CIE ‘special metamerism index: change in illuminant’ (CIE 2004a), sometimes abbreviated as the ‘illuminant metamerism index’. The index is obtained as follows.

7.4.1 Reference Illuminant As a rule, standard illuminant D65 is employed as the reference illuminant. When an illuminant other than D65 is used, it must be selected from the CIE standard and supplementary illuminants, and must be clearly described. Generally, standard illuminant A or a light from a representative fluorescent lamp having one of the spectral distributions shown in Figure 7.9 or Table A9 in the Appendix is used as the test light. The distributions marked with an asterisk, i.e., F2, F7, and F11 are preferred by the CIE. However, according to the JIS, F6, F8, and F10 are preferred.

7.4.2 Tristimulus Values of Metameric Pairs The tristimulus values (Xr1 , Yr1 , Zr1 ) and (Xr2 , Yr2 , Zr2 under the reference light (r) and those (Xt1 , Yt1 , Zt1 and (Xt2 , Yt2 , Zt2 under the test light (t) are computed in the usual way for samples 1 and 2 that constitute the metameric pair. Depending on the size of the viewing field, the color matching functions of the XYZ system or of the X10 Y10 Z10 system may be used.

245

DEGREE OF METAMERISM FOR CHANGE OF ILLUMINANT

F1 F2 F3

Spectral radiant power (µW/nm/lm)

40

30

30

20

20

10

10

0

400

500 600 (a) Standard

F4 F5 F6

40

700

0

400

500 600 (b) Standard

700

80 F7 F8 F9

40

F10 F11 F12

70 60

30

50 40

20 30 20

10

10 0 400

500 600 700 (c) High color rendering

0

400

500 600 700 (d) Three band

Wavelength (nm)

Figure 7.9 Spectral distribution of fluorescent lamps for use in the computation of the CIE metamerism index for change of illuminant

7.4.3 Correction of Tristimulus Values If the metameric pair does not match perfectly under the reference light, the tristimulus values under the test light are corrected according to Xt2 = Xt2 Xr1 /Xr2

Yt2 = Yt2 Yr1 /Yr2

Zt2 = Zt2 Zr1 /Zr2

(7.17)

where, Xt2 Yt2 , and Zt2 are the corrected tristimulus values of sample 2. If the sample pair match exactly under the reference

246


light, as a matter of course, it follows that Xt2 = Xt2 Yt2 = Yt2 , and Zt2 = Zt2 .

7.4.4 Computation of Color Difference The color difference E of the metameric pair under the test light is computed, as a rule, from the tristimulus values (Xt1 , Yt1 , Zt1 and (Xt2 Yt2 Zt2 ) using the CIE L ∗ a ∗ b ∗ color difference formula. If another color difference formula is used, this must be noted when the results are reported. The illuminant metamerism index Milm is determined as Milm = E

(7.18)

As an example of the illuminant metamerism index, consider the three object colors given in Figure 7.10 and Table 7.4. These are metameric with respect to standard illuminant D65 in the XYZ colorimetric system, and the tristimulus values (X0 Y0 Z0 ), (X1 Y1 Z1 , and (X2 Y2 Z2 ) are obtained as X0 = X1 = X2 = 425

Y0 = Y1 = Y2 = 330

Z0 = Z1 = Z2 = 151 (7.19)

1.0

1

0.8 Reflectance

0 0.6 2 0.4

0.2

0

400

500

600 Wavelength (nm)

700

Figure 7.10 Spectral reflectance of three object colors metameric under standard illuminant D65 (Wyszecki and Stiles1982). Reproduced by permission of Wiley

DEGREE OF METAMERISM FOR CHANGE OF ILLUMINANT

247

Table 7.4 Spectral reflectance of three object colors metameric under standard illuminant D65 (Wyszecki and Stiles 1982). Reproduced by the permission of Wiley Wavelength (nm)

380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 404.7∗ 435.8 546.1 577.8 ∗

Object color 0

1

1050 1200 1361 1427 1428 1409 1394 1386 1374 1368 1367 1360 1356 1377 1417 1467 1606 2032 2778 3747 4848 5735 6259 6568 6717 6818 6876 6931 6980 7040 7111 7186 7261 7314 7369 7445 7462 7482 7520 1392 1400 1866 4606

1592 1485 980 590 542 682 932 1239 1554 1907 2200 2301 2186 1926 1579 1199 985 1393 2447 3827 5258 6100 6387 6564 6690 6827 6927 7028 7120 7239 7337 7401 7506 7615 7743 7907 8009 8140 8313 797 827 1234 4943

Shows wavelengths of mercury emission lines.

2 1174 1344 1548 1614 1605 1560 1513 1464 1390 1303 1192 1037 878 784 784 873 1155 1918 3172 4697 6226 7098 7020 6310 5593 5109 4846 4763 4724 4741 4759 4781 4782 4754 4741 4771 4713 4652 4637 1579 1533 1620 5890

248


By illuminating these three objects with standard illuminant A and the three fluorescent lamps given in Table 7.5, colorimetric values shown in Table 7.6 are obtained. Then, the illuminant metamerism index is obtained as in Table 7.7. It should be noted Table 7.5 Spectral distribution of standard illuminant A and three fluorescent lamps (Wyszecki and Stiles 1982). Reproduced by permission of Wiley Wavelength (nm)

380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760

Spectral distribution A

F1

F2

F3

98 121 147 177 210 247 287 331 378 429 482 539 599 661 725 791 859 929 1000 1072 1144 1217 1290 1363 1436 1508 1580 1650 1720 1788 1854 1919 1983 2044 2104 2161 2217 2270 2321

54 56 58 61 74 106 172 265 336 383 397 398 407 436 499 574 678 823 1000 1132 1257 1129 1032 933 1098 1457 1436 2722 2965 869 356 212 124 80 52 35 20 10 02

107 120 139 168 208 280 379 488 585 644 665 670 666 677 699 732 787 884 1000 1104 1160 1153 1112 1046 1040 1049 1036 1169 1477 623 405 302 236 180 140 108 93 66 52

230 275 334 436 550 677 810 942 1046 1111 1143 1155 1142 1114 1076 1036 1010 998 1000 1011 1027 1027 1012 995 989 974 927 965 960 636 472 381 314 253 205 167 135 110 90

DEGREE OF METAMERISM FOR CHANGE OF OBSERVER

249

4047∗ 4358 5461 5778

777 1824 1008 291

272 840 777 237

423 1121 777 230

∗ Wavelengths of mercury emission lines and radiant powers that should be added to a continuum with 10 nm bandwidth.

Table 7.6 Colorimetric values of three metamers under test lights A, F1, F2, and F3 (Wyszecki and Stiles 1982). Reproduced by permission of Wiley Object color

0

1

2

Color difference (CIELUV)

Colorimetric values

D65

A

F1

F2

x0 y0 Y0 x1 y1 Y1 x2 y2 Y2 Euv (0, 1) Euv (0, 2)

0.4691 0.3643 33.0 0.4691 0.3643 33.0 0.4691 0.3643 33.0 0.0 0.0

0.5680 0.3847 40.25 0.5683 0.3810 40.23 0.5592 0.3941 40.36 2.7 11.2

0.5580 0.3877 39.73 0.5631 0.3847 39.36 0.5458 0.3991 40.34 4.7 13.8

0.5188 0.3872 38.86 0.5233 0.3856 36.55 0.5140 0.3928 37.88 3.3 5.1

F3

0.4695 0.3680 33.07 0.4743 0.3699 32.76 0.4691 0.3678 33.27 2.1 0.2

Table 7.7 Illuminant metamerism index for two metameric pairs using four different test illuminants. Reproduced by permission of Wiley Milm

Pair (0,1)

Pair (0,2)

MA MF1 MF2 MF3

2.7 4.7 3.3 2.1

112 138 51 02

that the color difference in this example is obtained in the CIE L ∗ u ∗ v ∗ color system.

7.5 EVALUATION OF DEGREE OF METAMERISM FOR CHANGE OF OBSERVER The color matching functions of the CIE standard colorimetric observers are the average of the color matching functions of many observers having normal color vision. Accordingly, the color

250


z10(λ)

Spectral tristimulus values

2.0

1.5 x10(λ) y10(λ) 1.0

0.5

0

400


700

Figure 7.11 Variations in color matching functions of 20 observers (Stiles and Burch 1959). Reprinted from ‘N. P. L. color matching investigation’ by W. S. Stiles and J. M. Burch from Optica Acta, 1959, 6, 1–26, with permission from Taylor and Francis Ltd

matching functions of any single real observer deviate individually from that of the standard. Figure 7.11 shows that there are considerable variations in the color matching functions of 20 observers measured in a 10 visual field (Stiles and Burch 1959). Accordingly, a pair of colors that are metameric for a reference observer may deviate from a match for another observer. The metamerism index in this case, in which the observer is changed, is called the CIE ‘special metamerism index: change in observer’ (CIE 1989), sometimes abbreviated as ‘observer metamerism index’. A virtual observer who provides the second observer for this index is called a standard deviate observer. The standard deviate observer used by the CIE predicts the average deviation of a plurality of observers having normal vision. In addition to the observer metamerism index, the CIE evaluation method also includes an ellipse that illustrates the range of chromaticity variations among normal observers (reliable metamerism ellipse) and the age dependence of observer metamerism (age metamerism index). However, the procedure of computing the observer metamerism index, which is most frequently used, is explained below (CIE 1989).


251

7.5.1 Color Matching Functions The color matching functions x d (), y d (), and z d () of the standard deviate observer are obtained according to x d = x s + x y d = y s + y

(7.20)

z d = z s + z where x s (), y s () and z s () are the color matching functions of a standard observer (either the XYZ colorimetric system or the X10 Y10 Z10 colorimetric system is used, depending on the size of viewing field); x (), y () and z() are the deviation functions defined in Table A10 in the Appendix. Figure 7.12 shows, as an example, the color matching functions of the standard observer and those of the standard deviate observer in the X10 Y10 Z10 colorimetric system.

z10(λ)

Spectral tristimulus values

2.0

1.5 x10(λ) y10(λ)

1.0

0.5

0

400


700

Figure 7.12 Color matching functions for standard observer (solid line) and standard deviate observer (dotted line) (10 viewing field)

252


7.5.2 Illuminant In the computation of observer metamerism index, as a rule, standard illuminant D65 is used. The spectral distribution S () of the illuminant is normalized according to Sn = 100S/

780

S y

(7.21)

=380

where Sn () is the normalized spectral distribution of the illuminant, y () is the color matching function for the tristimulus value Y or Y10 of the standard observer, and is the wavelength interval of the summation.

7.5.3 Tristimulus Values of Metameric Pairs The tristimulus values, Xri , Yri and Zri , with respect to the reference observer (r) and Xdi , Ydi and Zdi with respect to the deviate observer (d) are computed in the usual way for samples, i = 1, 2 that constitute the metameric pair. Color matching functions x s (), y s () and z s () of the reference observer (standard observer) are used for the computation of tristimulus values Xri , Yri and Zri , and color matching functions x d (), y d () and z d () of the standard deviate observer are used for the computation of tristimulus values Xdi , Ydi , and Zdi .

7.5.4 Correction of Tristimulus Values If the metameric pair is not completely matched with respect to the reference observer, the tristimulus values for the standard deviate observer are corrected according to Xd2 = Xd2 Xr1 /Xr2

Yd2 = Yd2 Yr1 /Yr2

Zd2 = Zd2 Zr1 /Zr2

(7.22)

where, Xd2 Yd2 , and Zd2 are the tristimulus values for sample 2 corrected for the standard deviate observer. If the sample pair is color matched for the reference observer, it follows that Xd2 = Xd2 , Yd2 = Yd2 , and Zd2 = Zd2 .

7.5.5 Computation of Color Difference The color difference E of the metameric pair is computed for the standard deviate observer from the tristimulus values (Xd1 , Yd1 , Zd1

253


and (Xd2 Yd2 Zd2 ) using a color difference formula, normally the CIE L ∗ a ∗ b ∗ formula. Then the observer metamerism index Mobs is obtained by Mobs = E

(7.23)

where, the type of illumination use in the evaluation is attached after Mobs , and, if the X10 Y10 Z10 colorimetric system is used, a subscript 10 is attached. The observer metamerism index is graded in accordance with Table 7.8. Incidentally, about 95% of the color differences obtained for real observers fall within 2Ewhere E is the value for the standard deviate observer. An example of evaluating the observer metamerism index is described below (JIS Z 8718). The tristimulus values for sample pair 1 and 2 shown in Figure 7.13, for the color matching functions of the reference observer (X10 Y10 Z10 colorimetric system), under standard illuminant D65 are Xr1 = 3029

Yr1 = 2441

Zr1 = 4325

Xr2 = 3031

Yr2 = 2442

Zr2 = 4320

(7.24)

Table 7.8 Evaluation grades of observer metamerism index (using CIELAB) (CIE 1989 ) Observer metamerism index

Grade

0.2 >0.2 and 0.5 >0.5

A B C

Spectral reflectance

1.0 2 1 0.5

0

400


700

Figure 7.13 Spectral reflectance of two object colors metameric under standard illuminant D65 for color matching functions in the X10 Y10 Z10 colorimetric system (JIS Z 8718)

254


It can be seen that the sample pair is close to a perfect metameric color match. However, the tristimulus values of the sample pair for the standard deviate observer are Xd1 = 2922

Yd1 = 2409

Zd1 = 4333

Xd2 = 2936

Yd2 = 2411

Zd2 = 4402

(7.25)

By applying the correction method of Equation 7.22, Xd2 Yd2 and Zd2 are obtained as Xd2 = Xd2 Xr1 /Xr2 = 2936 × 3029/3031 = 2934

Yd2 = Yd2 Yr1 /Yr2 = 2411 × 2441/ 2442

(7.26)

= 2411 Zd2 = Zd2 Zr1 /Zr2 = 4402 × 4325/4320 = 4406 Converting the tristimulus values (Xd1 , Yd1 , Zd1 ) and (Xd2 Yd2 , Zd2 ) into the L ∗ a ∗ b ∗ system, we obtain L1 ∗ = 5618

a1 ∗ = 2660

b1 ∗ = 5584

L2 ∗ = 5619

a2 ∗ = 2702

b2 ∗ = 5547

(7.27)

Thus, the color difference Eab ∗ is Eab ∗ = L1 ∗ − L2 ∗ 2 + a1 ∗ − a2 ∗ 2 + b1 ∗ − b2 ∗ 2 1/2 = 5618 − 56192 + 2660 − 27022 + 5584 − 55472 1/2 = 056

(7.28)

Thus, the observer metamerism index Mobs is Mobs10 D65 = 056 According to Table 7.8, this sample pair is graded C.

(7.29)

DESIGNING SPECTRAL DISTRIBUTIONS OF ILLUMINANTS

255

7.6 DESIGNING SPECTRAL DISTRIBUTIONS OF ILLUMINANTS By using linear programming, the technique introduced in Section 6.3, the spectral distribution of an illuminant having prescribed characteristics can be designed theoretically (Ohta and Wyszecki 1976). Only the results are given here, with the details of the computation being given in Note 7.1. Figure 7.14 shows the spectral reflectance of eight object colors used in the computation. These object colors were used previously in the computation of color rendering index. Note should be taken that the uv chromaticity diagram is used here.

7.6.1 Illuminant Giving Prescribed Chromaticity Coordinates to Object Colors The filled circles in Figure 7.15 correspond to the chromaticity coordinates of the eight object colors under standard illuminant D65 in the XYZ colorimetric system. If an illuminant that yields the chromaticity coordinates shown by the arrowheads is designed, the chromaticity coordinates of the illuminant are distributed within a region defined by the quadrilateral shown in Figure 7.15. An enlarged representation of the region is given in the right upper corner of the figure. By specifying the chromaticity coordinates for the eight colors, it can be seen that the chromaticity coordinates of the illuminant become almost fixed. Figure 7.16 shows the spectral distribution of an illuminant which provides the point M shown in the upper right of Figure 7.15. 0.8 Spectral reflectance

8

0.6 0.4

7 7 1

6 8 5 1

3 2 6

0.2

5 4

4 2 3

0.0

i =1 λ = 400

5

10 15 20 500 600 Wavelength (nm)

25

30 700

Figure 7.14 Spectral reflectances 8 object colors used in designing spectral distributions of illuminants (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

256


0.38

M

0.31155 v 0.31154

3

0.34

2

0.19079

u

0.19081

4

v

1

0.30

D65

5

8

6 7

0.26 0.12

0.16

0.20

0.24

0.28

u

Figure 7.15 Chromaticity coordinates (filled circles) under standard illuminant D65 and targeted chromaticity coordinates (arrowheads); the inset right upper diagram is the enlarged figure of chromaticity oordinates of the illuminant (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America


20 15 10 5 0

i=1 λ = 400

5

10 500

25 20 600 Wavelength (nm) 15

30 700

Figure 7.16 Spectral distribution of the illuminant that gives point M in Fig. 7.15 (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

7.6.2 Illuminant Having a General Color Rendering Index of 100 Let us consider a case in which the chromaticity coordinates of the 8 object colors in Figure 7.14 under the standard illuminant completely match with those under a sample illuminant. By


257

employing standard illuminant D65 as the standard, the chromaticity coordinates u and v of the sample illuminant which satisfy the above condition are computed to be in the following range 019765 ≤ u ≤ 019770

031231 ≤ v ≤ 031233

(7.30)

Figure 7.17 shows two examples (P1 and P2 of the spectral distribution of sample illuminants thus obtained. The chromaticity coordinates u0 and v0 of standard illuminant D65 are compared with those of P1 and P2 , i.e., u1 , v1 and u2 , v2 . u0 = 019768

v0 = 031232

u1 = 019770

v1 = 031231

u2 = 019767

v2 = 031232

(7.31)


40 P2 30 20 10 0


Wavelength (nm) 30

P1

20 10


0

Wavelength (nm) D65

10 0

i=1 λ = 400

5

10 15 20 600 500 Wavelength (nm)

25

30 700

Figure 7.17 Spectral distribution of standard illuminant D65 (lower) and illuminants P1 and P2 (upper and middle) having Ra of 100 (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

258


It can be seen that the values are extremely similar to each other. Thus, the average color rendering index Ra is evaluated as Ra = 100. However, because the spectral distributions of P1 and P2 differ considerably from that of D65 as is shown in Figure 7.17, it is expected that the color rendering properties will differ greatly for object colors other than the eight colors given in Figure 7.14. In practice, by considering all gray colors with Y = 10, metameric under standard illuminant D65 in the XYZ colorimetric system, and then obtaining the deviations from match under illuminants P1 and P2 , a wide chromaticity range, as shown in Figure 7.18, is obtained. This suggests that the method of evaluating illuminants using the CIE color rendering index is insufficient.

7.6.3 Illuminant Improving Excitation Purity The chroma of an object color is correlated with excitation purity. Accordingly, an illuminant which increases the excitation purity of object colors is expected to have the effect of increasing the chroma also. By designing the spectral distribution of an illuminant so that it increases the excitation purity of the eight object colors used above, as shown with arrows in Figure 7.19, a characteristic

0.4

520

540

560

580 600

500

620

640 660

D65

770 nm

P1 P2

v 0.2 480

470 460

0

450 440 420

400

0.2

0.4

0.6

u

Figure 7.18 Deviation obtained under illuminant P1 (broken line) and P2 (solid line) for gray colors metameric at Y = 10 under standard illuminant D65 (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America


259

0.38

3 0.34

2 4

v

1 D65 0.30

5

8 7

6 0.26 0.12

0.16

0.20

0.24

u

Figure 7.19 Chromaticity coordinates (filled circles) of eight object colors under standard illuminant D65 and improvement in excitation purity (arrowhead; length of arrow is enlarged by 5 times) (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

spectral distribution having 7 distinct emission lines, as shown in Figure 7.20, is obtained. If the excitation purity of only a single specified color is increased instead of increasing all the eight colors, the degree of increase is expected to be much higher, because there are less limiting


598.6 50 40 30 20 10 0.006 0

i=1 λ = 400

5

10 15 20 500 600 Wavelength (nm)

25

30 700

Figure 7.20 Spectral distribution of an illuminant which improves excitation purity as shown in Figure 7.19 (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

260


conditions. For example, if we consider illuminants having an infinite number of different spectral distributions and obtain a range of change of the chromaticity coordinates of a skin color, without changing the luminous reflectance (Y = 57.13), we obtain the result illustrated in Figure 7.21, where the chromaticity coordinates of the illuminants are assumed to be equal to those of standard illuminant D65 , and point C0 is the chromaticity of the skin color under standard illuminant D65 . From Figure 7.21, it can be seen that the excitation purity of the skin color covers a range of chromaticity from C2 to C1 , and that the excitation purity of skin color is increased by 44% at point C1 compared with C0 . Figure 7.22 shows the spectral distributions of illuminants P1 and P2 which give the points C1 and C2 respectively in Figure 7.21, and it can be seen that each of them consists of a plurality of emission lines. The spectral distributions of illuminants having these special effects yield, without exception, a plurality of emission lines. Figure 7.23 shows the spectral distribution of a commonly used daylight fluorescent lamp. It can be seen that it is somewhat similar to the spectral distribution obtained in the current study. As described in Section 3.8, a lamp obtained by adding metal halides into a high pressure mercury discharge (known as a metal halide lamp) can yield a specific spectral distribution relatively freely by selection of the type of metals. By combining the design method for spectral distributions introduced here with knowledge of the properties of metal halides, a variety of special-purpose illuminants may be developed in the future. 0.36

0.34

C1 C0

v

C2 0.32 D65 0.30 0.18

0.20

0.22 u

0.24

0.26

Figure 7.21 Range of chromaticity coordinates for skin color under illuminants having the same chromaticity coordinates as those of standard illuminant D65 (with luminous reflectance being constant) (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America

261

COMPUTER COLOR MATCHING

2819.5 1362.6

P2


120 P1 100 80 60 40 20 0

i=1 λ = 400

5

10 15 20 500 600 Wavelength (nm)

25

30 700

Figure 7.22 Spectral distribution of an illuminant that increases (P1 or decreases (P2 excitation purity of skin color (Ohta and Wyszecki 1976). Reproduced by permission of Optical Society of America


6 5 4 3 2 1 0 400


700

Figure 7.23 Example of spectral distribution of daylight fluorescent lamp (Wyszecki and Stiles 1982). Reproduced by permission of Optical Society of America

7.7 COMPUTER COLOR MATCHING Color matching comprises matching a target color, whether it be the result of reflection, transmission, or light emission, by combining component colors known as primaries. The color matching can

262


be realized either by additive color mixing (for example in the case of light emission) or by subtractive color mixing (for example in the case of dyes or pigments). Additive color mixing can usually be predicted accurately by a simple numerical computation as described in Section 6.1. However, for subtractive color mixing, as described in Section 6.2, the amounts of primaries needed to produce a match cannot be computed simply, because higher order nonlinear terms are included. In fact, many practical situations, such as the dying of textiles or the mixing of opaque pigments to produce paints are even more complex than the situation described in Section 6.2, because absorption and scattering are both involved. Accordingly, color matching depended historically on the experience of people, known as color matchers, who were skilled in the art of predicting dye or pigment recipes. However, even with the most experienced color matchers, the approach usually required several iterations and was not efficient, as shown in Figure 7.24, thus consuming considerable time. However, this problem is now easily solved with the aid of a computer, based on knowledge of the matching process. This method is known as computer color matching (CCM), and is utilized in a wide range of fields in the color industry. Figure 7.24 also shows a typical approach to a target color by means of CCM. It is apparent that the CCM approach is much more efficient than with a human color matcher. The principles and an example of CCM are described below.

0.435

0.430 y 0.425

0.420 0.545

0.550

0.555

0.560

0.565

0.570

x

Figure 7.24 Approach to the target color (filled circle) by a color matcher (solid line) and by CCM (broken line); the ellipse shows the color difference with respect to the target color for the color matcher, and the square shows the same for CCM (Berns 2000). Reproduced by permission of Wiley

263


Let us consider a simple case using cyan, magenta, and yellow dyes as the color primaries in subtractive mixing, with the amounts of the three dyes being c m and y respectively. The tristimulus values X Y and Z of the mixture can be expressed as X = fx c m y

Y = fy c m y

Z = fz c m y

(7.32)

using functions fx , fy , and fz . However, because the inverse functions cannot be obtained easily, the values of c m and y required to realize the desired tristimulus values cannot be directly obtained from Equations 7.32. Thus, methods of successive approximation such as the Regula Falsi (the rule of false position) or the Newton– Raphson method must be employed. The principles of these methods are explained below.

7.7.1 Regula Falsi To obtaining the solution for fx = 0 in the equation y = fx, for example, as is shown in Figure 7.25, the value of x is changed sequentially, for example, x = 0 1 2 , to obtain points a and b (where a < b) that satisfy the relation fa < 0 < fb. Then, a third point, c = a + b / 2 is examined to determine whether fc is over or under zero. If fc < 0, then a fourth point d = b + c/2 is considered, or, if fc > 0, a fourth point d = a + c / 2 is taken. Then fd is examined to determine whether it is negative or positive, and, depending on the result, a new value is taken to use in the subsequent computation. This procedure is repeated until the necessary precision is obtained. When the Regula Falsi method is used

y

y = f (x)

a 0

Figure 7.25 Principle of Regula Falsi

d c

b

x

264


for color matching, the amounts of the primaries, i.e., c m and y, which solve the simultaneous equations of Equation 7.32, are obtained. For example, when the Regula Falsi method is employed for color photographic films using the cyan, magenta, and yellow dyes of Figure 6.3 with a target color of u = 0.25, v = 0.25, and Y = 10, the color matching converges in the manner illustrated in Figure 7.26. Note should be taken that the Lambert–Beer Law is assumed here and that initial values are taken as c = m = y = 0. We note that the convergence is not very efficient, but that the color difference E decreases in the manner shown in Figure 7.27.

4 0.25

9 7 8 6

2

v

3

5

0.24

0.23

0.24

0.25

0.26

u

Figure 7.26 Approach to the target color (u = 0.25, v = 0.25) using Regula Ralsi (for color photographic film); numerals show number of repetitions (Ohta 1971). Reproduced by permission of Optical Society of America

2

Log (∆E )

1 0 –1 –2 –3 –4 0

100

200 300 400 500 Number of repetitions

Figure 7.27 Decrease of color difference E by Regula Falsi (Ohta 1971). Reproduced by permission of Optical Society of America

265


The Regula Falsi method is effective when the functions in Equation 7.32 are complicated, or when they are not given explicitly.

7.7.2 Newton–Raphson Method The Taylor expansion for the equation y = fx is obtained in the vicinity of x = x0 and the solution of fx = 0 is obtained by extracting the first-order terms, fx0 + x = fx0 + f x0 x

(7.33)

where x is a small change of x , and f x0 is obtained by substituting x0 into the first derivative f x of fx. Thus, a more accurate value x1 for the solution of fx = 0 can be obtained by setting fx0 + x = 0. This gives x1 = x0 + x = x0 − fx0 /f x0

(7.34)

Referring to Figure 7.28, the procedure above corresponds graphically to drawing a tangent to fx at the point (x0 y0 ), and obtaining the intersection x1 of the tangent line with the x axis. In applying this method to color matching, the initial values in Equation 7.32 are set to c0 , m0 and y0 , and the tristimulus values X0 , Y0 and Z0 are determined. Let the tristimulus values change by X , Y and Z when small changes c, m and y are added

y

y0

x1

x0

x

Figure 7.28 Principle of the Newton–Raphson method

266


to c0 , m0 and y0 . Thus, as in Equation 7.33, the following value is obtained for X0 + X X0 + X = fx c0 + c m + m y + y = fx c0 m y + cfx /c + mfx /m + yfx /y = X0 + cfx /c + mfx /m + yfx /y

(7.35)

where, for example, fx /c represents the partial derivative of function fx for parameter c. From Equation 7.35, we obtain X = cX/c + mX/m + yX/y

(7.36)

Similarly, Y and Z are obtained as Y = cY/c + mY/m + yY/y Z = cZ/c + mZ/m + yZ/y

(7.37)

Equations 7.36 and 7.37 can be expressed in matrix form as 

  −1   X X / c X / m X / y c  Y  =  Y / c Y / m Y / y   m  Z Z / c Z / m Z / y y

(7.38)

By multiplying both sides by the inverse of the first matrix on the right-hand side, we obtain 

  −1   c X / c X / m X / y X  m  =  Y / c Y / m Y / y   Y  y Z / c Z / m Z / y Z

(7.39)

A more accurate approximation can then be obtained by replacing the initial values c0 , m0 , and y0 by the following values c1 = c0 + c

m1 = m0 + m

y1 = y0 + y

(7.40)

Figure 7.29 is a flow chart showing the procedure of the computation. As an example, the same object color (u = 025, v = 025, Y = 10) as was used to illustrate the Regula Falsi method is used. Again, the Lambert–Beer law is assumed to hold, and color photographic films are used for the cyan, magenta, and yellow dyes, as shown in Figure 6.3. As shown in Figure 7.30, the color match is approached in an efficient manner. If reflection-type color photo-

267


Input tristimulus values Xg, Yg, and Zg of target color. Input spectral densities of cyan, magenta, and yellow dyes

Initial value c = m = y = 0

Computation of T (λ)

Computation of tristimulus values X, Y, and Z Computation of partial derivatives ∂X , ∂X , . . . , ∂X ∂c ∂m ∂y Computation of ∆ X = Xg – X, ∆ Y = Yg – Y, and ∆ Z = Zg – Z Pass

Fail Computation of

Output X, Y, Z, c, m, y, ∆ X, ∆ Y, and ∆ Z

∆ c, ∆ m, and ∆ y

Stop

Computation of new c, m, y

Figure 7.29 Flow chart for color matching using the Newton–Raphson method (Ohta 1971). Reproduced by permission of Optical Society of America

graphic prints are used, the functions in Equations 7.32 become far more complicated than the Lambert–Beer law, but a similar procedure is possible, and the color difference (E in both cases converges quite rapidly as is shown in Figure 7.31.

1 0.30

2

v 3

0.25

4 0.20

0.25 u

0.30

Figure 7.30 Approach to target color ( u = 025, v = 025) using the Newton–Raphson method (for color photographic film); numerals show the number of repetitions (Ohta 1971). Reproduced by permission of Optical Society of America

268


2 1

log (∆E )

0 –1 –2 –3 Color photographic film Color photographic print

–4 0

1

2

3 4 5 6 Number of repetitions

7

Figure 7.31 Decrease of color difference E by the Newton–Raphson method (Ohta 1971). Reproduced by permission of Optical Society of America

The colors obtained by either type of computer color matching (Regula Falsi or Newton Raphson) are all metameric, and hence, they deviate from matching when the viewing conditions (illuminant or observer) change. This fact is extremely inconvenient for industrial products but cannot be avoided. Hence, in many computer color matching programs, the metamerism index for change of illuminant is also computed to obtain a solution with the least deviations from color matching. It is also common to have computer programs that incorporate the price of the component elements into the computation of the optimum solution.

NOTE 7.1 COMPUTATION METHOD FOR PRESCRIBED SPECTRAL DISTRIBUTIONS For a general discussion, colors of m objects are considered, and their spectral reflectances are denoted by Rji j = 1 2 m ) i = 1 2 n), where i = 1 2 , n are the selected discrete wavelengths. To make the discussion simple, the tristimulus values X Y and Z are used in place of chromaticity coordinates, and the tristimulus values for the m object colors are denoted by ( Xj Yj Zj ) (j = 1 2 m ). By normalizing the spectral distribution of the illuminant Pi (i = 1 2 n) by n i −1

Pi y i = 100

(7.41)

PRESCRIBED SPECTRAL DISTRIBUTIONS

269

the constraints can be written as n i =1 n i =1

Rji Pi x i = Xj Rji Pi y i = Yj

n i =1

(7.42)

Rji Pi z i = Zj

where j = 1, 2, , m . The range of tristimulus values of the illuminant can then be treated sufficiently in the XZ plane because Y = 100. Accordingly, for a given value of X= Xs , the maximum and minimum values of Z are obtained, and this is repeated while sequentially changing the value of Xs . Thus the entire range of tristimulus values can be obtained. The condition for X = Xs is n i =1

Pi x i = Xj

(7.43)

Thus, the range within which the tristimulus values of the illuminant can lie, and corresponding spectral distributions, Pi , can be obtained under the constraints of Equations 7.41, 7.42 and 7.43 by obtaining the maximum and minimum values of the object function Z for an unknown Pi Z=

n i =l

Pi z i

(7.44)

Because arbitrary values can be selected for the object tristimulus values or chromaticity coordinates, the spectral distribution of an illuminant that, for example, increases the excitation purity of eight colors or of a specified color can be obtained. However, a practically feasible solution is not always available by linear programming due to the characteristics of the constraints. As an extreme example, if a constraint is introduced that an illuminant should yield blue chromaticity coordinates for a normally red object color and red chromaticity coordinates for a normally blue object color, a practically feasible solution cannot be obtained because such an illuminant is theoretically impossible.

Appendix I Basic Units and Terms

AI.1 SI UNITS Quantity

Unit

Symbol

Length Mass Time Electric current Thermodynamic temperature Luminous intensity Amount of substance Plane angle Solid angle Frequency Energy Power

meter kilogram second ampere kelvin

m kg s A K

candela mole radian steradian hertz joule watt

cd mol rad sr Hz or s−1 J or m2 kg s−2 W or J s−1 or m2 kg s−3


272

APPENDIX I

AI.2 PREFIXES FOR SI UNITS Factor

Name

Symbol

Factor

Name

Symbol

1024 1021 1018 1015 1012 109 106 103 102 10

yotta zetta exa peta tera giga mega kilo hecto deca

Y Z E P T G M k h da

10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18 10−21 10−24

deci centi milli micro nano pico femto atto zepto yocto

d c m n p f a z y

AI.3 FUNDAMENTAL CONSTANTS Quantity

Symbol c k h

Speed of light in vacuum Boltzmann constant Planck constant

Value 299 792 458 m s−1 1380 650 5 × 10−23 J K −1 6626 069 3 × 10−34 J s

AI.4 GREEK LETTERS Capital A B E Z H I K M N O

Lowercase

o

Pronunciation alpha beta gamma delta epsilon zeta eta theta iota kappa lambda mu nu xi omicron

273

APPENDIX I

P T Y X "

! #

pi rho sigma tau upsilon phi chi psi omega

Appendix II Matrix Algebra An (m ,n) matrix comprises m × n numbers a11 , a12 , … , amn arranged in the following rectangular form:    $A% =   

a11 a21 &&& &&& am 1

a12 a22 &&& &&& am 2

&&& &&& &&& &&& &&&

&&& &&& &&& &&& &&&

a1n a2n &&& &&& amn

     

[A1]

where aij (i = 1, 2, & & & , m ; j = 1, 2, & & & , n) are termed elements of the matrix; a row is a horizontal sequence of elements; and a column is a vertical sequence of elements. A matrix is expressed by symbols such as (A) or (aij ). When m = n, the matrix is said to be square and is of order n. If all the elements in the main diagonal (crossing from upper left to lower right) in a square matrix are unity and all other elements are zero, as in the example below, the matrix (E) is called the identity or unit matrix.    $E% =   

1 0 &&& &&& &&&

0 1 &&& &&& &&&

&&& 0 &&& &&& &&&

&&& &&& &&& 1 0

0 0 &&& 0 1

     


[A2]

276

APPENDIX II

A determinant resembles a matrix of order n, and is expressed as: a11 a21 D = &&& &&& a n1 = aij

a12 a22 &&& &&& an2

&&& &&& &&& &&& &&&

&&& &&& &&& &&& &&&

a1n a2n &&& &&& ann

[A3]

The determinant is a single value equal to ±a1i a2j · · · ank , where i , j, … k is a permutation of the n integers 1, 2, 3, … n in a defined order, with a plus or minus sign selected appropriately (the sign is plus if the permutation is even, and minus if the permutation is odd). The following are given as examples. a11 a12 a21 a22 = a11 a22 − a12 a21 a11 a12 a13 a21 a22 a23 a31 a32 a33 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 −a11 a23 a32 − a12 a21 a33 − a13 a22 a31

[A4]

Furthermore, each element aij has a determinant of order (n − 1) obtained by suppressing all the elements in row i and column j. A so-called cofactor of element aij , denoted Aij , is obtained by multiplying this determinant by (−1)i +j .

AII.1 ADDITION AND SUBTRACTION OF MATRICES When two matrices (A% and (B% are of the same order (m , n%, the following equation holds: $A% + $B% = $B% + $A%

[A5]

$A% + $B% = $C%

[A6]

and when

the elements aij , bij and cij are related by aij + bij = cij

[A7]

277

APPENDIX II

AII.2 MULTIPLICATION OF MATRICES A matrix multiplied by an arbitrary scalar, , is expressed as: $A% = $aij %

[A8]

When (A% and (B% are of the order (m ,n% and (n,l%, respectively, their multiplication is written as: $C% = $A%$B%

[A9]

and the resulting (m ,l% matrix (C% is called the product of (A% and (B%. The elements cij of matrix (C% are expressed by: cij =

n k=1

aik bkj

= ai 1 b1j + ai 2 b2j + · · · + ain bnj

[A10]

The following relations hold true: '$A% + $B%($C% = $A%$C% + $B%$C%

[A11]

'$A%$B%($C% = $A%'$B%$C%( It should be noted that, in general, $A%$B% = $B%$A%

[A12]

AII.3 INVERSE MATRIX For a given square matrix (A), if the following relation holds: $A%$B% = $E%

[A13]

where (E) is the unit matrix, (B) is called the inverse matrix of (A) and is written as (A)−1 . The elements, bij , of the inverse matrix (B) are given by bij = Aij / A

[A14]

where, Aij is the cofactor of aij and A is the determinant of (A). An inverse matrix has the following properties: '$A%−1 (−1 = A '$A%$B%(−1 = $B%−1 $A%−1 −1 $A% = A−1

[A15]

278

APPENDIX II

In the case of colorimetry, a (3, 3) matrix (aij ) (i = 1, 2, 3; j = 1, 2, 3) is mainly used. For example, the three equations a11 x + a12 y + a13 z = b1 a21 x + a22 y + a23 z = b2

[A16]

a31 x + a32 y + a33 z = b3 with unknowns x , y and z, can be expressed as a single matrix equation 

a11  a21 a31

    x b1 a13 a23   y  =  b2  z a33 b3

a12 a22 a32

[A17]

Then, by multiplying both sides of the equation from the lefthand side by the inverse of (aij ), the unknowns, x , y and z can be obtained as    x a11  y  =  a21 z a31

a12 a22 a32

−1   b1 a13 a23   b2  a33 b3

[A18]

Because the elements of the inverse matrix can be obtained by Eq. A14, x = D1 /D

y = D2 /D

z = D3 /D

[A19]

are obtained, where a11 D = a21 a31 a11 D2 = a21 a31

a12 a22 a32 b1 b2 b3

a13 a23 a33 a13 a23 a33

b1 a12 a13 D1 = b2 a22 a23 b3 a32 a33 a11 a12 b1 D3 = a21 a22 b2 a31 a32 b3

[A20]

AII.4 TRANSPOSE MATRIX The transpose of (A% is obtained by interchanging the rows and columns of the elements aij to aji . A symmetric matrix is a square

279

APPENDIX II

matrix whose transpose is the same as the original matrix. A transpose matrix has the following properties. '$A%T (T = $A% '$A% + $B%(T = $A%T + $B%T

[A21]

Although matrices and determinants have important additional features that are used in many fields, they are omitted here because they are rarely necessary in the field of color.

Appendix III Partial Derivatives For a function f (x , y, z, … ) having two or more independent parameters x , y, z, … , partial derivatives (also known as partial differential coefficients) can be obtained by changing only one parameter, e.g., x , while keeping the rest of the parameters constant. The partial derivative of f (x , y, z, … ) with respect to x is written as )f/)x or as fx . Partial derivatives with respect to y and z are obtained in a similar way. For example, let us consider a function f having three parameters x , y and z as follows: f$x* y* z% = x 2 + 2y 2 + 3z 2 + 4xy + 5yz + 6zx + 7x + 8y + 9z

[A22]

Then, for example, the partial derivative )f/)x is obtained by regarding parameters y and z as constants: )$x 2 %/)x = 2x )$5yz%/)x = 0*

)$2y 2 %/)x = 0

)$3z 2 %/)x = 0

)$4xy%/)x = 4y

)$6zx%/)x = 6z

)$7x%/)x = 7

)$8y%/)x = 0

)$9z%/)x = 0

[A23]

Thus )f/)x = 2x + 4y + 6z + 7


[A24]

282

APPENDIX III

Similarly, )f/)y and )f/)z are obtained as: )f/)y = 4y + 4x + 5z + 8 )f/)z = 6z + 5y + 6x + 9

[A25]

We can apply this to the problem of color matching. The illuminant P() is normalized by

P$%y$%d = 100

[A26]

vis

and the tristimulus values X , Y and Z are obtained from the spectral transmittance T$% by X=

T$%P$%x $%d

vis

Y=

T$%P$%y$%d

[A27]

vis

Z=

T$%P$%z$%d

vis

In the case of a color film, for instance, T$% is obtained as follows by assuming the Lambert–Beer law T$% = 10−'cDC $%+mDM $%+yD Y $%(

[A28]

where DC $%, DM $% and DY $% are the spectral densities of cyan, magenta and yellow color dyes, and c, m and y are the quantities of the color dyes. Then, a partial derivative, )T/)c can be obtained as follows by regarding m and c as constants: )T/)c = )+10−'cDC $%+mDM $%+yDY $%( ,/)c = −$ln10%DC $%10−'cDC $%+mDM $%+yDY $%( = −$ln10%DC $%T$%

[A29]

283

APPENDIX III

where ln is a logarithm to the base e, and ln 10 = 2.3026. Similarly, )T/)m and )T/)y are obtained as follows. )T/)m = −$ln10%DM $%T$% )T/)y = −$ln10%DY $%T$%

[A30]

Then, the partial derivatives )X/)c, )X/)m , … , )Z/)y are obtained as )X/)c = )' =

T$%P$%x $%d(/)c

vis

')T$%/)c(P$%x $%d

vis

= −$ln10%

DC $%T$%P$%x $%d

vis

)X/)m = −$ln10%

DM $%T$%P$%x $%d

vis

)Z/)y = −$ln10%

DY $%T$%P$%z$%d

[A31]

vis

In the calculation of color matching using the Newton–Raphson method (see Section 7.7), the partial derivatives obtained by Equation [A31] are used to obtain small correction values c, m , and y for the quantities c, m and y.

Appendix IV Tables Table A1 Color matching functions of the XYZ and X10 Y10 Z10 color systems (CIE 2004a, JIS Z 8701) Wavelength (nm)

XYZ colour system

X10 Y10 Z10 color system

x¯ $%

¯ y$%

¯ z$%

x¯ 10 $%

y¯ 10 $%

z¯ 10 $%

380 385 390 395 400

00014 00022 00042 00076 00143

00000 00001 00001 00002 00004

00065 00105 00201 00362 00679

00002 00007 00024 00072 00191

00000 00001 00003 00008 00020

00007 00029 00105 00323 00860

405 410 415 420 425

00232 00435 00776 01344 02148

00006 00012 00022 00040 00073

01102 02074 03713 06456 10391

00434 00847 01406 02045 02647

00045 00088 00145 00214 00295

01971 03894 06568 09725 12825

430 435 440 445 450

02839 03285 03483 03481 03362

00116 00168 00230 00298 00380

13856 16230 17471 17826 17721

03147 03577 03837 03867 03707

00387 00496 00621 00747 00895

15535 17985 19673 20273 19948

455 460 465 470

03187 02908 02511 01954

00480 00600 00739 00910

17441 16692 15281 12876

03430 03023 02541 01956

01063 01282 01528 01852

19007 17454 15549 13176


286

APPENDIX IV

Table A1 (Continued) Wavelength (nm)

XYZ color system

X10 Y10 Z10 color system

x¯ $%

¯ y$%

¯ z$%

x¯ 10 $%

y¯ 10 $%

z¯ 10 $%

475

01421

01126

10419

01323

02199

10302

480 485 490 495 500

00956 00580 00320 00147 00049

01390 01693 02080 02586 03230

08130 06162 04652 03533 02720

00805 00411 00162 00051 00038

02536 02977 03391 03954 04608

07721 05701 04153 03024 02185

505 510 515 520 525

00024 00093 00291 00633 01096

04073 05030 06082 07100 07932

02123 01582 01117 00782 00573

00154 00375 00714 01177 01730

05314 06067 06857 07618 08233

01592 01120 00822 00607 00431

530 535 540 545 550

01655 02257 02904 03597 04334

08620 09149 09540 09803 09950

00422 00298 00203 00134 00087

02365 03042 03768 04516 05298

08752 09238 09620 09822 09918

00305 00206 00137 00079 00040

555 560 565 570 575

05121 05945 06784 07621 08425

10000 09950 09786 09520 09154

00057 00039 00027 00021 00018

06161 07052 07938 08787 09512

09991 09973 09824 09555 09152

00011 00000 00000 00000 00000

580 585 590 595 600

09163 09786 10263 10567 10622

08700 08163 07570 06949 06310

00017 00014 00011 00010 00008

10142 10743 11185 11343 11240

08689 08256 07774 07204 06583

00000 00000 00000 00000 00000

605 610 615 620 625

10456 10026 09384 08544 07514

05668 05030 04412 03810 03210

00006 00003 00002 00002 00001

10891 10305 09507 08563 07549

05939 05280 04618 03981 03396

00000 00000 00000 00000 00000

630 635 640 645 650

06424 05419 04479 03608 02835

02650 02170 01750 01382 01070

00000 00000 00000 00000 00000

06475 05351 04316 03437 02683

02835 02283 01798 01402 01076

00000 00000 00000 00000 00000

655 660 665 670

02187 01649 01212 00874

00816 00610 00446 00320

00000 00000 00000 00000

02043 01526 01122 00813

00812 00603 00441 00318

00000 00000 00000 00000

287

APPENDIX IV

675 680 685 690 695 700

00636 00468 00329 00227 00158 00114

00232 00170 00119 00082 00057 00041

00000 00000 00000 00000 00000 00000

00579 00409 00286 00199 00138 00096

00226 00159 00111 00077 00054 00037

00000 00000 00000 00000 00000 00000

705 710 715 720 725

00081 00058 00041 00029 00020

00029 00021 00015 00010 00007

00000 00000 00000 00000 00000

00066 00046 00031 00022 00015

00026 00018 00012 00008 00006

00000 00000 00000 00000 00000

730 735 740 745 750

00014 00010 00007 00005 00003

00005 00004 00002 00002 00001

00000 00000 00000 00000 00000

00010 00007 00005 00004 00003

00004 00003 00002 00001 00001

00000 00000 00000 00000 00000

755 760 765 770 775

00002 00002 00001 00001 00001

00001 00001 00000 00000 00000

00000 00000 00000 00000 00000

00002 00001 00001 00001 00000

00001 00000 00000 00000 00000

00000 00000 00000 00000 00000

00000

00000

00000

00000

00000

00000

780 $ = 5nm%

213714 213711 213715 233294 233324 233343

$ = 10nm% 106836 106856 106770 116646 116644 116645

Table A2 Three eigenvectors S0 $%, S1 $%, and S2 $% for computing CIE D illuminants (CIE 2004a, JIS Z 8720) Wavelength (nm)

S0 $%

S1 $%

S2 $% Wavelength (nm)

300 305 310 315 320 325 330 335 340 345 350 355

004 302 600 1780 2960 4245 5530 5630 5730 5955 6180 6165

002 226 450 1345 2240 3220 4200 4130 4060 4110 4160 3980

000 100 200 300 400 625 850 815 780 725 670 600

360 365 370 375 380 385 390 395 400 405 410 415

S0 $%

S1 $%

S2 $%

6150 6515 6880 6610 6340 6460 6580 8030 9480 9980 10480 10535

3800 4020 4240 4045 3850 3675 3500 3920 4340 4485 4630 4510

530 570 610 455 300 210 120 005 −110 −080 −050 −060

288

APPENDIX IV

Table A2 (Continued) Wavelength (nm) 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 620 625

S0 $%

S1 $%

S2 $%

Wavelength (nm)

S0 $%

10590 10135 9680 10535 11390 11975 12560 12555 12550 12340 12130 12130 12130 11740 11350 11330 11310 11195 11080 10865 10650 10765 10880 10705 10530 10485 10440 10220 10000 9800 9600 9555 9510 9210 8910 8980 9050 9040 9030 8935 8840 8620

4390 4050 3710 3690 3670 3630 3590 3425 3260 3025 2790 2610 2430 2220 2010 1815 1620 1470 1320 1090 860 735 610 515 420 305 190 095 000 −080 −160 −255 −350 −350 −350 −465 −580 −650 −720 −790 −860 −905

−070 −995 −120 −190 −260 −275 −290 −285 −280 −270 −260 −260 −260 −220 −180 −165 −150 −140 −130 −125 −120 −110 −100 −075 −050 −040 −030 −015 000 010 020 035 050 130 210 265 320 365 410 440 470 490

630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730 735 740 745 750 755 760 765 770 775 780 785 790 795 800 805 810 815 820 825 830

8400 8455 8510 8350 8190 8225 8260 8375 8490 8310 8130 7660 7190 7310 7430 7535 7640 6985 6330 6750 7170 7435 7700 7110 6520 5645 4770 5815 6860 6680 6500 6550 6600 6350 6100 5715 5330 5610 5890 6040 6190

S1 $%

S2 $%

−950 510 −1020 590 −1090 670 −1080 700 −1070 730 −1135 795 −1200 860 −1300 920 −1400 980 −1380 1000 −1360 1020 −1280 925 −1200 830 −1265 895 −1330 960 −1310 905 −1290 850 −1175 775 −1060 700 −1110 730 −1160 760 −1190 780 −1220 800 −1120 735 −1020 670 −900 595 −780 520 −950 630 −1120 740 −1080 710 −1040 680 −1050 690 −1060 700 −1015 670 −970 640 −900 595 −830 550 −880 580 −930 610 −955 630 −980 650

289

APPENDIX IV

Table A3 Standard and supplementary illuminants (CIE 2004a, JIS Z 8720) Wavelength (nm) 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515

A

D65

093 113 136 162 193 227 266 310 359 414 474 541 614 695 782 877 980 1090 1209 1335 1471 1615 1768 1929 2099 2279 2467 2664 2870 3085 3309 3541 3781 4030 4287 4552 4824 5104 5391 5685 5986 6293 6606 6925

003 166 329 1177 2024 2864 3705 3850 3995 4243 4491 4578 4664 4936 5209 5103 4998 5231 5465 6870 8275 8712 9149 9246 9343 9006 8668 9577 10486 11094 11701 11741 11781 11634 11486 11539 11592 11237 10881 10908 10935 10858 10780 10630

C – – – – 001 020 040 155 270 485 700 995 1290 1720 2140 2750 3300 3992 4740 5517 6330 7181 8060 8953 9810 10580 11240 11775 12150 12345 12400 12360 12310 12330 12380 12409 12390 12292 12070 11690 11210 10698 10230 9881

D50

D55

D75

002 103 205 491 778 1126 1475 1635 1795 1948 2101 2248 2394 2545 2696 2572 2449 2718 2987 3959 4931 5291 5651 5827 6003 5893 5782 6632 7482 8104 8725 8893 9061 9099 9137 9324 9511 9354 9196 9384 9572 9617 9661 9687

002 105 207 665 1122 1594 2065 2227 2388 2585 2782 2922 3062 3246 3431 3345 3258 3534 3809 4952 6095 6475 6855 7007 7158 6975 6791 7676 8561 9180 9799 9923 10046 10019 9991 10133 10274 10041 9808 9938 10068 10069 10070 10034

004 259 513 1747 2981 4237 5493 5609 5726 6000 6274 6286 6298 6665 7031 6851 6670 6833 6996 8595 10193 10691 11189 11235 11280 10794 10309 11214 12120 12710 13301 13268 13236 12984 12732 12706 12680 12229 11778 11719 11659 11515 11370 11118

B – – – – 002 026 050 145 240 400 560 760 960 1240 1520 1880 2240 2685 3130 3618 4130 4662 5210 5770 6320 6837 7310 7731 8080 8344 8540 8688 8830 9008 9200 9375 9520 9623 9650 9571 9420 9237 9070 8965

290

APPENDIX IV

Table A3 (Continued) Wavelength (nm) 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730 735 740

A

D65

C

D50

D55

D75

7250 7579 7913 8252 8595 8941 9291 9644 10000 10358 10718 11080 11444 11808 12173 12539 12904 13270 13635 13999 14362 14724 15084 15442 15798 16152 16503 16851 17196 17538 17877 18212 18543 18870 19193 19512 19826 20136 20441 20741 21036 21327 21612 21892 22167

10479 10624 10769 10605 10441 10423 10405 10202 10000 9817 9633 9606 9579 9224 8869 8935 9001 8980 8960 8865 8770 8549 8329 8349 8370 8186 8003 8012 8021 8125 8228 8028 7828 7400 6972 7067 7161 7298 7435 6798 6160 6574 6989 7249 7509

9690 9678 9800 9994 10210 10395 10520 10567 10530 10411 10230 10015 9780 9543 9320 9122 8970 8883 8840 8819 8810 8806 8800 8786 8780 8799 8820 8820 8790 8722 8630 8530 8400 8221 8020 7824 7630 7436 7240 7040 6830 6630 6440 6280 6150

9713 9961 10210 10143 10075 10154 10232 10116 10000 9887 9774 9833 9892 9621 9350 9559 9769 9848 9927 9916 9904 9738 9572 9729 9886 9726 9567 9693 9819 10060 10300 10107 9913 9326 8738 8949 9160 9225 9289 8487 7685 8168 8651 8955 9258

9999 10210 10421 10316 10210 10253 10297 10148 10000 9861 9722 9748 9775 9459 9143 9293 9442 9478 9514 9468 9422 9233 9045 9139 9233 9059 8885 8959 9032 9213 9395 9195 8996 8482 7968 8126 8284 8384 8484 7754 7024 7477 7930 8215 8499

10866 10955 11044 10837 10629 10560 10490 10245 10000 9781 9562 9491 9421 9060 8700 8711 8723 8668 8614 8486 8358 8116 7875 7859 7843 7661 7480 7456 7432 7487 7542 7350 7158 6771 6385 6446 6508 6657 6807 6226 5644 6034 6424 6670 6915

B 8950 9043 9220 9446 9690 9916 10100 10220 10280 10292 10260 10190 10100 10007 9920 9844 9800 9808 9850 9906 9970 10036 10100 10156 10220 10305 10390 10459 10500 10508 10490 10455 10390 10284 10160 10038 9910 9770 9620 9460 9290 9110 8940 8800 8690

291

APPENDIX IV 745 750 755 760 765 770 775 780 785 790 795 800 805 810 815 820 825 830

22436 22700 22959 23212 23459 23701 23937 24168 24392 24612 24825 25033 25235 25431 25622 25807 25986 26160

6934 6359 5501 4642 5661 6681 6509 6338 6384 6430 6188 5945 5571 5196 5470 5744 5888 6031

6020 5920 5850 5810 5800 5820 5850 5910 – – – – – – – – – –

8540 7823 6796 5769 7031 8292 8060 7827 7891 7955 7648 7340 6866 6392 6735 7078 7261 7444

7844 7188 6234 5279 6436 7593 7387 7182 7238 7294 7014 6735 6304 5873 6186 6499 6665 6831

6389 5863 5062 4262 5198 6135 5984 5832 5873 5914 5694 5473 5132 4792 5042 5292 5423 5554

8590 8520 8480 8470 8490 8540 8610 8700 – – – – – – – – – –

Table A4 Tristimulus values and chromaticity coordinates of standard and supplementary illuminants (CIE, 2004a) (a) XYZ color space Tristimulus values

A D65 C D50 D55 D75 B


X

Y

Z

x

y

u

10985 9504 9807 9642 9568 9496 9909

10000 10000 10000 10000 10000 10000 10000

3558 10889 11823 8249 9214 12261 8531

04476 03127 03101 03457 03324 02990 03484

04074 03290 03162 03585 03474 03149 03516

02560 01978 02009 02092 02044 01935 02137

v 05243 04683 04609 04881 04807 04585 04852

(b) X10 Y10 Z10 color space Tristimulus values

A D65 C D50 D55 D75 B


X10

Y10

Z10

x10

y10

u 10

11115 9481 9728 9672 9579 9441 9919

10000 10000 10000 10000 10000 10000 10000

3520 10733 11614 8141 9093 12063 8436

04512 03138 03104 03478 03341 02997 03498

04059 03310 03191 03595 03488 03174 03527

02590 01979 02000 02102 02051 01930 02142

v 10 05242 04695 04626 04889 04816 04601 04859

0256 0254 0252 0248 0244

0240 0237 0232 0230 0226

0225 0222 0220 0218

400 405 410 415 420

425 430 435 440 445

450 455 460 465

0127 0128 0131 0134

0122 0122 0122 0123 0124

0111 0116 0118 0120 0121

0070 0079 0089 0101

Dark grayish yellow

Light grayish red

0219 0239 0252 0256

5Y6/4

2

7.5R6/4

1

380 385 390 395

Wavelength (nm)

Munsell notation Color name

Number

0074 0075 0077 0080

0073 0073 0073 0073 0073

0073 0073 0074 0074 0074

0065 0068 0070 0072

Deep yellow green

5GY6/8

3

0161 0172 0186 0205

0131 0135 0139 0144 0151

0116 0121 0124 0126 0128

0074 0083 0093 0105

Yellowish green

2.5G6/6

4

0360 0369 0381 0394

0330 0334 0339 0346 0352

0313 0315 0319 0322 0326

0295 0306 0310 0312

Light bluish green

10BG6/4

5

0556 0556 0554 0549

0524 0531 0538 0544 0551

0410 0464 0492 0508 0517

0151 0203 0265 0339

Light blue

5PB6/8

6

0522 0506 0488 0469

0558 0556 0551 0544 0535

0551 0555 0559 0560 0561

0378 0459 0524 0546

Light violet

2.5P6/8

7

8

0439 0426 0413 0397

0488 0482 0473 0462 0450

0319 0416 0462 0482 0490

0104 0129 0170 0240

Light reddish purple

10P6/8

Table A5 Spectral reflectances of test colors for computing the CIE color rendering index (CIE 1995, JIS Z 8726)

0216

0214 0214 0214 0216 0218

0223 0225 0226 0226 0225

0225 0227 0230 0236 0245

0253 0262 0272 0283 0298

0318 0341 0367 0390

470

475 480 485 490 495

500 505 510 515 520

525 530 535 540 545

550 555 560 565 570

575 580 585 590

0329 0335 0339 0341

0282 0289 0299 0309 0322

0264 0267 0269 0272 0276

0207 0225 0242 0253 0260

0143 0150 0159 0174 0190

0138

0332 0315 0299 0285

0400 0393 0380 0365 0349

0302 0339 0370 0392 0399

0198 0221 0241 0260 0278

0094 0109 0126 0148 0172

0085

0263 0247 0229 0214

0341 0327 0312 0296 0280

0392 0385 0377 0367 0354

0370 0383 0390 0394 0395

0254 0281 0308 0332 0352

0229

0271 0260 0247 0232

0331 0320 0308 0296 0284

0381 0372 0363 0353 0342

0413 0409 0403 0396 0389

0410 0415 0418 0419 0417

0403

0227 0225 0222 0221

0279 0265 0253 0241 0234

0358 0341 0325 0309 0293

0450 0431 0414 0395 0377

0531 0519 0504 0488 0469

0541

0254 0254 0259 0270

0259 0260 0260 0258 0256

0273 0265 0260 0257 0257

0324 0311 0301 0291 0283

0429 0408 0385 0363 0341

0448

0274 0278 0284 0295

0254 0258 0264 0269 0272

0262 0256 0251 0250 0251

0299 0289 0283 0276 0270

0366 0352 0337 0325 0310

0382

0409

0424 0435 0442 0448 0450

0451 0451 0451 0451 0451

0450 0450 0451 0451 0453

625 630 635 640 645

650 655 660 665 670

0336 0335 0334 0332 0332

0339 0339 0338 0338 0337

0342 0342 0342 0341 0341

0341

0216 0216 0219 0224 0230

0235 0229 0224 0220 0217

0264 0257 0252 0247 0241

0272

0148 0148 0148 0149 0151

0156 0154 0152 0151 0149

0185 0175 0169 0164 0160

0198

0175 0175 0175 0177 0180

0183 0180 0177 0176 0175

0210 0200 0194 0189 0185

0220

Light bluish green

Yellowish green

Deep yellow green

Dark grayish yellow

Light grayish red

595

10BG6/4

5

2.5G6/6

4

5GY6/8

3

5Y6/4

2

7.5R6/4

1

600 605 610 615 620

Wavelength (nm)


Number

Table A5 (Continued)

0258 0263 0268 0273 0278

0227 0233 0239 0244 0251

0220 0220 0220 0220 0223

0438 0445 0452 0457 0462

0389 0400 0410 0420 0429

0302 0324 0344 0362 0377

0284

Light violet

Light blue

0220

2.5P6/8

7

5PB6/8

6

0676 0685 0693 0700 0705

0568 0604 0629 0648 0663

0348 0384 0434 0482 0528

0316

Light reddish purple

10P6/8

8

0454 0455 0457 0458 0460 0462 0463 0464 0465 0466

0466 0466 0466 0467 0467

0467 0467 0467 0467 0467

0467 0467

675 680 685 690 695 700 705 710 715 720

725 730 735 740 745

750 755 760 765 770

775 780

0314 0314

0320 0318 0316 0315 0315

0324 0324 0323 0322 0321

0331 0331 0330 0329 0328 0328 0327 0326 0325 0324

0510 0516

0493 0497 0500 0502 0505

0447 0460 0472 0481 0488

0238 0251 0269 0288 0312 0340 0366 0390 0412 0431

0194 0197

0177 0181 0185 0189 0192

0164 0164 0165 0168 0172

0154 0158 0162 0165 0168 0170 0171 0170 0168 0166

0217 0219

0203 0205 0208 0212 0215

0195 0195 0196 0197 0200

0183 0186 0189 0192 0195 0199 0200 0199 0198 0196

0477 0485

0425 0436 0447 0458 0469

0364 0376 0389 0401 0413

0281 0283 0286 0291 0296 0302 0313 0325 0338 0351

0578 0581

0553 0559 0565 0570 0575

0518 0525 0532 0539 0546

0466 0468 0470 0473 0477 0483 0489 0496 0503 0511

0730 0730

0730 0730 0730 0730 0730

0727 0729 0730 0730 0730

0709 0712 0715 0717 0719 0721 0720 0719 0722 0725

0052 0052 0051 0050 0050

0049 0048 0047 0046 0044

0042 0041 0038 0035 0033

400 405 410 415 420

425 430 435 440 445

450 455 460 465 470

0083 0088 0095 0103 0113

0070 0072 0073 0076 0078

0066 0067 0068 0069 0069

0050 0054 0059 0063

Deep yellow

Deep red

0066 0062 0058 0055

5Y8/10

10

4.5R4/13

9

380 385 390 395

Wavelength (nm)


Number


0110 0115 0123 0134 0148

0105 0104 0104 0105 0106

0127 0121 0116 0112 0108

0111 0121 0127 0129

Deep green

4.5G5/8

11

0300 0331 0346 0347 0341

0093 0123 0160 0207 0256

0076 0068 0064 0065 0075

0120 0103 0090 0082

Deep blue

3PB3/11

12

0372 0374 0376 0379 0384

0361 0364 0365 0367 0369

0264 0313 0341 0352 0359

0104 0127 0161 0211

Caucasian skin

5YR8/4

13

0045 0046 0047 0048 0050

0042 0043 0044 0044 0045

0039 0039 0040 0041 0042

0036 0036 0037 0038

Foliage green

5GY4/4

14

0221 0229 0236 0243 0249

0175 0182 0192 0203 0212

0158 0162 0164 0167 0170

0131 0139 0147 0153

Japanese skin

1YR6/4

15

0031 0030 0029 0028 0028

0028 0029 0030 0030 0031

0031 0032 0032 0033 0034

0035 0037 0041 0044 0048

0052 0060 0076 0102 0136

475 480 485 490 495

500 505 510 515 520

525 530 535 540 545

550 555 560 565 570

575 580 585 590 595

0698 0701 0704 0705 0705

0653 0666 0678 0687 0693

0509 0546 0581 0610 0634

0262 0305 0365 0416 0465

0125 0142 0162 0189 0219

0138 0125 0114 0106 0100

0227 0206 0188 0170 0153

0333 0314 0294 0271 0248

0325 0347 0356 0353 0346

0167 0192 0219 0252 0291

0017 0017 0017 0016 0016

0035 0029 0025 0022 0019

0090 0075 0062 0051 0041

0204 0178 0154 0129 0109

0328 0307 0282 0257 0230

0651 0680 0701 0717 0729

0506 0526 0553 0582 0618

0471 0474 0476 0483 0490

0443 0454 0461 0466 0469

0389 0397 0405 0416 0429

0125 0118 0112 0106 0101

0155 0152 0147 0140 0133

0121 0133 0142 0150 0154

0075 0083 0092 0100 0108

0052 0055 0057 0062 0067

0280 0285 0314 0354 0398

0281 0286 0291 0289 0286

0289 0282 0276 0274 0276

0284 0291 0296 0298 0296

0254 0259 0264 0269 0276

0581 0641 0682 0717 0740

0758 0770 0781 0790 0797

0803 0809

625 630 635 640 645

650 655 660 665 670

675 680

0729 0731

0716 0718 0720 0722 0725

0708 0710 0711 0712 0714

0706 0707 0707 0707 0708

Deep yellow

Deep red

0190 0256 0336 0418 0505

5Y8/10

10

4.5R4/13

9

600 605 610 615 620

Wavelength (nm)


Number


0093 0102

0078 0078 0081 0083 0088

0082 0080 0079 0078 0078

0096 0092 0090 0087 0085

Deep green

4.5G5/8

11

0030 0035

0019 0020 0023 0024 0026

0016 0018 0018 0018 0018

0016 0016 0016 0016 0016

Deep blue

3PB3/11

12

0747 0747

0748 0748 0747 0747 0747

0748 0748 0748 0748 0748

0736 0742 0745 0747 0748

Caucasian skin

5YR8/4

13

0096 0102

0084 0084 0085 0087 0092

0087 0086 0085 0084 0084

0098 0095 0093 0090 0089

Foliage green

5GY4/4

14

0594 0599

0571 0576 0581 0586 0590

0535 0544 0552 0559 0565

0440 0470 0494 0511 0524

Japanese skin

1YR6/4

15

0814 0819 0824

0828 0830 0831 0833 0835

0836 0836 0837 0838 0839

0839 0839 0839 0839 0839 0839

0839

685 690 695

700 705 710 715 720

725 730 735 740 745

750 755 760 765 770 775

780

0759

0756 0757 0758 0759 0759 0759

0754 0755 0755 0755 0755

0746 0748 0749 0751 0753

0735 0739 0742

0338

0310 0314 0317 0323 0330 0334

0257 0270 0282 0292 0302

0161 0182 0203 0223 0242

0112 0125 0141

0645

0520 0551 0577 0599 0618 0633

0305 0354 0401 0446 0485

0097 0128 0166 0210 0257

0043 0056 0074

0747

0749 0748 0748 0747 0747 0747

0744 0745 0748 0750 0750

0746 0746 0745 0744 0743

0747 0747 0746

0399

0325 0339 0353 0366 0379 0390

0243 0260 0277 0294 0310

0152 0169 0188 0207 0226

0110 0123 0137

0611

0609 0608 0607 0607 0609 0610

0616 0615 0613 0612 0610

0612 0614 0616 0616 0616

0603 0606 0610

0048 0045 0043 0041 0040

0038 0037 0036 0035 0034

0034 0035 0036 0037 0039

0041 0045 0051 0058 0067

425 430 435 440 445

450 455 460 465 470

475 480 485 490 495

Standard

400 405 410 415 420

Wavelength (nm)

0054 0068 0085 0103 0121

0027 0028 0031 0035 0043

0024 0024 0025 0025 0026

0029 0028 0027 0026 0024

D50

0054 0068 0085 0103 0121

0028 0029 0032 0036 0044

0025 0025 0025 0025 0026

0029 0028 0027 0026 0025

D55

D65

0054 0068 0085 0104 0121

0029 0030 0033 0037 0044

0026 0026 0026 0026 0027

0029 0028 0027 0026 0026

Metameric pair 1

0055 0069 0086 0104 0121

0029 0031 0034 0038 0045

0026 0026 0026 0026 0027

0030 0029 0028 0027 0027

D75

0054 0052 0050 0048 0046

0070 0066 0063 0060 0057

0088 0085 0082 0078 0074

0080 0084 0087 0089 0089

Standard

0048 0045 0042 0039 0037

0076 0071 0066 0059 0052

0088 0089 0088 0083 0081

0044 0056 0063 0074 0081

D50

0048 0045 0042 0039 0037

0076 0071 0066 0059 0052

0088 0089 0088 0083 0081

0044 0056 0063 0074 0081

D55

Metameric pair 2

0048 0045 0043 0040 0037

0076 0071 0066 0059 0052

0088 0089 0088 0083 0081

0044 0056 0063 0073 0080

D65

0048 0046 0042 0039 0037

0076 0071 0066 0059 0052

0088 0089 0088 0083 0081

0044 0056 0063 0073 0080

D75

Table A6 Spectral reflectances of five metameric pairs for computing visible-range metamerism index (CIE 1999, JIS Z 8720)

0077 0089 0102 0115 0127

0139 0151 0162 0174 0185

0198 0213 0230 0251 0276

0305 0336 0369 0401 0431

0459 0482 0501

500 505 510 515 520

525 530 535 540 545

550 555 560 565 570

575 580 585 590 595

600 605 610

0427 0473 0515

0248 0275 0309 0345 0384

0182 0189 0196 0209 0226

0162 0164 0167 0172 0177

0136 0148 0156 0159 0160

0430 0476 0518

0247 0275 0309 0346 0386

0180 0187 0195 0208 0225

0161 0163 0166 0170 0175

0136 0147 0155 0158 0159

0434 0480 0523

0246 0275 0309 0347 0389

0177 0184 0193 0206 0223

0159 0161 0164 0167 0172

0136 0146 0153 0156 0157

0437 0483 0527

0245 0275 0309 0348 0391

0175 0182 0191 0204 0222

0157 0159 0162 0166 0170

0136 0145 0152 0154 0155

0046 0053 0061

0032 0033 0035 0037 0041

0032 0031 0031 0031 0031

0037 0036 0035 0034 0033

0044 0042 0041 0039 0038

0072 0076 0083

0041 0045 0049 0055 0063

0033 0032 0030 0032 0036

0032 0032 0032 0033 0033

0034 0035 0033 0032 0032

0071 0075 0082

0041 0045 0049 0055 0063

0033 0032 0030 0032 0036

0033 0033 0033 0033 0033

0034 0035 0033 0033 0033

0070 0074 0080

0041 0045 0049 0054 0062

0033 0032 0031 0032 0036

0034 0034 0034 0034 0033

0034 0035 0033 0033 0034

0069 0073 0079

0041 0045 0049 0055 0061

0033 0032 0031 0032 0036

0035 0035 0035 0034 0033

0034 0035 0033 0034 0035

0516 0528 0537 0544 0551 0557 0562

0567 0573 0579 0585 0592

0598 0605 0613 0621 0629

0637

650 655 660 665 670

675 680 685 690 695

700

Standard

615 620 625 630 635 640 645

Wavelength (nm)


0817

0751 0766 0781 0794 0806

0683 0695 0708 0722 0736

0552 0582 0608 0630 0646 0659 0671

D50

0822

0755 0771 0786 0799 0811

0687 0699 0712 0726 0740

0555 0586 0612 0634 0650 0663 0675

D55

D65

0830

0762 0778 0793 0807 0819

0693 0706 0719 0733 0747

0560 0593 0619 0641 0657 0669 0681

Metameric pair 1

0836

0767 0784 0799 0813 0825

0698 0711 0724 0738 0752

0564 0596 0622 0645 0661 0674 0686

D75

0326

0217 0236 0257 0279 0302

0156 0166 0177 0188 0201

0071 0082 0095 0109 0121 0133 0145

Standard

0088

0087 0086 0086 0086 0087

0085 0086 0088 0088 0088

0085 0087 0087 0086 0085 0084 0084

D50

0087

0086 0085 0085 0085 0086

0084 0085 0087 0087 0087

0084 0086 0086 0085 0084 0083 0083

D55

Metameric pair 2

0086

0085 0084 0084 0084 0085

0083 0084 0086 0086 0086

0082 0084 0085 0084 0083 0082 0082

D65

0085

0084 0083 0083 0083 0084

0082 0083 0085 0085 0085

0081 0083 0084 0083 0082 0081 0081

D75

0044 0042 0040 0038 0037

0036 0035 0034 0033 0032

0032 0032 0032 0033 0034

0036 0038 0041 0045 0049

0055 0062

425 430 435 440 445

450 455 460 465 470

475 480 485 490 495

500 505

Standard

400 405 410 415 420

Wavelength (nm)


0092 0095

0045 0056 0067 0077 0086

0027 0029 0031 0034 0037

0026 0026 0024 0025 0026

0029 0028 0028 0027 0027

D50

0092 0095

0046 0056 0067 0077 0086

0027 0029 0031 0034 0038

0026 0026 0025 0026 0027

0029 0028 0028 0027 0027

D55

D65

0092 0095

0047 0057 0067 0077 0086

0028 0029 0031 0034 0039

0026 0026 0026 0027 0028

0029 0028 0028 0027 0027

Metameric pair 3

0092 0094

0048 0057 0067 0077 0086

0028 0029 0032 0035 0040

0026 0026 0027 0027 0028

0029 0028 0028 0027 0027

D75

0348 0331

0400 0396 0387 0376 0363

0369 0381 0391 0398 0401

0318 0326 0335 0346 0357

0298 0298 0300 0305 0311

Standard

0296 0288

0337 0328 0319 0312 0304

0372 0366 0360 0353 0345

0400 0397 0392 0384 0378

0402 0402 0402 0402 0401

D50

0297 0290

0335 0327 0319 0312 0304

0370 0364 0358 0351 0343

0398 0395 0390 0382 0376

0400 0400 0400 0400 0399

D55

D65

0299 0292

0333 0326 0319 0312 0305

0367 0361 0355 0348 0340

0395 0392 0387 0379 0373

0397 0397 0397 0397 0396

Metameric pair 4

0299 0294

0331 0324 0318 0311 0305

0365 0359 0353 0346 0338

0393 0390 0385 0378 0371

0395 0395 0395 0395 0394

D75

0070 0078 0086

0092 0097 0101 0104 0106

0107 0107 0106 0104 0101

0099 0096 0093 0090 0089

0089 0089 0090

525 530 535 540 545

550 555 560 565 570

575 580 585 590 595

600 605 610

Standard

510 515 520

Wavelength (nm)


0102 0103 0104

0094 0096 0097 0097 0100

0084 0086 0087 0088 0091

0090 0089 0088 0086 0084

0097 0095 0092

D50

0103 0104 0104

0094 0096 0097 0098 0101

0084 0085 0086 0088 0091

0089 0088 0087 0086 0084

0096 0094 0091

D55

D65

0104 0105 0104

0094 0096 0098 0100 0102

0083 0083 0084 0088 0091

0088 0087 0086 0085 0083

0095 0093 0090

Metameric pair 3

0105 0106 0106

0094 0096 0098 0100 0103

0083 0083 0084 0087 0090

0087 0086 0085 0084 0083

0094 0092 0089

D75

0203 0209 0220

0208 0206 0206 0204 0202

0220 0213 0208 0207 0208

0272 0262 0251 0241 0230

0313 0297 0283

Standard

0238 0240 0242

0239 0236 0234 0234 0235

0246 0244 0244 0243 0242

0265 0260 0255 0250 0248

0281 0276 0271

D50

0235 0237 0239

0239 0235 0233 0232 0233

0248 0246 0246 0244 0242

0267 0262 0257 0253 0250

0283 0278 0273

D55

Metameric pair 4

0230 0232 0234

0238 0234 0231 0229 0229

0251 0249 0248 0246 0243

0270 0265 0260 0257 0253

0286 0281 0276

D65

0226 0228 0230

0238 0233 0229 0227 0226

0254 0252 0250 0247 0243

0273 0268 0263 0259 0256

0288 0284 0279

D75

0091 0092

0092 0092 0093 0096 0101

0109 0120 0134 0154 0177

0202 0228 0252 0275 0296

0316

615 620

625 630 635 640 645

650 655 660 665 670

675 680 685 690 695

700

0250

0154 0170 0188 0206 0227

0109 0114 0120 0129 0140

0103 0103 0104 0104 0106

0104 0104

0252

0155 0171 0189 0208 0229

0110 0115 0121 0130 0141

0103 0103 0104 0105 0107

0104 0104

0255

0157 0173 0191 0211 0232

0111 0116 0123 0132 0143

0104 0104 0105 0106 0108

0105 0105

0257

0158 0174 0192 0213 0234

0112 0117 0124 0133 0144

0105 0105 0106 0107 0109

0106 0106

0835

0647 0702 0749 0787 0816

0391 0430 0476 0531 0589

0277 0298 0317 0337 0361

0236 0256

0647

0404 0451 0502 0554 0601

0245 0262 0288 0319 0359

0234 0228 0226 0228 0234

0240 0238

0638

0398 0445 0495 0546 0593

0242 0259 0284 0315 0354

0230 0225 0223 0225 0231

0237 0234

0624

0390 0436 0484 0534 0580

0237 0254 0278 0309 0347

0224 0220 0218 0221 0226

0232 0228

0614

0383 0429 0476 0525 0570

0233 0250 0273 0304 0341

0221 0217 0215 0217 0222

0228 0225

0106 0106 0109 0114 0120

0127 0136 0146 0156 0166

0176 0184 0191 0195 0197

425 430 435 440 445

450 455 460 465 470

Standard

400 405 410 415 420

Wavelength (nm)


0174 0170 0165 0160 0156

0187 0187 0186 0181 0178

0175 0177 0179 0182 0184

D50

0173 0169 0164 0159 0155

0185 0185 0184 0180 0177

0174 0176 0178 0181 0183

D55

0171 0167 0162 0157 0153

0183 0183 0182 0179 0175

0173 0175 0177 0180 0182

D65

Metameric pair 5

0170 0166 0161 0156 0152

0181 0181 0180 0178 0174

0172 0174 0176 0179 0180

D75

525 530 535 540 545

500 505 510 515 520

475 480 485 490 495

Wavelength (nm)

0115 0110 0107 0105 0105

0155 0146 0137 0129 0122

0195 0191 0183 0174 0165

Standard

0125 0122 0121 0121 0121

0137 0135 0135 0132 0129

0151 0148 0143 0141 0139

D50

0126 0123 0122 0122 0122

0137 0135 0135 0133 0130

0150 0147 0143 0141 0139

D55

D65

0127 0124 0123 0123 0123

0138 0137 0136 0134 0131

0149 0146 0143 0141 0139

Metameric pair 5

0128 0125 0124 0124 0124

0138 0137 0137 0135 0132

0148 0145 0143 0141 0139

D75

0105 0105 0105 0105 0107

0111 0120 0135 0156 0183

0214 0250 0285 0313 0333

550 555 560 565 570

575 580 585 590 595

600 605 610 615 620

0200 0228 0258 0286 0316

0113 0119 0131 0149 0174

0121 0119 0116 0110 0108

0199 0227 0258 0286 0315

0113 0119 0131 0149 0173

0122 0119 0116 0111 0109

0198 0226 0256 0285 0313

0113 0119 0131 0149 0172

0123 0120 0117 0113 0111

0197 0225 0255 0284 0311

0113 0119 0131 0149 0171

0124 0121 0118 0113 0111

0667 0721 0766 0803 0830 0849

700

0391 0434 0487 0547 0609

0340 0342 0341 0345 0362

675 680 685 690 695

650 655 660 665 670

625 630 635 640 645

0621

0522 0543 0564 0584 0603

0437 0451 0466 0482 0502

0342 0366 0387 0405 0422

0619

0520 0541 0562 0582 0601

0436 0450 0465 0481 0500

0341 0365 0386 0404 0421

0616

0517 0538 0559 0579 0598

0434 0448 0463 0479 0497

0339 0363 0384 0402 0419

0613

0514 0535 0556 0576 0595

0432 0446 0461 0477 0494

0337 0361 0382 0400 0417

0182 0194 0205 0214 0220

0226 0230 0232 0232 0230

0224 0216 0204 0177 0145

0117 0088 0056 0028 0016

325 330 335 340 345

350 355 360 365 370

375 380 385 390 395

Q$ %

300 305 310 315 320

Wavelength (nm)

s $%

Sample 1 F$%

0228 0220 0196 0164 0134

0174 0204 0218 0227 0229

0023 0050 0075 0102 0137

0001 0001 0006

Q$ % s $%

Sample 2 F$%

0218 0235 0244 0245 0245

0055 0082 0111 0152 0191

0001 0001 0002 0025

Q$ % s $%

Sample 3 F$%

Table A7 Spectral characteristics of three pairs of fluorescent samples for computing ultraviolet-range metamerism index (CIE 1999, JIS Z8720)

0766 0770 0774 0778 0782

0788 0794 0799 0804 0808

0812 0817

450 455 460 465 470

475 480 485 490 495

500 505

0638 0661 0683 0704 0722

0734 0742 0750 0756 0761

0011 0009 0006 0002

425 430 435 440 445

400 405 410 415 420

0011 0009

0033 0028 0022 0018 0014

0071 0064 0056 0048 0039

0070 0085 0092 0090 0081

0008 0011 0022 0036 0051 0011 0007 0004 0001

0104 0068 0038 0023 0016

0812 0817

0788 0794 0799 0804 0808

0766 0770 0774 0778 0782

0718 0730 0744 0755 0762

0490 0570 0640 0678 0701

0011 0009

0033 0028 0022 0018 0014

0071 0064 0056 0048 0039

0070 0085 0092 0090 0081

0008 0011 0022 0036 0051

0009 0005 0001

0082 0046 0028 0019 0013

0237 0213 0182 0153 0120

0812 0817

0788 0794 0799 0804 0808

0753 0764 0773 0778 0782

0603 0676 0712 0731 0744

0194 0270 0357 0437 0517

0011 0009

0033 0028 0022 0018 0014

0071 0064 0056 0048 0039

0070 0085 0092 0090 0081

0008 0011 0022 0036 0051

0831 0831 0831 0832 0832

0833 0833 0834 0834 0835

0835 0836 0837 0837 0837

0838

525 530 535 540 545

550 555 560 565 570

575 580 585 590 595

600

s $%

Sample 1

0822 0827 0830

Q$ %

510 515 520

Wavelength (nm)


0001 0001 0001 0001 0001

0004 0004 0003 0003 0003

0008 0006 0005

F$%

Q$ %

0838

0835 0836 0837 0837 0837

0833 0833 0834 0834 0835

0831 0831 0831 0832 0832

0822 0827 0830

s $%

Sample 2

0001 0001 0001 0001 0001

0004 0004 0003 0003 0003

0008 0006 0005

F$%

Q$ %

0838

0835 0836 0837 0837 0837

0833 0833 0834 0834 0835

0831 0831 0831 0832 0832

0822 0827 0830

s $%

Sample 3

0001 0001 0001 0001 0001

0004 0004 0003 0003 0003

0008 0006 0005

F$%

0839 0840 0842 0844

0846 0848 0850 0852 0854

0856 0857 0857 0857 0858

0859 0860 0861 0862 0863

0864

605 610 615 620

625 630 635 640 645

650 655 660 665 670

675 680 685 690 695

700

0864

0859 0860 0861 0862 0863

0856 0857 0857 0857 0858

0846 0848 0850 0852 0854

0839 0840 0842 0844

0864

0859 0860 0861 0862 0863

0856 0857 0857 0857 0858

0846 0848 0850 0852 0854

0839 0840 0842 0844

400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490

Wavelength (nm)

0647 0672 0704 0738 0768 0799 0822 0825 0821 0815 0810 0809 0808 0807 0805 0807 0810 0812 0815

1

0502 0585 0669 0723 0763 0805 0838 0846 0843 0835 0826 0823 0819 0817 0813 0814 0816 0816 0818

2

D50 sample

0212 0293 0401 0507 0613 0737 0842 0868 0866 0857 0845 0845 0843 0837 0830 0828 0827 0826 0826

3 0646 0674 0705 0740 0773 0806 0832 0835 0831 0825 0818 0815 0814 0812 0810 0811 0814 0815 0817

1 0501 0587 0669 0725 0769 0813 0848 0856 0854 0846 0835 0830 0827 0823 0819 0818 0820 0820 0821

2

D55 sample

0210 0295 0400 0507 0618 0744 0852 0878 0878 0869 0855 0853 0851 0845 0838 0833 0832 0830 0829

3 0648 0676 0709 0748 0785 0824 0855 0860 0855 0846 0836 0832 0829 0825 0822 0821 0822 0822 0823

1 0502 0589 0672 0731 0778 0828 0868 0878 0876 0866 0852 0846 0841 0836 0831 0828 0828 0827 0827

2

D65 sample

0211 0296 0402 0511 0625 0755 0868 0898 0899 0888 0872 0869 0866 0859 0850 0843 0841 0838 0836

3 0649 0679 0713 0754 0795 0838 0875 0881 0875 0865 0852 0846 0842 0837 0832 0829 0830 0828 0828

1

0503 0590 0675 0736 0786 0839 0885 0896 0894 0882 0866 0859 0853 0847 0840 0836 0835 0833 0832

2

D75 sample

0211 0297 0403 0514 0630 0764 0881 0913 0915 0903 0886 0882 0878 0869 0859 0852 0849 0844 0841

3

Table A8 Spectral radiance factors of three pairs of non-fluorescent samples for computing ultraviolet-range metamerism index (CIE 1999, JIS Z8720)

495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600

0816 0818 0822 0826 0830 0833 0833 0833 0833 0834 0834 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0819 0820 0824 0828 0832 0834 0834 0834 0833 0834 0834 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0825 0825 0828 0831 0834 0836 0836 0835 0834 0835 0835 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0818 0820 0824 0828 0831 0833 0833 0833 0833 0834 0834 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0821 0823 0826 0830 0832 0834 0834 0834 0833 0834 0834 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0828 0828 0830 0833 0835 0837 0836 0836 0834 0835 0835 0835 0835 0836 0836 0837 0835 0836 0837 0837 0837 0838

0823 0824 0827 0831 0833 0835 0835 0835 0834 0835 0835 0834 0834 0835 0835 0836 0835 0836 0837 0837 0837 0838

0826 0827 0829 0833 0835 0836 0836 0836 0834 0835 0835 0835 0835 0836 0836 0837 0835 0836 0837 0837 0837 0838

0833 0833 0834 0837 0838 0839 0837 0837 0835 0836 0836 0835 0835 0836 0836 0837 0835 0836 0837 0837 0837 0838

0827 0828 0830 0834 0835 0837 0836 0836 0834 0836 0836 0835 0835 0836 0836 0837 0835 0836 0837 0837 0837 0838

0830 0831 0832 0835 0837 0838 0837 0837 0835 0836 0836 0835 0835 0836 0836 0837 0835 0836 0837 0837 0837 0838

0838 0837 0837 0840 0840 0841 0839 0839 0836 0837 0837 0836 0836 0837 0837 0838 0835 0836 0837 0837 0837 0838

605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700

Wavelength (nm)

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

1

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

2

D50 sample


0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

3 0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

1 0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

2

D55 sample

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

3 0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

1 0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

2

D65 sample

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

3 0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

1

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

2

D75 sample

0839 0840 0842 0844 0846 0848 0850 0852 0854 0856 0857 0857 0857 0858 0859 0860 0861 0862 0863 0864

3

380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485

Wavelength (nm)

187 236 294 347 517 1949 613 624 701 779 856 4367 1694 1072 1136 1189 1237 1275 1300 1315 1323 1317

F1

118 148 184 215 344 1569 385 374 419 462 506 3498 1181 627 663 693 719 740 754 762 765 762

F2*

082 102 126 144 257 1436 270 245 273 300 328 3185 947 402 425 444 459 472 480 486 487 485

F3 057 070 087 098 201 1375 195 159 176 193 210 3028 803 255 270 282 291 299 304 308 309 309

F4

Standard

187 235 292 345 510 1891 600 611 685 758 831 4076 1606 1032 1091 1140 1183 1217 1240 1254 1258 1252

F5 105 131 163 190 311 1480 343 330 368 407 445 3261 1074 548 578 603 625 641 662 658 659 656

F6 256 318 384 453 615 1937 737 705 771 841 915 4414 1752 1135 1200 1258 1308 1345 1371 1388 1395 1393

F7* 121 150 181 213 317 1308 383 345 386 442 509 3410 1242 768 860 946 1024 1084 1133 1171 1198 1217

F8 090 112 136 160 259 1280 305 256 286 330 382 3262 1077 584 657 725 786 835 875 906 931 948

F9

High color rendering

Table A9 Spectral distributions of typical fluorescent lamps (CIE 2004a, JIS Z 8719)

111 080 062 057 148 1216 212 270 374 514 675 3439 1486 1040 1076 1067 1011 927 829 729 791 1664

F10 091 063 046 037 129 1268 159 179 246 333 449 3394 1213 695 719 712 672 613 546 479 566 1429

F11*

Three band

096 064 045 033 119 1248 112 094 108 137 178 2905 790 265 271 265 249 233 210 191 301 1083

F12

490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595

Wavelength (nm)

1313 1285 1252 1220 1183 1150 1122 1105 1103 1118 1153 2774 1705 1355 1433 1501 1552 1829 1955 1548 1491 1415

F1


762 745 728 715 705 704 716 747 804 888 1001 2488 1664 1459 1616 1756 1862 2147 2279 1929 1866 1773

F2*

488 477 467 462 462 473 499 548 625 734 878 2382 1614 1459 1663 1840 1995 2311 2469 2141 2085 1993

F3 314 306 300 298 301 314 341 390 469 581 732 2259 1511 1388 1633 1868 2064 2428 2626 2328 2294 2214

F4

Standard

1247 1220 1189 1161 1133 1110 1096 1097 1116 1154 1212 2778 1773 1447 1520 1577 1610 1854 1950 1539 1464 1372

F5 656 642 628 620 619 630 660 712 794 907 1049 2522 1746 1563 1722 1853 1943 2197 2801 1941 1856 1742

F6 1382 1364 1343 1325 1308 1293 1278 1260 1244 1233 1226 2952 1705 1244 1258 1272 1283 1546 1675 1283 1267 1245

F7* 1228 1232 1235 1244 1255 1268 1277 1272 1260 1243 1222 2896 1651 1179 1176 1177 1184 1461 1611 1234 1253 1272

F8 961 968 974 988 1004 1026 1048 1063 1076 1096 1118 2771 1629 1228 1274 1321 1365 1657 1814 1455 1465 1466

F9


1673 1044 594 334 235 188 159 147 180 571 4098 7369 3361 824 338 247 214 486 1145 1479 1216 897

F10 1496 897 472 233 147 110 089 083 118 490 3959 7284 3261 752 283 196 167 443 1128 1476 1273 974

F11*

Three band

1188 688 343 149 092 071 060 063 110 456 3440 6540 2948 716 308 247 227 509 1196 1532 1427 1186

F12

600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730

1322 1219 1112 1003 895 796 702 620 542 473 415 364 320 281 247 218 193 172 167 143 129 119 108 096 088 081 077

1654 1521 1380 1236 1095 965 840 732 631 543 468 402 345 296 255 219 189 164 153 127 110 099 088 076 068 061 056

1867 1722 1565 1404 1245 1095 951 827 711 609 522 445 380 323 275 233 199 170 155 127 109 096 083 071 062 054 049

2091 1943 1774 1600 1442 1256 1093 952 818 701 600 511 436 369 313 264 224 191 170 139 118 103 088 074 064 054 049

1269 1157 1045 935 829 732 641 563 490 426 372 325 283 249 219 193 171 152 143 126 113 105 096 085 078 072 068

1609 1464 1315 1168 1025 895 774 669 571 487 416 355 302 257 220 187 160 137 129 105 091 081 071 061 054 048 044

1219 1189 1160 1135 1112 1095 1076 1042 1011 1004 1002 1011 987 865 727 644 583 541 504 457 412 377 346 308 273 247 225

1292 1312 1334 1361 1387 1407 1420 1416 1413 1434 1450 1446 1400 1258 1099 998 922 862 807 739 671 616 563 503 446 402 366

1461 1450 1439 1440 1447 1462 1472 1455 1440 1458 1488 1551 1547 1320 1057 918 825 757 703 635 572 525 480 429 380 343 312

652 831 4412 3455 1209 1215 1052 443 195 219 319 277 229 200 152 135 147 179 174 102 114 332 449 205 049 024 021

733 972 5527 4258 1318 1316 1226 511 207 234 358 301 248 214 154 133 146 194 200 120 135 410 558 251 057 027 023

928 1231 6853 5302 1467 1438 1471 646 257 275 418 344 281 242 164 136 149 214 234 142 161 504 698 319 071 030 026

Note Note Note Note

1. 2. 3. 4.

054 051 047 047 043 046 047 040 033 027 03721

F2*

046 043 039 039 035 038 039 033 028 021 04091

F3 046 042 037 037 033 035 036 031 026 019 04402

F4 067 065 061 062 059 062 064 055 047 040 03138

F5 043 040 037 038 035 039 041 033 026 021 03779

F6 206 190 175 162 154 145 132 117 099 081 03129

F7* 336 309 285 265 251 237 215 189 161 132 03458

F8 286 264 243 226 214 202 183 161 138 112 03741

F9


021 024 024 021 017 021 022 017 012 009 03458

F10 021 024 024 020 024 032 026 016 012 009 03805

F11*

023 028 028 021 017 021 019 015 010 005 04370

F12

Three band

4230

64

6430

76

57

3450

51

2940

72

6350

59

4150

90

6500

95

5000

90

4150

81

5000

83

4000

83

3000

03371 03751 03941 04031 03452 03882 03292 03586 03727 03588 03769 04042

075 073 068 069 064 068 069 061 052 043 03131

F1

Standard

The spectral distributions are obtained by dividing the spectral radiant flux in W/nm by the luminous flux in lumens. The spectral distributions should be taken as 0.00 outside the range defined in the table. Mercury emission lines are assigned proportionally to the neighboring wavelengths. The CIE gives priority to those lamps marked with an asterisk (∗ ) while JIS gives priority to those listed in bold.

735 740 745 750 755 760 765 770 775 780 Chromaticity coordinate x Chromaticity coordinate y Correlated color temperature General Color rendering Index

Wavelength (nm)


319

APPENDIX IV Table A10 Standard deviate functions for computing observer metamerism index (CIE 1989, JIS Z 8718) Wavelength (nm) 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585

x¯ $% −00001 −00003 −00009 −00026 −00069 −00134 −00197 −00248 −00276 −00263 −00216 −00122 −00021 00036 00092 00186 00263 00256 00225 00214 00205 00197 00187 00167 00146 00133 00118 00094 00061 00017 −00033 −00085 −00139 −00194 −00247 −00286 −00334 −00426 −00517 −00566 −00600 −00637

¯ y$%

¯ z$%

00000 00000 −00001 −00004 −00009 −00015 −00019 −00022 −00021 −00017 −00009 00005 00015 00008 −00003 −00005 −00011 −00036 −00060 −00065 −00060 −00045 −00031 −00037 −00047 −00059 −00060 −00025 00010 00005 −00011 −00020 −00028 −00039 −00044 −00027 −00022 −00073 −00127 −00129 −00126 −00162

−00002 −00010 −00036 −00110 −00294 −00558 −00820 −01030 −01140 −01079 −00872 −00455 −00027 00171 00342 00703 00976 00859 00641 00547 00475 00397 00319 00228 00150 00117 00096 00062 00029 00005 −00012 −00020 −00022 −00024 −00024 −00021 −00017 −00015 −00014 −00013 −00013 −00011

320

APPENDIX IV Table A10 (Continued) Wavelength (nm) 590 595 600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730 735 740 745 750 755 760 765 770 775 780

x¯ $% −00656 −00638 −00595 −00530 −00448 −00346 −00242 −00155 −00085 −00044 −00019 −00001 00010 00016 00019 00019 00017 00013 00009 00006 00004 00003 00002 00001 00001 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

¯ y$%

¯ z$%

−00196 −00199 −00187 −00170 −00145 −00112 −00077 −00048 −00025 −00012 −00006 00000 00003 00005 00006 00006 00006 00005 00003 00002 00001 00001 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

−00009 −00008 −00006 −00005 −00004 00000 00002 00000 −00002 −00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

References The following books and papers are referred to in this book. They are listed in alphabetical order of authors. Although numerous other books and papers were used by the authors, only selected ones are included below and in the Bibliography, taking ready availability into consideration. Agoston, G. A., Color Theory and Its Application in Art and Design, SpringerVerlag, Berlin, 1987. Baba, G., Shikisai no Jiten (Editors, Kawakami, G., Kodama, A., Tomiie, T., Ohta, N), Section 1.3 Colorimetry, pp.53–81, Asakura Shoten, Tokyo, 1987. Berman, S. M., Jewett, D. L., Bingham, L. R., Nahass, R. M., Perry, F. and Fein, G., Pupillary size differences under incandescent and high pressure sodium lamps, J. Illum. Eng. Soc. 16, 3–20 (1987). Berns, R. S., Billmeyer and Saltzman’s Principles of Color Technology, (3rd edition), John Wiley & Sons, New York, 2000. Billmeyer, Jr., F. W. and Saltzman, M., Principles of Color Technology (2nd edition), John Wiley & Sons, New York, 1981. Borges, F. C., Trichromatic approximation method for surface illumination, J. Opt. Soc. Am. A 8, 1319–1323 (1991). Bowmaker, J. K., Mollon, J. D. and Jacobs, G. H., Microspectrophotometric results for old and new world primates, in Colour Vision: Physiology and Psychophysics (J. D. Mollon and L. T. Sharpe, Editors.), pp.57–68, Academic Press, 1983. BSI, Calculation of Small Colour Differences, BS 6923, British Standards Institute, London, 1988. Brown, P. K. and Wald, G., Visual pigments in single rods and cones of the human retina, Science 144, 45–51 (1964). Burnham, R. W., Haynes, R. M. and Bartleson, C. J., Color: A Guide to Basic Facts and Concepts, John Wiley & Sons, New York, 1963.


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Index Abney’s Law 28–9 Abnormal color vision 217–19 Absolute temperature 82 Achromatic color 53 Adams–Nickerson color difference formula 132, 147–8 Adaptation 7–9 Adaptation, chromatic 188–97, 232–4 Adaptation, luminance 188–9 Adaptive color shift 190 Additive color mixing 57–61, 176–80 Additivity 28–9 Alychne 71 Aperture color 51 Basic stimulus 59, 64 Beer’s Law 213–14 BFD color difference formula 143 Black-body radiation 88–9, 113–14 Blind spot 7 Brightness 17, 32–4, 36–8, 49–50 Candela 23–5, 114 CCM, see Computer color matching Chroma 49, 51, 53, 61–2

Chromatic adaptation 188–97, 232–4 Chromatic color 53 Chromaticity 67 Chromaticity coordinates 66–7, 75–6, 165–6 Chromaticity diagram 66–7, 75–6 CIE 12 CIE 1924 spectral luminous efficiency function 12–17, 26–30 CIE 1931 standard colorimetric system 63–71 CIE 1960 UCS diagram 110, 119 CIE 1964 standard colorimetric system 71–4 CIE 1964 U ∗ V ∗ W ∗ color space 128–30 CIE 1976 chroma 133 CIE 1976 hue angle 133 CIE 1976 hue difference 134, 150–1 CIE 1976 L ∗ a ∗ b ∗ color space 130–42 CIE 1976 L ∗ u ∗ v ∗ color space 130–42 CIE 1976 lightness 130, 133, 146–7 CIE 1976 saturation 133


330 CIE 1976 UCS diagram 120–1 CIE 1994 color difference formula 143 CIE chromatic adaptation transforms 194–7, 225–6 CIE daylight illuminants 85–8, 92–6 CIE illuminant B 96 CIE illuminant C 55, 92 CIE illuminants D50 , D55 and D75 95 CIE illuminants F 248 CIE source C 95 CIE standard illuminant A 92, 94–5 CIE standard illuminant D65 92 CIE standard illuminants 92–6 CIE standard photometric observers 12–17, 26–30 CIE standard source A 94–5 CIE standard sources 92–6 CIE supplementary illuminants 93 CIEV function 12–17, 26–30 CIEV function 12–17, 26–30 CIECAM02 color appearance model 201–4 CIECAM97s color appearance model 200–1 CIEDE2000 color difference formula 143–4 CIELAB color space 130–42 CIELAB/CIELUV calculation 161–71 CIELAB/CIELUV comparison 134–40 CIELAB/CIELUV conversion 140–3 CIELUV color space 130–42 CMC (l c color difference formula 132, 143, 149–50 Color appearance models 198–204 Color appearance system 48–51 Color appearance value 49 Color blindness 217–19 Color constancy 189–91, 229–30

INDEX Color coordinates 131 Color difference 115–21 Color differences in CIELAB and CIELUV 132 Color equation 58–61 Color gamut 180, 185–8 Color matcher 262 Color matching 57–61, 261–8 Color matching coefficient 59 Color matching functions 59–61, 64–5 Color measurement 153–61 Color mixing 176–83 Color mixing systems 48–51 Color names 57 Color perception 49 Color rendering 229–36 Color rendering index 229–36, 256–8 Color sensation 50 Color solid 53 Color space 66 Color specification value 48 Color stimulus 50 Color system 48 Color temperature 82–5, 111–12 Color temperature altering power 112 Color temperature conversion filter 111–12 Color vision models 44–5, 197–8 Color vision, abnormal 217–19 Colorfulness 61–2 Colorimeters 153–6 Colorimetric calculations 161–71 Colorimetric purity 78–82 Colorimetric shift 189–90 Colorimetric values 50–1, 153 Colorimetry 153–61 Commission Internationale de l’Eclairage, see CIE Complementary wavelength 79 Computer color matching 261–8 Cones 5–6, 42–5, 216–21 Confusion colors 217–19 Correlated color temperature 82–5, 110–11, 230–1 Corresponding color 190–1

331

INDEX Cosine law 31 CRI, see Color rendering index Davis–Gibson filter 95 Daylight locus 85–6 Daylight simulators 95–6, 237–42 Defective color vision 217–19 Degree of metamerism 205–11 Degree of metamerism for change of illuminant 244–9 Degree of metamerism for change of observer 249–54 Designing spectral distributions 255–61, 268–9 Determinant 276 Diffuse reflectance 160 Diffuse transmittance 161 DIN color system 56 Direct comparison photometry 10 Dominant wavelength 78–82, 166–7 Excitation purity 78–82, 166–7, 258–61 Eye, human 4–7 Flag, Algerian 169 Flag, American 169 Flag, Canadian 169 Flag, Japanese 162–71 Flicker photometry 11 Fluorescent lamps 89–90, 235–6 Fluorescent materials, colorimetry of 172–4 Fluorescent samples for evaluating metamerism 239–42 Four-color theory 42 Fovea 71 Full radiator 89 Fundamental constants 272 Fundamental spectral responsivity 216–21 General color rendering index 235 Geometric conditions for measurement 158–61 Geometrical metamerism 205–6 Grassmann, Hermann 58

Grassmann’s laws 58 Greek letters 272–3 Guild, John 64 Helmholtz–Kohlrausch effect 36–8 Helmholtz, Hermann von 40 Helson–Judd effect 194 Hering, Ewald 41 Hue 49, 51, 52–4 Hue circle 54 Hunt effect 194 Hunter color difference formula 132, 148–9 Illuminance 17–22 Illuminance, retinal 26, 33 Illuminant 92 Illuminant color 52 Illuminant color shift 189–90 Illuminant metamerism 205–6 Illuminant metamerism index 244–9 Imaginary colors 67, 105–8 Incandescent lamp 88 Integrating sphere 157–8 Inverse matrix 277–8 Isotemperature line 83–4, 110–11 Jnd, see Just-noticeable difference Judd, Deane Brewster 69 Just-noticeable difference 81, 116–19 Kubelka–Munk model

183

Lambert–Beer law 181, 183, 211 Lambert–Beer law 213–14 Lambert’s cosine law 31 Lambert’s law 213–14 Light 1 Light-source color 52 Lightness 49–51, 122 Linear programming 210–11, 214–16, 224–5, 226–8, 255, 268–9 Lumen 23–5 Luminance 17–22, 36–8

332 Luminance adaptation 188–9 Luminous color 51 Luminous efficacy 11–17 Luminous efficacy, maximum value of 11–17, 26–30, 184–8, 214–15 Luminous exitance 17–22 Luminous flux 17–22 Luminous intensity 17–22 Luminous reflectance 75 Luminous transmittance 75 Luminous unit 59 Luther condition 153–4 Lux 23–5 MacAdam ellipses 117–21 Macular pigment 71 Matrix algebra 275–9 Maximum luminous efficacy 11–17, 26–30, 184–8, 214–15 Maxwell spot 71–2 Mercury lamp 91 Mesopic vision 8–9 Metal halide lamps 92 Metamer 76–8 Metamerism 76–8, 109–10, 204–11, 224–8, 244–54 Metamerism index 205–6, 237–42, 244–54 Metamerism index for change of illuminant 244–9 Metamerism index for change of observer 249–54 Modes of appearance 51–2 Monochromatic light 3 Monochromatic stimulus 59 Monochromator 156 Monte Carlo method 207–11 Munsell chroma 53–4 Munsell color system 52–7 Munsell hue 52–4 Munsell renotation system 53 Munsell value 53–4, 122–7, 144–5 Munsell, Albert Henry 52 Nayatani, Yoshinobu 194 Neuroepithelial cells 5–7 Newton–Raphson method 265–8

INDEX Newton, Isaac 1–3 Nickerson, Dorothy 148 Non-luminous color 51 Non-luminous line 71 Object color 51 Object-color metamerism 205–6 Observer metamerism 205–6 Observer metamerism index 249–54 Opponent-colors theory 39–46 Optical Society of America, see OSA color system Optimal colors 184–8, 215–16 OSA color system 56–7 Ostwald color system 56 Ostwald, Wilhelm 56 Partial derivatives 281–3 Pauli extension 131–2, 146–7 Perceived color 49 Perfect diffusing planar light source 20, 31–2, 34–6 Photoelectric colorimeter 154 Photographic dyes 182–3, 264, 266–7 Photographic film 46 Photometric calculations 26–9 Photometric quantities 17–22 Photometric quantities in X10 Y10 Z10 system 108–9 Photometry, direct comparison 10 Photometry, flicker 11 Photometry, physical 29–30 Photometry, step-by-step 9–10 Photometry, visual 9–11, 29 Photopic vision 8–9 Physical photometry 29–30 Planar light source 20, 31–2, 34–6 Planckian locus 83, 110 Planckian radiator 89 Planck’s law of radiation 88–9, 111 Platinum black body 23–4, 114 Point light source 32 Power law, Stevens 125 Prescribed chromaticity coordinates 255–6

INDEX Prescribed spectral distributions 255–61, 268–9 Primaries, additive 59 Primaries, subtractive 181 Psychometric lightness 127 Psychophysical color 51 Psychophysical quantity 18 Purity 78–82 Purkinje phenomenon 15 Purple boundary 67 Quantity of light 17, 22 Radiance factor 158–61 Radiation, electromagnetic 1 Radiometric quantities 17–18, 22 Reciprocal color temperature 84–5 Reciprocal correlated color temperature 84–5 Reference color stimulus 59, 64 Reflectance 158–61 Reflectance factor 158–61 Reflectance standard 174 Regula Falsi method 263–5 Regular transmittance 161 Related color 52 Responsivity, spectral 9–17 Resultant color shift 190–1 Retina 5–7 Retinal illuminance 26, 32 RGB color specification system 63–7, 96, 99–105 Rhodopsin 47–8 Rods 5–6, 44–5 Saturation 61–2 Scotopic vision 8–9 Sensitivity 9 SI 22–3 Sodium lamp 91–2 Special color rendering index 234–5 Spectral chromaticity coordinates 67 Spectral colorimetry 156–8 Spectral concentration 27 Spectral distribution 27

333 Spectral emissivity 89 Spectral luminous efficacy 11–17, 26–30 Spectral luminous efficiency 11–17, 26–30 Spectral stimulus 59 Spectral tristimulus value 59 Spectrophotometry 153, 156–8 Spectrum locus 67 Spectrum, electromagnetic 2–3 Specular reflection 160 Stage theory 44–6, 198 Standard deviate observer 249–54 Step-by-step photometry 9–10 Steradian 21–3 Stevens effect 194 Stevens power law 125 Stiles, Walter Stanley 72–4 Stimulus sum 80 Subtractive color mixing 180–3, 211–13 Subtractive primaries 181 Surface color 51 Système Internationale d’Unités, see SI Template colorimetry 154 Threshold of vision 15–16 Tint 242–3 Total reflectance 160 Total transmittance 161 Transmittance 161 Transmitted color 51 Transpose matrix 278–9 Trichromatic system 51 Trichromatic theory 39–46 Tristimulus colorimeter 153–6 Tristimulus filters 154–6 Tristimulus values 18, 50–1, 59–61, 74–6, 163–6 Troland 26 UCS diagrams 119 Ulbricht sphere 157–8 ULS, see Uniform lightness scales Ultraviolet-range metamerism index 239–42

334 Uniform chromaticity diagrams 115–21 Uniform-chromaticity-scale diagrams, see UCS diagrams Uniform color spaces 127–34 Uniform lightness scales 122–7 Unit matrix 275 Units, International System 22–3, 271–2 Units, photometric 21–6 Units, prefixes 272 Units, radiometric 22 Units, SI 22–3, 271–2 Unrelated color 52 uv chromaticity diagram 119 u‘v chromaticity diagram 120–1 V function 12–17, 26–30 V function 12–17, 26–30 Visible-range metamerism index 237–9 Visual colorimeter 153 Visual density 125 Visual photometry 9–11, 29

INDEX Visual pigment 46–8 Visual purple 46–8 von Helmholtz, Hermann 40 von Kries’ chromatic adaptation transform 193, 221–5 von Kries, Johannes 192 Weber–Fechner law 124 Weber’s Law 124 White balance 191 Whiteness 242–3 Wien’s displacement law 113 Wien’s law 111 Wright, William David 65 Wyszecki, Günter 128 x10 y10 chromaticity diagram 75–6 X10 Y10 Z10 system 71–4 xy chromaticity diagram 75–6 XYZ system 68–71, 99–105 Young, Thomas Zone theory

40

198

400

500

Wavelength (nm)

600

Plate 1MSpectrum of white light

Plate 2MColor tree of Munsell color system

700

Plate 3MCIE xy chromaticity diagram (colors are for illustration only and are limited by the gamut of the printing process)

Plate 4MTest colors for evaluating the color rendering properties of light sources

Colorimetry Fundamentals and Applications

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