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11-11-2016
Canara Engineering College 1
Digital Image Processing Unit 7:
Image Restoration
11/11/2016 ajaybolar.weebly.com 1
Preview • Goal of image restoration
– Improve an image in some predefined sense
– Difference with image enhancement ?
• Features
– Image restoration v.s image enhancement
– Objective process v.s. subjective process
– A prior knowledge v.s heuristic process
– A prior knowledge of the degradation phenomenon is considered
– Modeling the degradation and apply the inverse process to recover the original image
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Preview (cont.)
• Target
– Degraded digital image
– Sensor, digitizer, display degradations are less
considered
• Spatial domain approach
• Frequency domain approach
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Unit Outline
• A model of the image degradation / restoration process
• Noise models
• Restoration in the presence of noise only – spatial filtering
• Periodic noise reduction by frequency domain filtering
• Linear, position-invariant degradations
• Estimating the degradation function
• Inverse filtering
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(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Concept of Image Restoration
Image restoration is to restore a degraded image back to
the original image while image enhancement is to
manipulate the image so that it is suitable for a specific
application.
Degradation model:
),(),(),(),( yxyxhyxfyxg
where h(x,y) is a system that causes image distortion and
(x,y) is noise. 11/11/2016 ajaybolar.weebly.com 5
Noise models • Source of noise
– Image acquisition (digitization)
– Image transmission
• Spatial properties of noise
– Statistical behavior of the gray-level values of pixels
– Noise parameters, correlation with the image
• Frequency properties of noise
– Fourier spectrum
– Ex. white noise (a constant Fourier spectrum)
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Noise Models
Noise cannot be predicted but can be approximately described in
statistical way using the probability density function (PDF)
Gaussian noise: 22 2/)(
2
1)(
zezp
Rayleigh noise
azfor 0
for )(2
)(/)( 2
azeazbzp
baz
Erlang (Gamma) noise
0zfor 0
0for )()!1()(
1
zeazb
za
zpaz
bb
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Noise Models (cont.)
Exponential noise
Uniform noise
Impulse (salt & pepper) noise
azaezp )(
otherwise 0
afor a-b
1
)(bz
zp
otherwise 0
for
for
)( bzP
azP
zp b
a
11/11/2016 ajaybolar.weebly.com 8
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
PDF: Statistical Way to Describe Noise
PDF tells how much
each z value occurs.
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Noise probability density
functions
• Noises are taken as random variables
• Random variables
– Probability density function (PDF)
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Gaussian noise
• Math. tractability in spatial and frequency
domain
• Electronic circuit noise and sensor noise
22 2/)(
2
1)(
zezp
mean variance
1)( dzzpNote:
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Gaussian noise (PDF)
70% in [(), ()]
95% in [(2), (2)]
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Canara Engineering College 3
Uniform noise
otherwise 0
if 1
)(bza
abzp
12
)(
22
2 ab
ba
Mean:
Variance:
• Less practical, used for random number
generator
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Uniform Noise PDF
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Impulse (salt-and-pepper) nosie
otherwise 0
for
for
)( bzP
azP
zp b
a
If either Pa or Pb is zero, it is called unipolar.
Otherwise, it is called bipoloar.
•In practical, impulses are usually stronger than image
signals. Ex., a=0(black) and b=255(white) in 8-bit image.
• Quick transients, such as faulty switching during
imaging
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Impulse (salt-and-pepper) nosie PDF
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Test for noise behavior
• Test pattern
Its histogram:
0 255
11/11/2016 ajaybolar.weebly.com 17 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Image Degradation with Additive Noise
Original image
Histogram
Degraded images
),(),(),( yxyxfyxg
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Canara Engineering College 4
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Original image
Histogram
Degraded images
Image Degradation with Additive Noise (cont.)
),(),(),( yxyxfyxg
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Periodic noise • Source: electrical or electromechanical interference
during image acquisition
• Characteristics
– Spatially dependent
– Periodic – easy to observe in frequency domain
• Processing method
– Suppressing noise component in frequency domain
11/11/2016 ajaybolar.weebly.com 20
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Periodic Noise
Periodic noise
looks like dots
In the frequency
domain 11/11/2016 ajaybolar.weebly.com 21
Estimation of noise parameters • Periodic noise
– Observe the frequency spectrum
• Random noise with unknown PDFs
– Case 1: imaging system is available
• Capture images of “flat” environment
– Case 2: noisy images available
• Take a strip from constant area
• Draw the histogram and observe it
• Measure the mean and variance
11/11/2016 ajaybolar.weebly.com 22
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Estimation of Noise
We cannot use the image
histogram to estimate
noise PDF.
It is better to use the
histogram of one area
of an image that has
constant intensity to
estimate noise PDF.
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Restoration in the presence of Noise Only- Spatial Filtering
Degradation model:
),(),(),(),( yxyxhyxfyxg
To remove this part
When only degradation present in an image is noise,
Spatial filtering is the method of choice in situations when only
additive noise is present.
Enhancement and Restoration become almost indistinguishable
disciplines in this particular case.
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Canara Engineering College 5
Arithmetic mean filter or moving average filter
xySts
tsgmn
yxf),(
),(1
),(ˆ
Geometric mean filter
mn
Sts xy
tsgyxf
1
),(
),(),(ˆ
mn = size of moving window
Mean Filters (Noise Reduction Spatial Filters)
Achieves smoothing comparable to arithmetic mean filter, but it
tends to lose less image detail in the process. 11/11/2016 ajaybolar.weebly.com 25
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Geometric Mean Filter: Example
Original
image
Image
corrupted
by AWGN
Image
obtained
using a 3x3
geometric
mean filter
Image
obtained
using a 3x3
arithmetic
mean filter
AWGN: Additive White Gaussian Noise 11/11/2016 ajaybolar.weebly.com 26
Harmonic and Contraharmonic Filters
Harmonic mean filter
xySts tsg
mnyxf
),( ),(
1),(ˆ
Contraharmonic mean filter
xy
xy
Sts
Q
Sts
Q
tsg
tsg
yxf
),(
),(
1
),(
),(
),(ˆ
mn = size of moving window
Works well for salt noise
but fails for pepper noise
Works well for Gaussian Noise
Q = the filter order
Reduces salt and pepper noise,
Positive Q is suitable for
eliminating pepper noise.
Negative Q is suitable for
eliminating salt noise.
Cannot do both simultaneously
For Q = 0, the filter reduces to an arithmetic mean filter.
For Q = -1, the filter reduces to a harmonic mean filter. 11/11/2016 ajaybolar.weebly.com 27
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Contraharmonic Filters: Example
Image
corrupted
by pepper
noise with
prob. = 0.1
Image
corrupted
by salt
noise with
prob. = 0.1
Image
obtained
using a 3x3
contra-
harmonic
mean filter
With Q = 1.5
Image
obtained
using a 3x3
contra-
harmonic
mean filter
With Q=-1.5
11/11/2016 ajaybolar.weebly.com 28
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Contraharmonic Filters: Incorrect Use Example
Image
corrupted
by pepper
noise with
prob. = 0.1
Image
corrupted
by salt
noise with
prob. = 0.1
Image
obtained
using a 3x3
contra-
harmonic
mean filter
With Q=-1.5
Image
obtained
using a 3x3
contra-
harmonic
mean filter
With Q=1.5
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Order-Statistic Filters: Revisit
subimage
Original image
Moving
window
Statistic parameters
Mean, Median,
Min, Max, Etc.
Output image 11/11/2016 ajaybolar.weebly.com 30
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Canara Engineering College 6
Order-Statistics Filters
Median filter
),(median),(ˆ),(
tsgyxfxySts
Max filter
),(max),(ˆ),(
tsgyxfxySts
Min filter
),(min),(ˆ),(
tsgyxfxySts
Midpoint filter
),(min),(max2
1),(ˆ
),(),(
tsgtsgyxfxyxy StsSts
Reduce “dark” noise (pepper noise)
Reduce “bright” noise (salt noise)
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Median Filter : How it works
A median filter is good for removing impulse, isolated noise
Degraded image
Salt noise
Pepper noise
Moving
window
Sorted
array
Salt noise Pepper noise
Median
Filter output
Normally, impulse noise has high magnitude
and is isolated. When we sort pixels in the
moving window, noise pixels are usually
at the ends of the array.
Therefore, it‟s rare that the noise pixel will be a median value. 11/11/2016 ajaybolar.weebly.com 32
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Median Filter : Example
Image
corrupted
by salt-
and-pepper
noise with
pa=pb= 0.1
Images obtained using a 3x3 median filter
1
4
2
3
11/11/2016 ajaybolar.weebly.com 33 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Max and Min Filters: Example
Image
corrupted
by pepper
noise with
prob. = 0.1
Image
corrupted
by salt
noise with
prob. = 0.1
Image
obtained
using a 3x3
max filter
Image
obtained
using a 3x3
min filter
11/11/2016 ajaybolar.weebly.com 34
Alpha-trimmed Mean Filter
xySts
r tsgdmn
yxf),(
),(1
),(ˆ
where gr(s,t) represent the remaining mn-d pixels after
removing the d/2 highest and d/2 lowest values of g(s,t).
This filter is useful in situations involving multiple types of noise
such as a combination of salt-and-pepper and Gaussian noise.
Formula:
11/11/2016 ajaybolar.weebly.com 35 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Alpha-trimmed Mean Filter: Example
Image
corrupted
by additive
uniform
noise
Image
obtained
using a 5x5
arithmetic
mean filter
Image
additionally
corrupted
by additive
salt-and-
pepper
noise
1 2
2 Image
obtained
using a 5x5
geometric
mean filter
2
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Canara Engineering College 7
Alpha-trimmed Mean Filter: Example (cont.)
Image
corrupted
by additive
uniform
noise
Image
obtained
using a 5x5
median filter
Image
additionally
corrupted
by additive
salt-and-
pepper
noise
1 2
2
Image
obtained
using a 5x5
alpha-
trimmed
mean filter
with d = 5
2
11/11/2016 ajaybolar.weebly.com 37
Alpha-trimmed Mean Filter: Example (cont.)
Image
obtained
using a 5x5
arithmetic
mean filter
Image
obtained
using a 5x5
geometric
mean filter
Image
obtained
using a 5x5
median filter
Image
obtained
using a 5x5
alpha-
trimmed
mean filter
with d = 5
11/11/2016 ajaybolar.weebly.com 38
Adaptive Filters
• The filters discussed so far are applied to an entire image
without any regard for how image characteristics vary from
one point to another
• The behaviour of adaptive filters changes depending on the
characteristics of the image inside the filter region
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Adaptive Filter
-Filter behavior depends on statistical characteristics of local areas
inside mxn moving window
- More complex but superior performance compared with “fixed” filters
Statistical characteristics:
General concept:
xySts
L tsgmn
m),(
),(1
Local mean:
Local variance:
xySts
LL mtsgmn ),(
22 )),((1
2
Noise variance:
11/11/2016 ajaybolar.weebly.com 40
Adaptive, Local Noise Reduction Filter
Purpose: want to preserve edges
1. If 2 is zero, No noise
the filter should return g(x,y) because g(x,y) = f(x,y)
2. If L2 is high relative to
2, Edges (should be preserved),
the filter should return the value close to g(x,y)
3. If L2 =
2, Areas inside objects
the filter should return the arithmetic mean value mL
L
L
myxgyxgyxf ),(),(),(ˆ2
2
Formula:
Concept:
11/11/2016 ajaybolar.weebly.com 41 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Adaptive Noise Reduction Filter: Example
Image
corrupted
by additive
Gaussian
noise with
zero mean
and 2=1000
Image
obtained
using a 7x7
arithmetic
mean filter
Image
obtained
using a 7x7
geometric
mean filter
Image
obtained
using a 7x7
adaptive
noise
reduction
filter
11/11/2016 ajaybolar.weebly.com 42
11-11-2016
Canara Engineering College 8
Adaptive Median Filtering
• The median filter performs relatively well on impulse noise as
long as the spatial density of the impulse noise is not large
• The adaptive median filter can handle much more spatially dense
impulse noise, and also performs some smoothing for non-
impulse noise
• The key insight in the adaptive median filter is that the filter size
changes depending on the characteristics of the image
11/11/2016 ajaybolar.weebly.com 43
Adaptive Median Filtering
(cont…)
The key to understanding the algorithm is to
remember that the adaptive median filter has
three purposes:
– Remove impulse noise
– Provide smoothing of other noise
– Reduce distortion
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Algorithm: Level A: A1= zmedian – zmin
A2= zmedian – zmax
If A1 > 0 and A2 < 0, goto level B
Else increase window size
If window size <= Smax repeat level A
Else return zxy
Level B: B1= zxy – zmin
B2= zxy – zmax
If B1 > 0 and B2 < 0, return zxy
Else return zmedian
Adaptive Median Filter
zmin = minimum gray level value in Sxy
zmax = maximum gray level value in Sxy
zmedian = median of gray levels in Sxy
zxy = gray level value at pixel (x,y)
Smax = maximum allowed size of Sxy
where
Purpose: want to remove impulse noise while preserving edges
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Level A: A1= zmedian – zmin
A2= zmedian – zmax
Else Window is not big enough
increase window size
If window size <= Smax repeat level A
Else return zxy
zmedian is not an impulse
B1= zxy – zmin
B2= zxy – zmax
If B1 > 0 and B2 < 0, zxy is not an impulse
return zxy to preserve original details
Else return zmedian to remove impulse
Adaptive Median Filter: How it works
If A1 > 0 and A2 < 0, goto level B
Level B:
Determine
whether zmedian
is an impulse or not
Determine
whether zxy
is an impulse or not
11/11/2016 ajaybolar.weebly.com 46
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Adaptive Median Filter: Example
Image corrupted
by salt-and-pepper
noise with
pa=pb= 0.25
Image obtained
using a 7x7
median filter
Image obtained
using an adaptive
median filter with
Smax = 7
More small details are preserved
11/11/2016 ajaybolar.weebly.com 47 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Periodic Noise Reduction by Freq. Domain Filtering
Band reject filter Restored image
Degraded image DFT
Periodic noise
can be reduced by
setting frequency
components
corresponding to
noise to zero.
11/11/2016 ajaybolar.weebly.com 48
11-11-2016
Canara Engineering College 9
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Band Reject Filters
Use to eliminate frequency components in some bands
Periodic noise from the
previous slide that is
Filtered out.
11/11/2016 ajaybolar.weebly.com 49 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Notch Reject Filters
A notch reject filter is used to eliminate some frequency components.
11/11/2016 ajaybolar.weebly.com 50
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Notch Reject Filter:
Degraded image DFT Notch filter
(freq. Domain)
Restored image Noise 11/11/2016 ajaybolar.weebly.com 51
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Example: Image Degraded by Periodic Noise
Degraded image
DFT
(no shift)
Restored image Noise DFT of noise 11/11/2016 ajaybolar.weebly.com 52
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Estimation of Degradation Model
Degradation model:
),(),(),(),( yxyxhyxfyxg
Purpose: to estimate h(x,y) or H(u,v)
),(),(),(),( vuNvuHvuFvuG
Methods:
1. Estimation by Image Observation
2. Estimation by Experiment
3. Estimation by Modeling
or
Why? If we know exactly h(x,y), regardless of noise, we can do
deconvolution to get f(x,y) back from g(x,y).
11/11/2016 ajaybolar.weebly.com 53
Estimation by Image Observation
f(x,y) f(x,y)*h(x,y) g(x,y)
Subimage
Reconstructed
Subimage
),( vuGs ),( yxgs
),(ˆ yxf s
DFT
DFT ),(ˆ vuFs
Restoration process by estimation
Original image (unknown) Degraded image
),(ˆ
),(),(),(
vuF
vuGvuHvuH
s
ss
Estimated Transfer
function
Observation
This case is used when we
know only g(x,y) and cannot
repeat the experiment! 11/11/2016 ajaybolar.weebly.com 54
11-11-2016
Canara Engineering College 10
Estimation by Experiment
Used when we have the same equipment set up and can repeat the
experiment.
Input impulse image
System
H( )
Response image from the system
),( vuG
),( yxg),( yxA
AyxADFT ),(
A
vuGvuH
),(),(
DFT DFT
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
11/11/2016 ajaybolar.weebly.com 55 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Estimation by Modeling
Used when we know physical mechanism underlying the image
formation process that can be expressed mathematically.
Atmospheric
Turbulence model
6/522 )(),( vukevuH
Example: Original image Severe turbulence
k = 0.00025 k = 0.001
k = 0.0025
Low turbulence Mild turbulence
11/11/2016 ajaybolar.weebly.com 56
Estimation by Modeling: Motion Blurring
Assume that camera velocity is ))(),(( 00 tytx
The blurred image is obtained by
dttyytxxfyxg
T
))(),((),( 00
0
where T = exposure time.
dtdxdyetyytxxf
dxdyedttyytxxf
dxdyeyxgvuG
T
vyuxj
vyuxj
T
vyuxj
0
)(2
00
)(2
0
00
)(2
))(),((
))(),((
),(),(
11/11/2016 ajaybolar.weebly.com 57
Estimation by Modeling: Motion Blurring (cont.)
dtevuF
dtevuF
dtdxdyetyytxxfvuG
T
tvytuxj
T
tvytuxj
T
vyuxj
0
))()((2
0
))()((2
0
)(2
00
00
00
),(
),(
))(),(( ),(
Then we get, the motion blurring transfer function:
dtevuH
T
tvytuxj
0
))()((2 00),(
For constant motion ),())(),(( 00 btattytx
)(
0
)(2 ))(sin()(
),( vbuaj
T
vbuaj evbuavbua
TdtevuH
11/11/2016 ajaybolar.weebly.com 58
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Motion Blurring Example
For constant motion
)())(sin()(
),( vbuajevbuavbua
TvuH
Original image Motion blurred image
a = b = 0.1, T = 1 11/11/2016 ajaybolar.weebly.com 59
Restoration Model
f(x,y) Degradation
Model f(x,y)
Restoration
Filter
Unconstrained Constrained
• Inverse Filter
• Pseudo-inverse Filter • Wiener Filter
demos/demo5blur_invfilter/ 11/11/2016 ajaybolar.weebly.com 60
11-11-2016
Canara Engineering College 11
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Inverse Filter
after we obtain H(u,v), we can estimate F(u,v) by the inverse filter:
),(
),(),(
),(
),(),(ˆ
vuH
vuNvuF
vuH
vuGvuF
From degradation model:
),(),(),(),( vuNvuHvuFvuG
Noise is enhanced
when H(u,v) is small. To avoid the side effect of enhancing
noise, we can apply this formulation
to freq. component (u,v) with in a
radius D0 from the center of H(u,v).
In practical, the inverse filter is not popularly used. 11/11/2016 ajaybolar.weebly.com 61
Inverse Filtering
Limitations:
• Even if the degradation function is known the undegraded image cannot be
recovered exactly because N(u,v) is the random function which is not known.
• If the degradation function has „0‟ or small value the ratio easily dominates
the estimate F(u,v)
• One approach to get rid of 0 or small value problem is to limit the filter
frequency to the value near the origin.
11/11/2016 ajaybolar.weebly.com 62
Inverse Filter: Example
6/522 )(0025.0),( vuevuH
Original image
Blurred image
Due to Turbulence
Result of applying
the full filter
Result of applying
the filter with D0=70
Result of applying
the filter with D0=40
Result of applying
the filter with D0=85
11/11/2016 ajaybolar.weebly.com 63
WIENER FILTERING • Inverse filtering has no explicit provision for handling noise
• The wiener filter incorporates both degradation function, and statistical
characteristics of noise in the restoration process.
• Objective of the wiener filter is to find the estimate of uncorrupted image f,
such that the mean square error is minimum.
• The wiener filter is an optimum filter
• Conditions
• (1) Noise and image are uncorrelated
• (2) One or the other has zero mean
• (3) Gray levels in ˆ f are linear function of gray levels in g 11/11/2016 ajaybolar.weebly.com 64
65
Norbert Wiener (1894-1964)
The renowned MIT professor Norbert Wiener
was famed for his absent-mindedness. While
crossing the MIT campus one day, he was
stopped by a student with a mathematical
problem. The perplexing question answered,
Norbert followed with one of his own: "In which
direction was I walking when you stopped me?"
he asked, prompting an answer from the
curious student. "Ah," Wiener declared,
"then I've had my lunch”
Anecdote of Norbert Wiener
11/11/2016 ajaybolar.weebly.com (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Wiener Filter: Minimum Mean Square Error Filter
Objective: optimize mean square error: 22 )ˆ( ffEe
),(),(/),(),(
),(
),(
1
),(),(/),(),(
),(
),(),(),(),(
),(),(),(ˆ
2
2
2
*
2
*
vuGvuSvuSvuH
vuH
vuH
vuGvuSvuSvuH
vuH
vuGvuSvuHvuS
vuSvuHvuF
f
f
f
f
Wiener Filter Formula:
where
H(u,v) = Degradation function
S(u,v) = Power spectrum of noise Sf(u,v) = Power spectrum of the undegraded image 11/11/2016 ajaybolar.weebly.com 66
11-11-2016
Canara Engineering College 12
Wiener Filter: Minimum Mean Square Error Filter…
• H(u, v) = degradation function
• H∗(u, v) = complex conjugate of H(u, v)
• |H(u, v)|2 = H∗(u, v) H(u, v)
• Sη(u, v) = |N(u, v)|2 = power spectrum of the noise
• Sf (u, v) = |F(u, v)|2 = power spectrum of undegraded image
• No problem with zeros unless H(u, v) and Sη(u, v) are both zero
• When noise is zero, Wiener filter = inverse filter
11/11/2016 ajaybolar.weebly.com 67
Approximation of Wiener Filter
),(),(/),(),(
),(
),(
1),(ˆ
2
2
vuGvuSvuSvuH
vuH
vuHvuF
f
Wiener Filter Formula:
Approximated Formula:
),(),(
),(
),(
1),(ˆ
2
2
vuGKvuH
vuH
vuHvuF
Difficult to estimate
Practically, K is chosen manually to obtained the best visual result!
Since Sη(u, v) = |N(u, v)| 2 and Sf (u, v) = |F(u, v)|2 are seldom
known, the Wiener filter is frequently approximated
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Wiener Filtering
ADVANTAGES:
1.The wiener filter does not have zero value problem.
2.The result obtained by wiener filter is more closer to the original image
than inverse filter.
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Image Example
motion blurred image deblurred image after wiener filtering
(K=0.01)
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71
Image Example (Con’t)
K=0.1 K=0.001 K=0.01
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Wiener Filter: Example
Original image
Blurred image
Due to Turbulence
Result of the
full inverse filter Result of the inverse
filter with D0=70
Result of the
full Wiener filter 11/11/2016 ajaybolar.weebly.com 72
11-11-2016
Canara Engineering College 13
Wiener Filter: Example (cont.)
Original image
Result of the inverse
filter with D0=70
Result of the
Wiener filter
Blurred image
Due to Turbulence 11/11/2016 ajaybolar.weebly.com 73
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Example: Wiener Filter and Motion Blurring
Image
degraded
by motion
blur +
AWGN
Result of the
inverse filter
Result of the
Wiener filter
2=650
2=325
2=130
Note: K is
chosen
manually
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Degradation model:
),(),(),(),( yxyxhyxfyxg
Written in a matrix form
Constrained Least Squares Filter
ηHfg
Objective: to find the minimum of a criterion function
1
0
1
0
22 ),(M
x
N
y
yxfC
Subject to the constraint 22
ˆ ηfHg
),(),(),(
),(),(ˆ
22
*
vuGvuPvuH
vuHvuF
We get a constrained least square filter
where
P(u,v) = Fourier transform of p(x,y) =
010
141
010
where www T2
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Constrained Least Squares Filter: Example
),(),(),(
),(),(ˆ
22
*
vuGvuPvuH
vuHvuF
Constrained least square filter
is adaptively adjusted to achieve the best result.
Results from the previous slide obtained from the constrained least square filter
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Constrained Least Squares Filter: Example (cont.)
Image
degraded
by motion
blur +
AWGN
Result of the
Constrained
Least square
filter
Result of the
Wiener filter
2=650
2=325
2=130
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Constrained Least Squares Filter:Adjusting
Define fHgr ˆ It can be shown that 2
)( rrr T
We want to adjust gamma so that a22
ηr
where a = accuracy factor 1. Specify an initial value of
2. Compute
3. Stop if is satisfied
Otherwise return step 2 after increasing if
or decreasing if Use the new value of to recompute
1
1
2r
a22
ηr
a22
ηr
),(),(),(
),(),(ˆ
22
*
vuGvuPvuH
vuHvuF
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11-11-2016
Canara Engineering College 14
Constrained Least Squares Filter:Adjusting (cont.)
),(),(),(
),(),(ˆ
22
*
vuGvuPvuH
vuHvuF
),(ˆ),(),(),( vuFvuHvuGvuR
1
0
1
0
22),(
1 M
x
N
y
yxrMN
r
1
0
1
0
22 ),(1 M
x
N
y
myxMN
1
0
1
0
),(1 M
x
N
y
yxMN
m
mMN 22η
2η
2rFor computing
For computing
11/11/2016 ajaybolar.weebly.com 79
(Images from Rafael C.
Gonzalez and Richard E.
Wood, Digital Image
Processing, 2nd Edition.
Constrained Least Squares Filter: Example
Original image
Blurred image
Due to Turbulence
Results obtained from constrained least square filters
Use wrong noise
parameters
Correct parameters:
Initial = 10-5
Correction factor = 10-6
a = 0.25
2 = 10-5
Wrong noise parameter
2 = 10-2
Use correct noise
parameters
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Geometric Mean filter
),(
),(
),(),(
),(
),(
),(),(ˆ
1
2
*
2
*
vuG
vuS
vuSvuH
vuH
vuH
vuHvuF
f
This filter represents a family of filters combined into a
single expression
= 1 the inverse filter
= 0 the Parametric Wiener filter
= 0, = 1 the standard Wiener filter
= 1, < 0.5 More like the inverse filter = 1, > 0.5 More like the Wiener filter
Another name: the spectrum equalization filter 11/11/2016 ajaybolar.weebly.com 81
Geometric Transformation
These transformations are often called rubber-sheet transformations:
Printing an image on a rubber sheet and then stretch this sheet according to some predefine set of rules.
A geometric transformation consists of 2 basic operations:
1. A spatial transformation :
Define how pixels are to be rearranged in the spatially
transformed image.
2. Gray level interpolation :
Assign gray level values to pixels in the spatially
transformed image.
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Geometric Transformation : Algorithm
Distorted image g
1. Select coordinate (x,y) in f to be restored
2. Compute
),( yxrx
),( yxsy
3. Go to pixel
in a distorted image g
),( yx
Image f to be
restored
4. get pixel value at
By gray level interpolation
),( yxg
5. store that value in pixel f(x,y)
1 3
5
),( yx ),( yx
11/11/2016 ajaybolar.weebly.com 83 (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Spatial Transformation
To map between pixel coordinate (x,y) of f and pixel coordinate (x‟,y‟) of g
),( yxrx ),( yxsy
For a bilinear transformation mapping between a pair of Quadrilateral regions
4321),( cxycycxcyxrx
8765),( cxycycxcyxsy ),( yx ),( yx
To obtain r(x,y) and s(x,y), we need
to know 4 pairs of coordinates
and its corresponding
which are called tiepoints.
),( yx ),( yx
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11-11-2016
Canara Engineering College 15
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
Gray Level Interpolation: Nearest Neighbor
Since may not be at an integer coordinate, we need to Interpolate the value of
),( yx
),( yxg
Example interpolation methods that can be used:
1. Nearest neighbor selection
2. Bilinear interpolation 3. Bicubic interpolation
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Geometric Distortion and Restoration Example
Original image and
tiepoints
Tiepoints of distorted
image
Distorted image Restored image
Use nearest
neighbor
intepolation
(Images from Rafael C.
Gonzalez and Richard E.
Wood, Digital Image
Processing, 2nd Edition. 11/11/2016 ajaybolar.weebly.com 86
Geometric Distortion and Restoration Example (cont.)
Original image and
tiepoints
Tiepoints of distorted
image
Distorted image Restored image
Use bilinear
intepolation
(Images from Rafael C.
Gonzalez and Richard E.
Wood, Digital Image
Processing, 2nd Edition. 11/11/2016 ajaybolar.weebly.com 87
Example: Geometric Restoration
(Images from Rafael C.
Gonzalez and Richard E.
Wood, Digital Image
Processing, 2nd Edition.
Original image Geometrically distorted image
Difference between 2 above images
Restored image
Use the same
Spatial Trans.
as in the previous example
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