Resmi N.G. Reference: Digital Image Processing,2 nd Edition Rafael C. Gonzalez Richard E. Woods Woods
Image Restoration
A Model of the Image Degradation/ Restoration Process
Noise Models
Important Noise Probability Density Functions
Gaussian Noise
Rayleigh Noise
Erlang or Gamma Noise
Exponential Noise
Uniform Noise
Impulse or Salt-and-Pepper Noise
Periodic Noise
Estimation of Noise Parameters
Spatial Filtering – Filtering – Restoration Restoration in the presence of noise only
Mean Filters
Arithmetic Mean Filter
Geometric Mean Filter Harmonic Mean Filter
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Image Restoration
An objective process where it attempts to reconstruct or recover an image that has been degraded by using a priori knowledge of degradation phenomenon.
Image Degradation/Restoration model
f(x,y) : Input Image
η(x,y) : Additive Noise
g(x,y) : Degraded Image ^
f(x,y) : Estimate of the Original Image
The more we know about the degradation function H and the additive noise η, the closer is the estimate to the original image. Degraded image in spatial domain: If H is a linear, position-invariant process, then the degraded image is given by
g(x,y) = h(x,y) * f(x,y) + η(x,y)
h(x,y) is the spatial representation of the degradation function.
* indicates spatial convolution.
Frequency Domain Representation: G(u,v) = H(u,v)F(u,v) + N(u,v) (Hint: Convolution in spatial domain is equal to multiplication in frequency domain.)
Image Restoration
A Model of the Image Degradation/ Restoration Process
Noise Models
Important Noise Probability Density Functions
Gaussian Noise
Rayleigh Noise
Erlang or Gamma Noise
Exponential Noise
Uniform Noise
Impulse or Salt-and-Pepper Noise
Periodic Noise
Estimation of Noise Parameters
Spatial Filtering – Filtering – Restoration Restoration in the presence of noise only
Mean Filters
Arithmetic Mean Filter
Geometric Mean Filter Harmonic Mean Filter
Noise Models
The principal sources of noise in digital images arise during image acquisition and transmission.
Most types of noise are modeled as probability density functions (PDFs) represented as p(z) for gray levels z.
Parameters can be estimated based on histogram on small flat area of an image.
1. Gaussian noise Arises in image from factors like electronic circuit noise, sensor noise due to poor illumination or high temperature
Where
z : Gray level μ : Mean average value of z σ : Standard deviation of μ
• 70% of values are in [(μ -σ),(μ+σ)]
2. Rayleigh noise
Where a,b are positive integers. Mean and variance are
Helpful in range imaging.
3.Erlang (Gamma) noise
Where a>0; b is a positive integer. If the equation includes Gamma function then the density is more Erlang appropriately called density.
Application in laser imaging.
4.Exponential noise
Where a > 0 and and b=1. It is a special case of Erlang PDF with b=1. Mean and Variance are given by
Application in laser imaging.
5.Uniform noise
Basis for random number generators that are used in simulators.
6.Impulse (salt-and-pepper) noise
Found where quick transients take place during imaging (as in faulty switching).
• If b > a, gray-level gray-level b will will appear as a light light dot in the image. • Conversely, level a will appear like a dark dot. • If either Pa or Pb is zero, impulse noise is called unipolar. •If neither is zero and are approx. equal, noise
Original Image
Periodic Noise
Arises from electrical or electromechanical interference during image acquisition.
Spatially dependent noise.
Can be reduced significantly by frequency domain filtering.
Image Restoration
A Model of the Image Degradation/ Restoration Process
Noise Models
Important Noise Probability Density Functions
Gaussian Noise
Rayleigh Noise
Erlang or Gamma Noise
Exponential Noise
Uniform Noise
Impulse or Salt-and-Pepper Noise
Periodic Noise
Estimation of Noise Parameters
Spatial Filtering – Filtering – Restoration Restoration in the presence of noise only
Mean Filters
Arithmetic Mean Filter
Geometric Mean Filter Harmonic Mean Filter
Estimation of Noise parameters
Parameters of periodic noise – estimated by inspecting the Fourier spectrum of the image.
Parameters of noise PDFs – known partially from sensor specifications.
When only sensor images are available, the parameters of the PDF can also be estimated from small patches of reasonably constant gray level.
Histogram can also be used to identify the PDF.
Estimation of noise parameters 1.
2.
Experimentally we can usually choose a small patch of an image that is i s relatively uniform and compute a histogram of the image over that region.
The shape of the histogram identifies the closest PDF match.
Estimation of noise parameters 3. Using the histogram, we can estimate the noise mean and variance as follows:
where z where zi s are the gray-level values of pixels in strip S, and p(z ) i are the corresponding normalized histogram values. 5. The mean and variance are used to solve for the parameters and b in the density function. a and b
Image Restoration
A Model of the Image Degradation/ Restoration Process
Noise Models
Important Noise Probability Density Functions
Gaussian Noise
Rayleigh Noise
Erlang or Gamma Noise
Exponential Noise
Uniform Noise
Impulse or Salt-and-Pepper Noise
Periodic Noise
Estimation of Noise Parameters
Spatial Filtering – Filtering – Restoration Restoration in the presence of noise only
Mean Filters
Arithmetic Mean Filter
Geometric Mean Filter Harmonic Mean Filter
1.a Arithmetic Mean Filter
Let Sxy represent the set of coordinates in a rectangular sub-image window of size mn, centered at point (x,y).
The arithmetic mean filter computes the average value of the corrupted image g(x,y) in the area defined by S xy.
The value of the restored image f at any point (x,y) is given by 1 f ( x, y) g( s, t) ˆ
ˆ
mn (
)
1.b Geometric Mean Filter
Achieves smoothing comparable to arithmetic mean filter but tends to lose image detail deta il in the process.
1.c Harmonic Mean Filter
Works well for salt noise but fails for pepper noise. Works well with other types of noise like Gaussian noise.
1.d Contraharmonic Mean Filter
g (s, t )
Q 1
f ( x, y) ˆ
( s ,t )S xy
g (s, t )
Q
( s ,t )S xy
• • • • •
where Q is called the order of the filter. Well-suited for eliminating the effects of salt-andpepper noise. For positive values of Q, it eliminates Pepper noise. For negative values of Q, it eliminates Salt noise. Cannot work simultaneously. Reduces to Arithmetic Mean filter if Q=0 and
Arithmetic and Geometric Mean filters – well suited for random noise like Gaussian or uniform noise.
Contraharmonic filter – well suited for impulse noise.
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
2.a Median filter f ( x, y) median g( s, t)
ˆ
( s ,t )S xy
•Replaces the value of a pixel by the median of graylevels in the neighborhood of that pixel. •Median represents the 50th percentile of a ranked set of numbers. •For random noise, it provides excellent noise-reduction with lesser blurring than linear smoothing filters of similar size.
2.b Max and Min Filters f ( x, y) max
g( s, t)
f ( x, y) min
g( s, t)
ˆ
( s ,t )S xy
ˆ
( s ,t )S xy
Max filter –
Uses 100th percentile. Used for finding the brightest points in an image. Reduces pepper noise.
Min filter –
Uses 0th percentile. Used for finding the darkest points in the image. Reduces salt noise.
2.c Midpoint Filter max g( s, t) min g( s, t) f ( x, y) ( s ,t )S 2 ( s ,t )S
ˆ
1
xy
xy
Computes midpoint between the maximum and minimum values in the area encompassed by the filter.
Works best for randomly distributed noise (Gaussian or uniform noise).
2.d Alpha-t Alpha-trimmed rimmed Filters
If d/2 lowest and d/2 highest gray-level values of g(s,t) in the neighborhood Sxy are deleted, and if gr(s,t) represents the remaining mn-d pixels, then the alpha-trimmed mean filter formed by averaging the remaining pixels is given by
f ( x, y)
ˆ
1
mn d ( s ,t )S xy
gr ( s, t)
Where, d ranges from 0 to mn-1. When d=0, the filter reduces to arithmetic mean filter. filter. When d= (mn-1)/2, the filter reduces to median filter.
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Adaptive, Local Noise Reduction Filter
Mean – Mean – measure measure of average gray-level in a region. Variance – Variance – measure measure of average contrast in a region. Response of a filter at (x,y) operating on a region S is based on: g(x,y) – g(x,y) – the the value of noisy image at (x,y) σ2η – the variance of the noise corrupting f(x,y) to form g(x,y) – the local mean of the pixels in S xy. mL – the σ2L – the – the local variance of pixels in Sxy.
The filter expression is
f ( x, y) g ( x, y) ˆ
2 2 L
g ( x, y) m L
Zero-noise case: If σ2η is zero, the filter returns the value of g(x,y) which is equal to f(x,y). If local variance is high relative to σ2η, the filter should return a value close to g(x,y). If the two variances are equal, the filter returns the arithmetic mean value of pixels in the neighborhood.
Adaptive Median Filter
Can handle impulse noise with larger probabilities.
Preserves detail while smoothing non-impulse noise.
Adaptive median filter increases the size of subimage during filter operations.
Output of any filter is a single value used to replace the value of the pixel at (x,y), the point on which the window is centered at a given time.
zmin = minimum gray level value in S xy.
zmax = maximum gray level value in S xy.
zmed = median of gray levels in S xy.
zxy = gray level at coordinates (x,y).
Smax = maximum allowed size of S xy.
Two-level algorithm Level A: A1 = z med - zmin A2 = z med - zmax AND A2<0, goto level B. If A1>0 AND Else increase the window size. If window size ≤ S max, repeat level A. Else output zxy.
Level B: B1 = z xy - zmin B2 = z xy - zmax If B1>0 AND B2<0, output z xy. Else output zmed.
Three main purposes:
To remove salt-and-pepper noise
To provide smoothing of other noise that may not be impulsive
To reduce distortion(excessive thinning or thickening of object boundaries).
Zmin and zmax are considered impulse-like noise components.
Purpose of level A is to determine if the median filter output zmed is an impulse (black or white) noise or not.
If the condition z min < zmed < zmax holds, then zmed cannot be an impulse. Goto level B and test to see if the point z xy in the centre of the window is itself an impulse.
If B1>0 AND B2<0, then z min < zxy < zmax. zxy cannot be an impulse. Algorithm outputs z
If the condition B1>0 AND B2<0 does not hold, then either zxy = zmin or zxy = zmax. In either case, the value of pixel is an extreme value and the algorithm outputs the median value, zmed.
Suppose, A finds an impulse. Then, it increases the size of the window and repeats level A. Continues until the algorithm either finds a median value that is not an impulse or maximum window size is reached.
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Bandreject Filters
Bandreject filters remove (or attenuate) a band of frequencies, around some frequency, frequency, say D0 . An ideal bandreject filter is given by: W if D(u, v) D0 1 2 W W H( u, v) 0 if D0 D( u, v) D0 2 2 W if D(u, v) D0 1 2
where Du , v u 2 v 2
W - width of the band D0 is the radial centre.
D(u,v) is the distance from the origin of the centered frequency rectangle.
Butterworth Bandreject filters
A Butterworth bandreject filter of order n is given by
H( u, v)
1
D(u, v)W 1 2 2 D ( u, v) D0
2n
A Gaussian bandreject filter is given by 1 D ( u, v )
2 D( u , v )W 2
H( u, v) 1 e
2 D0
2
Bandreject Filters
Example: Bandreject Filters Filters
Bandpass Filters
Bandpass filters perform the opposite operation of bandreject filters. They pass a band of frequencies, around some frequency, say D0 (rejecting the rest).
The transfer function of a bandpass filter is obtained from a corresponding bandreject filter as: H bp(u,v) = 1 - H br (u,v)
Bandpass filter is usually used to isolate components of an image that correspond to a band of frequencies.
It can also be used to isolate noise pattern, so that a more detailed analysis of the noise can be performed, independent of the image.
Notch Filter Filterss
It is a kind of bandreject/bandpass filter that rejects/passes a very narrow set of frequencies, around a center frequency.
Due to symmetry of Fourier transform, the notch filters must occur in symmetric pairs about the origin of the frequency plane.
The transfer function of an ideal notch-reject filter of radius D0 with centers at (u0 ,v0) and (-u0 ,-v0) is given by:
or D2 (u, v) D0 0 if D1 (u, v) D0 or H( u, v) otherwise 1 1
Where
2 2 2 M N u0 v v0 D1 ( u, v) u 2 2 1
And
2 2 2 M N u0 v v0 D2 ( u, v) u 2 2
The transfer function of a Butterworth notch-reject filter of order n is given by: 1 H( u, v) n 2 D0 1 ( , ) ( , ) D u v D u v 2 1
A Gaussian notch reject filter has the form
1 D1 ( u , v ) D2 ( u , v )
2
H( u, v) 1 e
2 D0
Notch-pass filter passes the frequencies contained in the notch areas.
Performs exactly the opposite function as notch-reject filters.
Transfer function is given by:
H np(u,v) = 1 - H nr (u,v)
Optimum Notch Filtering
When several interference patterns are present, filtering may remove much image information.
Solution - first filter out the noise interference by placing a notch pass filter H(u,v) at the location of each spike: N(u,v) = H(u,v)G(u,v) is the Fourier transform of the corrupted G(u,v) image.
Corresponding pattern in the spatial domain is obtained as (x,y) (x,y) = F -1{N(u,v)} = F -1{H(u,v) G(u,v)}
(x,y) We can then subtract off a weighted portion of (x,y) x,y) to obtain our restored image: from the image g( x,y
f ( x, y) g ( x, y ) w( x, y ) ( x, y )
w(x,y) is called weighting or modulation function. It can be chosen so as to minimize the variance of the estimate
f ( x, y) over a specified neighbourhood of every point
(x,y).
Refer word doc for derivation) derivation ) ( Refer
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Linear,, Position-Invariant Degradations Linear
Input-output relationship before restoration stage g(x,y) = H[f(x,y)] + η(x,y)
Assume η(x,y) = 0.
Therefore, g(x,y) = H[f(x,y)]
H is linear if :
H[af 1(x,y)+bf 2(x,y)] = aH[f 1(x,y)]+bH[f 2(x,y)]
If a = b = 1, H[f 1(x,y)+f 2(x,y)] = H[f 1(x,y)]+H[f 2(x,y)]
This property is called additivity called additivity..
If f 2(x,y) = 0, H[af 1(x,y)] = aH[f 1(x,y)]
This property is called homogeneity called homogeneity..
An operator satisfying g(x,y) = H[f(x,y)] is said to be position be position invariant if:
H[ f (x-α (x-α, y-β y-β) ] = g( x-α x-α, y-β y-β )
That is, response at any point in the image depends only on the value of input at that point, not in its position.
In terms of continuous impulse function,
f ( x, y)
f ( , ) ( x , y ) d d
g ( x, y ) H [ f ( x, y )]
H f ( , ) ( x , y ) d d
H f ( , ) ( x , y ) d d
f ( , ) H ( x , y ) d d
h (x,α, y, y,β) is the impulse response of H.
If η(x,y) = 0,
g ( x, y)
f ( , )h( x, , y, )dd
This is called the superposition integral of the first kind.
A linear system h is completely characterized by its impulse response.
If H is position invariant,
H ( x , y ) h( x , y )
g ( x, y)
f ( , )h( x , y )dd
f ( , ) h( x , y ) d d
is called the convolution the convolution integral .
That is, the response g is the convolution of impulse response and the input function.
Convolution
Presence of Noise
g ( x, y)
f ( , )h( x, , y, )dd ( x, y)
If H is position invariant,
g ( x, y )
f ( , ) h( x , y ) d d ( x , y )
h( x, y ) * f ( x, y) ( x, y) G (u, v) H (u, v) F (u, v) N (u, v)
Degradation is modeled as convolution.
Restoration is modeled as deconvolution.
Restoration filters are hence called deconvolution filters.
Estimating the degr degradation adation function
Estimation by observation
Estimation by experimentation
Estimation by mathematical modeling
Estimation by Observation
No knowledge about the degradation function H.
Gather information from image itself.
Let: gs(x,y) be the observed subimage.
f ( x, y)
be the constructed subimage.
Assume negligible noise (choose strong signal area).
Then,
Hs ( u, v)
Gs (u, v)
F s (u, v)
From the characteristics of Hs, deduce the complete function H(u,v) assuming position invariance.
Estimation by Experimentation
Accurate estimate of the degradation can be obtained if device similar to the one used for capturing degraded image is available.
Obtain the impulse response of degradation by imaging an impulse using the same system settings.
Linear space invariant system is completely described by its impulse response.
Assume negligible noise.
H(u, v)
G (u , v) A
Fourier transform of an impulse is a constant which describes the strength of the impulse.
Estimation Estima tion by Modeling
A) Takes Takes into account the environmental conditions 5
k u 2 v2
H ( u, v) e
6
Where k is a constant that depends on nature of atmospheric turbulence.
B) Image blurred due to uniform and linear motion between image and sensor during acquisition.
Let f(x,y) undergo planar motion.
x0(t) and y0(t) be the time varying components of motion in x and y directions.
Total exposure at any point of recording medium is obtained by integrating instantaneous exposure over the time interval during which the shutter is open.
Let T be the duration of exposure.
T
g ( x, y)
f ( x x (t ), y y (t ))dt 0
0
0
G(u, v)
g ( x, y)e
j 2 ( ux vy )
dxdy
T f ( x x0 ( t), y y0 ( t)) dt e 0
j2 ( ux vy)
dxdy
T f ( x x0 ( t), y y0 ( t)) e 0 shif sh ifte ted d F ( u ,v ) T
j2 ( ux ( t) vy ( t)) dt F (u , v)e 0
0
0
T
j2 ( ux ( t) vy ( t)) dt F (u , v) e 0
0
0
j2 ( ux vy) dxdy dt
Def D efine ine the the trans transfe ferr func functio tion n, T
j2 ( ux0 ( t) vy0 ( t))
H( u, v) e 0
G (u, v) H (u, v) F (u , v)
dt
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Inverse Inv erse Filtering Filtering
The simplest method to restore images degraded by a degradation function H is direct inverse filtering.
It computes an estimate original image as
F (u, v)
F (u, v )
G(u, v) H(u, v)
of the transform of the
G (u, v) H (u , v) F (u, v) N (u , v )
F (u, v) H (u , v) H (u, v) F (u , v) N (u , v )
F (u, v) F (u, v)
N ( u , v) H(u, v)
Cannot restore the image exactly with knowledge of H(u,v) because N(u,v) is unknown. If H(u,v) has values 0,
N(u, v) H(u, v)
will dominate F(u,v).
Inverse filtering is hence very sensitive to noise and has no provision to handle noise.
One way to avoid values of H(u,v) that tend to zero is to limit the filter frequencies to values near the origin.
H(0,0), the average value of h(x,y) is the highest value of H(u,v).
Blurring (degradation) corresponds to lowpass filtering and inverse filtering corresponds to highpass filtering.
Order-Statistics Filters Median Filter Max and Min Filters Mid-point Filter Alpha-trimmed Mean Filter Adaptive Filters Adaptive, local noise reduction filter Adaptive median filter Frequency Domain Filtering - Periodic Noise Reduction Bandreject Filters Bandpass Filters Notch Filters Linear Position-Invariant Degradations Inverse Filtering Minimum Mean Square Error (MMSE) or o r Weiner Weiner Filtering Constrained Least Squares Filtering Geometric Mean Filter
Wiener (MMSE) Filtering
Refer Page: 284-286
Constrained Least Squares Filtering
Refer Page: 288-291
Laplacian Operator 2 2 f f 2 f 2 2 x y
2 f x 2 2 f y 2
f ( x 1, y) f ( x 1, y) 2 f ( x, y) f ( x, y 1) f ( x, y 1) 2 f ( x, y)
2 f f ( x 1, y) f ( x 1, y) f(
1)
f(
1) 4 f (
)
coeff f ( x 1, y 1) coeff f ( x, y 1) coeff f ( x 1, y 1) p( x, y) coeff f( x 1, y) coeff f( x, y) coeff f( x 1, y) coeff f ( x 1, y 1) coeff f ( x, y 1) coeff f ( x 1, y 1) 0 1 0 1 4 1 0 1 0
Geometric Mean Filter
Refer Page: 292
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