Image Restoration

 The main objective of restoration is to improve
the quality of a digital image which has been
degraded due to Various phenomena like:
 Motion
 Improper focusing of Camera during image
acquisition.
 Noise
Modified from restoration.ppt by
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 The purpose of image restoration is to restore a degraded/distorted
image to its original content and quality.
 Restoration involves following process:-
 Modeling of Degradation
 Applying the inverse process to recover the original image
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 We will assume that a degradation function exists,
which, together with additive noise, operates on the
input image f(x,y) to produce a degraded image g(x,y).
 The objective of restoration is to obtain an estimate for
the original image from its degraded version g(x,y)
while having some knowledge about the degradation
function H and the noise
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 the degraded image in the spatial domain is
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Therefore, in the frequency domain it is

 The Principal source of noise in digital images arise
during image Acquisition and transmission.
 In Acquiring images with Camera and Light levels are
major factor affecting the amount of noise in resulting
image.
 Images are corrupted during transmission due to
interferences in the channel used for transmission.for
example:image is transmitted using a wireless network
might be corrupted due to atmospheric disturbances.
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 Some important noise probability density functions
 – Gaussian noise
 – Rayleigh noise
 -Erlang gamma noise
 – Exponential noise
 – Impulse
),(),(),(
),(),(),(
vuNvuFvuG
yxyxfyxg

 
Modified from restoration.ppt by Yu Hen Hu
Degradation models :
noise only

 The PDF of a Gaussian random variable z given by:
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z represents intensity,
ž is the mean (average) value of z ,
is its standard deviation.
is the variance of z.

 The Gaussian noise arises in an image due to factors
such as electronic circuit noise and sensor noise
due to poor illumination. The images acquired by
image scanners exhibit this phenomenon.
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 Rayleigh noise is specified as
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 Rayleigh noise PDF is helpful in characterizing
noise phenomena in range imaging. (e.g-X-
ray,ultraviolet imaging which depend upon the
frequency of light )
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 Erlang noise is specified as
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Here a > 0 and b is a positive integer. The mean and variance are
given by

 Gamma noise finds in laser imaging.
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Exponential noise is specified as
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Here a > 0. The mean and variance are given by
Exponential pdf is a special case of Erlang pdf with b =1.Used in laser imaging.

 Impulse (salt-and-pepper) noise (bipolar) is
specified as
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If b>a, intensity b will appear as a light dot on the image and a appears as a
dark dot If either Pa or Pb is zero the noise is called unipolar.If neither
probability is zero, and especially if they are approximatly equal, impulse noise
value will resemble salt and peeper granules randomely distributed over the
image.for this reason , bipolar impulse noise is called salt and peeper noise.

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 Mean filters
› Arithmetic mean filter
› Geometric mean filter
› Harmonic mean filter
› Contra-harmonic mean
filter
 Order statistics filters
› Median filter
› Max and min filters
› Mid-point filter
› alpha-trimmed filters
 Adaptive filters
› Adaptive local noise
reduction filter
› Adaptive median filter
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 Arithmetic mean filter:
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The arithmetic mean filter computed the average value of the corrupted
image g(x,y) in the area defined by Sxy. Let Sxy represent the set of
coordinates in a rectangular neighborhood of size m x n, centered at the
point (x,y).
Effect:The Mean filter simply smoothes the variations in an
image.noise is reduced as a result of blurring

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Effect:
Geometric mean filter achieves smoothing comparable to the
arithmetic mean filter but it preserves more details (It means
loss less image detail in the process)

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Effect:
Harmonic mean filter works well for salt noise and other types of
noise (such as Gaussian) but fails for pepper noise.

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Here Q is the order of the filter. This filter is well suited for
reducing the effects of salt-pepper noise. For positive values of Q,
eliminates pepper noise; for negative values of Q, it eliminates salt
noise. This filter cannot reduce both simultaneously.
Notice that contraharmonic filter reduces to the arithmetic mean
filter when Q = 0 and to the harmonic mean filter if Q = -1.

 Median filter:It replaces the pixel value by the median of the
intensity levels in the neighborhood of that pixel:
 Effect:
 Median filters provide excellent results for certain types of
noise with considerably less blurring than linear smoothing
filters of the same size. These filters are very effective against
both bipolar and unipolar noise. The same filter can be
 applied more than once to yield better results.
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 
,( , )
ˆ( , ) ( , )
x ys t S
f x y median g s t


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Effective for
removing salt-
and-paper
(impulsive)
noise.

 Max filters:
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This filter is useful for finding the brightest points in an image;
therefore, its effective against pepper noise.
Min filters:
This filter is useful for finding the darkest points in an
image;therefore, its effective against salt noise.( it reduces the salt
noise because it will eliminate the higher gray values in the
subimage.

 It computes the midpoint between the maximum and
minimum values of intensities:
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This filter is a combination of order statistics and averaging and
works best for Gaussian and uniform noise contaminations.

 if we delete d/2 highest intensity values and d/2 lowest
intensity values of g(s,t) in the neighbourhood sxy., denote the
rest as gr(s.t), a filter that averages what is left is alpha-
trimmed mean filter:
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d can range from 0 to mn-1. When d = 0, this filter reduces to
the arithmetic mean filter, when d = mn-1, this filter reduces
to a median filter. For other values of d, the filter is useful in
situation with noise of multiple types, such as a combination
of salt-and-pepper and Gaussian noise.

 Adaptive filters are those filters whose behavior
changes based on the statistical characteristics of the
image inside the filter region defined by a
rectangular window size Sxy.
 It is better than the mean filter and order statistics
filter.
 Two types of filter
› Adaptive local noise reduction filter
› Adaptive median filter
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 It uses two statistical parameters, mean and variance for the
elimination of noise.
 Mean Parameter:It gives the average gray value.
 Variance:It provides the estimate of the contrast in the image.
 the adaptive filter is:
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 The response of filter is based on four quantities:-
 G(x,y) the value of noisy image at (x,y).
 the variance of noise corrupting f(x,y) to form g(x,y).
 mL local mean of pixels in the Sxy
 The local variance of the pixels in Sxy
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 The behavior of the Adaptive filter is
obtained as:

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 Adaptive filter achieves approximately the same
performance in noise cancellation but adds much less
blurring than the mean filters.
 Adaptive filtering typically yields considerably
better results in overall performance at the price of
filter complexity.
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 It can handle impulse noise with larger probabilities than
traditional median filter. It operates on a rectangular region
Sxy, whose size is changing. Window size is variable to
improve efficiency
 Adaptive median filter has 3 goals:
 to remove impulse noise,
 To provide smoothing
 to reduce distortion
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 Three cases were implemented:
 With Salt and Pepper noise alone
 With non impulsive noise alone
 With both included
 Variations in the window size were introduced
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Salt and Pepper
Standard Median output
Adaptive median Output

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Gaussian Noise Standard Median output
Adaptive Median Output

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Gaussian and impulsive Noise
Standard Median output
Adaptive Median output

 The adaptive median filter successfully removes
impulsive noise from images. It does a reasonably
good job of smoothening images that contain non-
impulsive noise.
 When both types of noise are present, the
algorithm is not as successful in removing
impulsive noise and its performance deteriorates.
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 This noise typically comes from electrical or
electromechanical interference during image acquisition.
 It can be reduced via frequency domain filtering.
 The image is corrupted by sinusoidal noise of various
frequencies.
 The parameters of periodic noise are estimated by inspection
of Fourier spectrum of the image.
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 Bandreject filters
 Bandpass filters
 Notch aFilters
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 Bandreject filters remove or attenuate frequencies
about the origin of the Fourier transform.
 Ideal Bandreject Filter:
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Where
D(u,v) :distance from the origin of the centered freq.
Do :Radial centre
W- width of the band

Image Restoration

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