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EEM 463 Introduction to Image Processing
Week 5: Image Restoration and Reconstruction
Fall 2013
Instructor: Hatice Çınar Akakın, Ph.D.
Anadolu University
12.11.2013
Image Restoration
• Image restoration: recover an image that has been degraded by using a prior knowledge of the degradation phenomenon.
• Model the degradation and applying the inverse process in order torecover the original image.
• The principal goal of restoration techniques is to improve an image insome predefined sense.
• Although there are areas of overlap, image enhancement is largely asubjective process, while restoration is for the most part an objectiveprocess.
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A Model of the Image Degradation/Restoration Process
• The degraded image in the spatial domain:𝑔 𝑥, 𝑦 = ℎ 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦 + 𝜂 𝑥, 𝑦
• Frequency domain representation𝐺 𝑢, 𝑣 = 𝐻 𝑢, 𝑣 𝐹 𝑢, 𝑣 + 𝑁(𝑢, 𝑣)
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Noise Models• The principal sources of noise in digital images arise during image
acquisition and/or transmission • Light levels and sensor temperature during acquisition
• Lightning or other atmospheric disturbance in wireless network duringtransmission
• White noise: Fourier spectrum this noise is constant• carryover from the physical proporties of White light, which contains nearly
all frequencies in the visible spectrum in equal proportions.
• With the exception of spatially periodic noise, we assume• Noise is independent of spatial coordinates
• Noise is uncorrelated with respect to the image itself
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Gaussian Noise
• The pdf of a Gaussian random variable, z, is given by
𝑝 𝑧 =1
2𝜋𝜎𝑒−(𝑧−𝑧)2/2𝜎2
where z represents intensity, 𝑧 is the mean (average) value of z ,
and σ is its standard deviation.• 70% of its values will be in the range
• 95% of its values will be in the range
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)(),(
)2(),2(
Rayleigh Noise
• The pdf of Rayleigh noise is given by
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2( ) /
The PDF of Rayleigh noise is given by
2( ) for
( )
0 for
z a bz a e z ap z b
z a
2
The mean and variance of this density are given by
/ 4
(4 )
4
z a b
b
Erlang (Gamma) Noise
• The pdf of Erlang noise is given by
• where a > 0 and b is a positive integer. The mean and variance
of this density are given by
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1
The PDF of Erlang noise is given by
for 0 ( ) ( 1)!
0 for
b baza z
e zp z b
z a
2 2
The mean and variance of this density are given by
/
/
z b a
b a
Exponential Noise
• The PDF of exponential noise is given by
where a > 0 . The mean and variance of this density are given by
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The PDF of exponential noise is given by
for 0 ( )
0 for
azae zp z
z a
2 2
The mean and variance of this density are given by
1/
1/
z a
a
Uniform Noise
• The PDF of uniform noise is given by
• The mean and variance of this density function are given by
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The PDF of uniform noise is given by
1 for a
( )
0 otherwise
z bp z b a
2 2
The mean and variance of this density are given by
( ) / 2
( ) /12
z a b
b a
Impulse (Salt-and-Pepper) Noise
• The PDF of impulse noise is given by
• If b > a , intensity b appears as a light dot in the image. Conversely, intensity a will appear like a dark dot.
• If either Pa or Pb is zero, the impulse noise is called unipolar.
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The PDF of (bipolar) impulse noise is given by
for
( ) for
0 otherwise
a
b
P z a
p z P z b
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• Example: Original test image
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Periodic Noise• Periodic noise in an image
arises typically fromelectrical orelectromechanicalinterference during imageacquisition.
• It can be reducedsignificantly by usingfrequency domain filtering
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Estimation of Noise Parameters
• The parameters of periodic noise can be estimated by inspection ofthe Fourier spectrum of the image.
• Periodic noise tends to produce frquency spikes, which are detectableeven by visual analysis
• Automated analysis is possible if the noise spikes are eitherexceptionally pronounced, or when knowledge is available about thegeneral location of the frequency components of the interference.
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• It is often necessary to estimate the noise probability densityfunctions for a particular imaging arrangement.
• When images already generated by a sensor are available, it may bepossible to estimate the parameters of the probability densityfunctions from small patches of reasonably constant backgroundintensity.
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The shape of the histogram identifies the closest PDF match
• If the image stripe (subimage) S is given,• Calculate mean and variance of intensity levels
where S denote a stripe and pS (zi ) , i = 0,1,2,...,L −1, denote the probability estimates of the intensities of the pixels in S
• The shape of the histogram identifies the closest probability density functionmatch.
• The Gaussian probability density function is completely specified by these twoparameters.
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Consider a subimage denoted by , and let ( ), 0, 1, ..., -1,
denote the probability estimates of the intensities of the pixels in .
The mean and variance of the pixels in :
s iS p z i L
S
S
1
0
12 2
0
( )
and ( ) ( )
L
i s i
i
L
i s i
i
z z p z
z z p z
Restoration in the Presence of Noise Only̶ Spatial Filtering • When the only degradation present in an image is noise,
• Since the noise terms are unknown, subtracting them from g(x,y ) or G(u,v ) is not a realistic option.
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Noise model without degradation
( , ) ( , ) ( , )
and
( , ) ( , ) ( , )
g x y f x y x y
G u v F u v N u v
Mean Filters
• Arithmetic mean filter
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Let represent the set of coordinates in a rectangle
subimage window of size , centered at ( , ).
xyS
m n x y
( , )
Arithmetic mean filter
1 ( , ) ( , )
xys t S
f x y g s tmn
• Geometric mean filter
• A geometric mean filter achieves smoothing comparable to thearithmetic mean filter, but it tends to lose less image detail in theprocess.
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1
( , )
Geometric mean filter
( , ) ( , )xy
mn
s t S
f x y g s t
• Harmonic mean filter
which works well for some types of noise like Gaussian noise and salt noise, but fails for pepper noise.
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( , )
Harmonic mean filter
( , )1
( , )xys t S
mnf x y
g s t
• Contraharmonic mean filter
where Q is called the order of the filter.• It is well suited for reducing or eliminating the effects of salt-and-pepper
noise.• When Q = 0 , the contraharmonic mean filter reduces to the arithmetic
mean filter.• When Q = −1 , the contraharmonic mean filter becomes the harmonic
mean filter.
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1
( , )
( , )
Contraharmonic mean filter
( , )
( , )( , )
xy
xy
Q
s t S
Q
s t S
g s t
f x yg s t
Q>0 for pepper noise and Q<0 for salt
noise.
Example
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corrupted version with additive
Gaussian noise of zero
mean and variance of 400.
In general, the arithmetic and geometric mean filters aresuited for random noise like Gaussian or uniform noise.
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• The positive-order filter did a better job
of cleaning the background, at the expense
of slightly thinning and blurring the dark
areas.
• The opposite was true of the negative-
order filter.
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The contraharmonic mean filter is well suited forimpulse noise, with the disadvantage that it mustbe known whether the noise is dark or light in order to select Q .
Order Statistic Filters
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( , )
Max filter
( , ) max ( , )xys t S
f x y g s t
( , )
Median filter
( , ) ( , )xys t S
f x y median g s t
( , )
Min filter
( , ) min ( , )xys t S
f x y g s t
The median filters are particularly effective in the presence ofboth bipolar and unipolar impulse noise.
• Suppose that, we delete d /2 lowest and the d /2 highest intensity values of g(s,t) in Sxy . Let gr (s,t) represent the remaining mn −d pixels, an alpha-trimmed mean filter is given by
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( , )( , )
Midpoint filter
1 ( , ) max ( , ) min ( , )
2 xyxy s t Ss t Sf x y g s t g s t
( , )
Alpha-trimmed mean filter
1 ( , ) ( , )
xy
r
s t S
f x y g s tmn d
The midpoint filter works best for random distributed noise,like Gaussian or uniform noise.
When d = 0 , the alpha-trimmedmean filter is reduced to thearithmetic mean filter.If d = mn −1 , the alpha-trimmed mean filter becomes a median filter.
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Note that, repeated passesof a median filter will blur
the image, so keep thenumber of passes as low as
possible!!
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The min filter did a better job on noise removal, but it removessome white points around the border of light objects.
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Adaptive filters
• Adaptive filters are capable of performance superior to that of thefilters discussed thus far. However, the price paid for improvedfiltering power is an increase in filter complexity.
• The behavior changes based on statistical characteristics of the imageinside the filter region defined by the mхn rectangular window.
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Adaptive, Local Noise Reduction Filters
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2
: local region
The response of the filter at the center point (x,y) of
is based on four quantities:
(a) ( , ), the value of the noisy image at ( , );
(b) , the variance of the noise corrupti
xy
xy
S
S
g x y x y
2
ng ( , )
to form ( , );
(c) , the local mean of the pixels in ;
(d) , the local variance of the pixels in .
L xy
L xy
f x y
g x y
m S
S
The mean gives a measure of average intensity in the region over which the mean is computed, and thevariance gives a measure of contrast in that region.
• We want to have the following behaviours for the filter:
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2
2
The behavior of the filter:
(a) if is zero, the filter should return simply the value
of ( , ).
(b) if the local variance is high relative to , the filter
should return a value cl
g x y
ose to ( , );
(c) if the two variances are equal, the filter returns the
arithmetic mean value of the pixels in .xy
g x y
S
This is the zero-noise case in which g(x,y) is equal to f (x,y) .
A high local variance typically is associated with edges,which should be preserved.
It occurs when the local area has the same properties as the overall image, and local noise is to bereduced simply by averaging.
• Based on these assumptions, an adaptive expression for obtaining
𝑓(𝑥, 𝑦)may be written as
𝑓 𝑥, 𝑦 = 𝑔 𝑥, 𝑦 −𝜎𝜂
2
𝜎𝐿2 𝑔 𝑥, 𝑦 − 𝑚𝐿
• The only quantity needed to be estimated is the variance of theoverall noise, 𝜎𝜂
2, and other parameters can be computed from thepixels in Sxy .
• A tacit assumption is 𝜎𝜂2≤ 𝜎𝐿
2 , which is reasonable because Sxy is a subset of g(x,y) .
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𝜎𝜂2 = 1000
Adaptive Median Filter
• The adaptive median filtering can handle impulse noise withprobabilities larger than median filter (i.e. 0.2)
• Unlike other filters, the adaptive median filter changes the size of Sxyduring operation, depending on certain conditions.
• Consider the following notations
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min
max
med
max
The notation:
minimum intensity value in
maximum intensity value in
median intensity value in
intensity value at coordinates ( , )
maximum all
xy
xy
xy
xy
z S
z S
z S
z x y
S
owed size of xyS
• The adaptive median filtering algorithm works in two stages:
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med min med max
max
The adaptive median-filtering works in two stages:
Stage A:
A1 = ; A2 =
if A1>0 and A2<0, go to stage B
Else increase the window size
if window size , re
z z z z
S
med
min max
med
peat stage A; Else output
Stage B:
B1 = ; B2 =
if B1>0 and B2<0, output ; Else output
xy xy
xy
z
z z z z
z z
• Note that this algorithm has three main purposes:
1. to remove salt-and-pepper (impulse) noise;
2. to provide smoothing of other noise that may not be impulsive;
3. to reduce the distortion of object boundaries.
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It can be observed that with the similarnoise removal performance, the adaptive median filter did abetter job of preserving sharpness and detail.
Periodic Noise Reduction by Frequency Domain Filtering • Periodic noise appears as concentrated bursts of energy in the Fourier
transform, at locations corresponding to the frequencies of the periodic interference
• Periodic noise can be analyzed and filtered effectively by usingfrequency domain techniques.
• A selective filter is used to isolate the noise
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Perspective plots of ideal, Butterworth, and Gaussian bandreject filters
• One of the principal applications of bandreject filtering is for noise removal inapplications where the general location of the noise component(s) in thefrequency domain is approximately known
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The noise components can be seen as symmetric pairs of
bright dots in the Fourier spectrum
Since the component lie on an approximate circle about the origin of the transform, so a
circularly symmetric bandrejectfilter is a good choice!
Bandpass Filters
• A bandpass filter performs the opposite operation of a bandrejectfilter.
• The transfer function HBP (u,v) of a bandpass filter is obtained from a corresponding bandreject filter transfer function HBR(u,v) by using the equation
• HBP (u,v) = 1 - HBR (u,v)
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Notch Filters
• A notch filter rejects/passes frequencies in predefined neighbourhoods about a center frequency.
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• When several interference components are present, the methodsdiscussed previously are not always acceptable because they mayremove too much image information in the filtering process.
• Alternative filtering methods that reduce these degradations arenecessary!
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Optimum Notch Filtering
• Solution: first filter out the noise interference byplacing a notch pass filter H(u,v) at the location of each spike:
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( , ) ( , ) ( , )NPN u v H u v G u v Fourier transform of the corrupted image
Fourier transform of the interference noise
pattern
• Since the formation of HNP(u,v) requires judgment about what is or is not an interference spike, the notch pass filter generally is constructed interactively by observing the spectrum of G(u,v) on a display.
• After a particular filter has been selected, the corresponding pattern in the spatial domain is obtained from the expression
• Since the corrupted image is assumed to be formed by the addition of the uncorrupted image f(x,y) and the interference, if 𝜂(𝑥, 𝑦) were known, to obtain f(x,y) would be a simple matter
• The filtering procedure usually yields only an approximation of the true pattern. The effect of components not present in the estimate of 𝜂(𝑥, 𝑦) can be minimized by subtracting from g( x,y ) a weighted portion of 𝜂(𝑥, 𝑦) to obtain an estimate of f(x,y ):
𝑓 𝑥, 𝑦 = 𝑔 𝑥, 𝑦 − 𝑤(𝑥, 𝑦)𝜂(𝑥, 𝑦)
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1( , ) ( , ) ( , )NPx y H u v G u v Denklemi buraya yazın.
Refer your book forderivation!
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The origin was not shifted to the center of
the frequency plane in this case, so u v = = 0 is at the top left corner
in Figure 5.21.
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Optimum Notch Filtering
• It is optimum, in the sense that it minimizes local variances of the restored estimate 𝑓(𝑥, 𝑦)
• Apply the following procedure for restoration:• Isolate the principal contributions of the interference pattern
• Subtract a variable, weighted portion of the pattern from the corrupted image
1. Extract the principal frequency components of the interference pattern • Place a notch pass filter, HNP(u,v) at the location of each spike.
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( , ) ( , ) ( , )NPN u v H u v G u v Fourier transform of the corrupted image
Fourier transform of the interference noise
pattern
• Since the formation of HNP(u,v) requires judgment about what is or is not an interference spike, the notch pass filter generally is constructed interactively by observing the spectrum of G(u,v) on a display.
• After a particular filter has been selected, the corresponding pattern in the spatial domain is obtained from the expression
• Since the corrupted image is assumed to be formed by the addition of the uncorrupted image f(x,y) and the interference, if 𝜂(𝑥, 𝑦) were known, to obtain f(x,y) would be a simple matter
• The filtering procedure usually yields only an approximation of the true pattern. The effect of components not present in the estimate of 𝜂(𝑥, 𝑦) can be minimized by subtracting from g( x,y ) a weighted portion of 𝜂(𝑥, 𝑦) to obtain an estimate of f(x,y ):
𝑓 𝑥, 𝑦 = 𝑔 𝑥, 𝑦 − 𝑤(𝑥, 𝑦)𝜂(𝑥, 𝑦)12.11.2013
1( , ) ( , ) ( , )NPx y H u v G u v Denklemi buraya yazın.
