Image Restoration

1. IMAGE
RESTORATION
MS.S.Y.SALUNKHE
SITCOE

2. To reconstruct or recover an image that has been distorted by a known
degradation phenomenon. Restoration techniques thus try to model the
degradation process and apply the inverse process in order to reconstruct the
original image.
These techniques are oriented toward modeling the degradation and then
applying the inverse process in order to recover the original image.
Image Restoration refers to a class of methods that aim to remove or reduce
the degradations that have occurred while the digital image was being
obtained. All natural images when displayed have gone through some sort of
degradation:
a)During display mode
b)Acquisition mode.
c)Processing mode.
The degradations may be due to
a)Sensor noise, b)Blur due to camera mis focus, c)Relative object-camera
motion , d)Random atmospheric turbulence , e)Others

3. Image Degradation &
Restoration Model
The degradation process is generally modelled as a H degradation
function and the additive noise term η(x, y) together they yield g(x,y)
Given g(x,y , some knowledge about H , and η(x,y) it is the objective of
restoration to estimate f(x,y) Of course this estimate should be as close as
possible to original f(x,y)

4. Degradation process operates on a degradation function that operates on an
input image with an additive noise term. Input image is represented by using
the notation f(x,y), noise term can be represented as η(x,y).These two terms
when combined gives the result as g(x,y).
If we are given g(x,y), some knowledge about the degradation function H and
some knowledge about the additive noise teem η(x,y), the objective of
restoration is to obtain an estimate f'(x,y) of the original image. We want the
estimate to be as close as possible to the original image. The more we know
about h and η , the closer f(x,y) will be to f'(x,y).
If it is a linear position invariant process, then degraded image is given in the
spatial domain by
g(x,y)=f(x,y)*h(x,y)+η(x,y)
h(x,y) is spatial representation of degradation function and symbol * represents
convolution. In frequency domain we may write this equation as
G(u,v)=F(u,v)H(u,v)+N(u,v)
The terms in the capital letters are the Fourier Transform of the corresponding
terms in the spatial domain.

5. Noise Models
The principal source of noise in digital images arises during image acquisition
and /or transmission.
The performance of imaging sensors is affected by a variety of factors, such as
environmental conditions during image acquisition and by the quality of the
sensing elements themselves.
Images are corrupted during transmission principally due to interference in the
channels used for transmission. Since main sources of noise presented in
digital images are resulted from atmospheric disturbance and image sensor
circuitry, following assumptions can be made:
1The noise model is spatial invariant, i.e., independent of spatial location.
2The noise model is uncorrelated with the object function.

6. 1.Gaussian Noise
Because of its mathematical tractability in both the spatial and frequency domain,
Gaussian noise models (aka normal distribution) are used frequently in practice.
The PDF (Probability density function) of a Gaussian random variable Z is given by
z= gray level
μ= mean of average value of z
σ= standard deviation
The application of the Gaussian model is so convenient that it is often used in situations
in which they are marginally applicable at best.

7. 2.Rayleigh Noise
Unlike Gaussian distribution, the Rayleigh distribution is no symmetric. It is
given by the formula
As the shape of the Rayleigh density function is skewed it is useful for
approximating skewed histograms.

8. 3. Erlang Noise
The PDF of the Erlang noise is given by
where a>0, b is a positive integer.
The mean and variance are then given by
Its shape is similar to Rayleigh disruption.
This equation is referred to as gamma density it is correct only
when the denominator is the gamma function.

9. 4. Exponential Noise
Exponential distribution has an exponential shape.
The PDF of exponential noise is given as
Where a>0 . The mean and variance of the density function are
The exponential noise model is a special case of the Erlang
noise model with b>1

10. 5. Uniform Noise
The PDF of the uniform noise is given by
The mean and variance of the density function are given by

11. 6. Impulse (Salt and- Pepper) Noise
In this case, the noise is signal dependent, and is multiplied to the image.
The PDF of bipolar impulse noise
model is given by
If b>a , grey-level b appears as a light dot (salt) in the image. Conversely,
a will appear as dark dot (pepper). If Either Pa, Pb is zero, the PDF is
called unipolar.
Because impulse corruption is generally large compared to the signal
strength, the assumption is usually that a and b are digitized as saturated
values thus black (pepper) and white (salt).

12. The presented noise models provide a useful tool for approximating a broad
range of noise corruption situations found in practice.
For example:
Gaussian noise: arises in images due to sensor noise caused by poor
illumination and/or high temperature, and electronic circuit noise
The Rayleigh noise model is used to characterize noise phenomena in range
images
Exponential and Erlang noise models find their application in Laser
imaging
Impulse noise takes place in situations where high transients (faulty
switching) occurs

13. Visual Comparison of Different Noise Sources
Although the corresponding histograms show close resemblance to the PDF
of the respective noise model, it is virtually impossible to select the type of
noise causing the degradation from the image alone.

14. Periodic Noise
Periodic noise typically arises from electrical or electromechanical
interference during image acquisition and is spatial dependent.

15. Estimation of Noise Parameters
In rare cases the parameters of noise PDFs may be known from sensor
specifications, but it is often necessary to estimate them for a particular imaging
arrangement.
If the imaging system is available, one simple way to study noise characteristics
would be to capture multiple images of homogeneous environments, such
as solid grey cards.
In many cases the noise parameters have to be estimated from already captured
images. The simplest way is to estimate the mean and variance of the grey levels
from small patches of reasonably constant grey level.

16. Estimation of Periodic Noise Parameters
The parameters of periodic noise are typically estimated by inspection of
the Fourier spectrum. The frequency spikes can often be detected by visual
analysis.
Automatic analysis is possible if the noise spikes are either exceptionally
pronounced or a priori knowledge about their location is known.

17. Restoration in the Presence of
Noise Only
When the only degradation present in an image is noise, i.e.
g(x,y)= f(x,y)+ η(x,y)
or
G(u,v)= F(u,v)+ N(u,v)
The noise terms are unknown so subtracting them from g(x,y) or G(u,v)
is not a realistic approach. In the case of periodic noise it is possible to
estimate N(u,v) from the spectrum G(u,v). So N(u,v) can be subtracted
from G(u,v) to obtain an estimate of original image.
Spatial filtering can be done when only additive noise is present.

18. Periodic Noise Reduction by Frequency Domain
Filtering
1.Band reject Filter-Band reject filters remove or attenuate a band of
frequencies around the origin in the Fourier domain.
Butterworth Bandreject Filter

19. These filters are mostly used when the location of noise component in the
frequency domain is known. Sinusoidal noise can be easily removed by
using these kinds of filters because it shows two impulses that are mirror
images of each other about the origin of the frequency transform.

20. 2. Bandpass Filter-The function of a band pass filter is opposite to that
of a band reject filter It allows a specific frequency band of the image to
be passed and blocks the rest of frequencies. The transfer function of a
band pass filter can be obtained from a corresponding band reject filter
with transfer function Hbr(u,v) by using the equation-
These filters cannot be applied directly on an image because it may
remove too much details of an image but these are effective in isolating
the effect of an image of selected frequency bands.

21. Bandpass Filter Example

22. 3. Notch Filter:-A Notch filter rejects or passes frequencies in predefined
neighborhoods about a Centre frequency. Due to the symmetry of the
Fourier transform, notch filters must appear in symmetric pairs about the
origin (except the one at the origin).

23. Where
Butterworth notch reject filter of order n is given by
A Gaussian notch reject filter
These filter become high pass rather than suppress. The frequencies
contained in the notch areas. These filters will perform exactly the opposite
function as the notch reject filter.
The transfer function of this filter may be given as
Hnp(u,v)- transfer function of the pass filter
Hnr(u,v)- transfer function of a notch reject filter

24. Ideal Notch Filter D_0=16 Example

25. Optimum Notch Filtering-
Images derived from electro-optical scanners, such as those used in space and
aerial imaging, are sometimes corrupted by coupling and amplification of low-
level signals in the scanners' circuitry. The resulting images tend to contain
substantial periodic noise. The interference pattern are, however, in practice not
always as clearly defined as in the samples shown thus far.
The first step in Optimum notch filtering is to find the principal frequency
components and placing notch pass filters at the location of each spike, yielding
H(u,v) . The Fourier transform of the interference pattern is thus given by
Where G(u,v) is the Fourier transform of the corrupted image. The
corresponding interference pattern in the spatial domain is obtained with the
inverse Fourier transform

26. As the corrupted image g(x,y) is assumed to be formed by the addition of
the uncorrupted image f(x,y) and the interference noise η(x,y)
If the noise term η(x,y) were known completely, subtracting it from g(x,y)
would yield f(x,y) .

27. Estimating the Degradation
Function
So far we concentrated mainly at the additive noise term η(x,y) . In the
remainder of this presentation we concentrate on ways to remove the
degradation function H.

28. Estimating the Degradation Function
There are three principle methods to estimate the degradation function H
1. Observation
2. Experimentation
3. Mathematical modelling
The process of restoring a corrupted image using the estimated degradation
function is sometimes called blind deconvolution, as the true degradation is
seldom known completely.
In recent years a forth method based on maximum likelihood estimation
(MLE) often called blind deconvolution has been proposed.

29. Estimating by Image Observation
Given
A degraded image without any information about the degradation function
H.
Solution
1. Select a sub image with strong signal content →
2. By estimating the sample grey levels of the object and background in
we can construct an unblurred image of the same size and characteristics as
the observed sub image →
3. Assuming that the effect of noise is negligible (thus the choice of a strong
signal area),
4. Thanks to the shift invariance we can deduce from

30. Estimating by Experimentation
Given
Equipment similar to the equipment used to acquire the degraded image
Solution
Images similar to the degraded image can be acquired with various system
settings until it closely matches the degraded image
1. The system can then be characterized by measuring the impulse (small
bright dot of light) response. A linear, shift invariant system can be fully
described by its impulse response.
2. As the Fourier transform of an impulse is a constant, it follows from
Where G(u,v) is the Fourier transform of the observed image, and A is a
constant describing the strength of the impulse

31. Estimating by Mathematical Modelling
Degradation modelling has been used for many years. It, however,
requires an in-depth understanding of the involved physical
phenomenon. In some cases it can even incorporate environnemental
conditions that cause degradations. Hufnagel and Stanly proposed in
1964 a model to characterize atmospheric turbulence.
where k is a constant that depends on the nature of the turbulence.
Except of the 5/6 power the above equation has the same form as a
Gaussian lowpass filter.

32. Illustration of the Atmospheric Turbulence
Model

33. Inverse Filtering
Direct Inverse Filtering
The simplest approach to restore a degraded image is to form an estimate of the
form
and then obtain the corresponding image by inverse Fourier transform. This
method is called direct inverse filtering. With the degradation model Equation
we get

34. This simple equation tells us that, even if we knew H(u,v) exactly we
could not recover the undegraded image f(x,y)completely because:
1. the noise component is a random function whose Fourier transform
N(u,v) is not known, and
2. in practice H(u,v) can contain numerous zeros. Direct Inverse filtering
is seldom a suitable approach in practical applications

35. Two Practical Approaches for Inverse Filtering
1. Low Pass Filter
where L(u,v)is a low-pass filter that should eliminate the very low (or
even zero) values often experienced in the high frequencies.
2. Constrained Division
gives values, where H(u,v)is lower than a threshold d a special treatment.
These approaches are also known as Pseudo-Inverse Filtering.

36. Artificial Inverse Filtering Example

37. Artificial Inverse Filtering Example
Although very tempting from its simplicity, the direct inverse filtering
approach does not work for practical applications. Even if the degradation
is known exactly H.

38. Wiener Filtering
The best known improvement to inverse filtering is the Wiener Filter. The Wiener
filter seeks an estimate that minimizes the statistical error function
where E is the expected value and f the undegraded image. The solution to this
equation in the frequency domain is

39. Geometric Mean Filter
The Geometric Mean Filter is a generalization of inverse filtering and the Wiener
filtering. Through its parameters α β it allows access to an entire family of filters.

40. Constrained Least Squares (Regularized) Filtering
The Constrained Least Squares Filtering approach only requires
knowledge of the mean and variance of the noise. As shown in
previous lectures these parameters can be usually estimated from
the degraded image. The definition of the 2D discrete convolution
is

41. Constrained Least Squares Filtering
It seems obvious that the restoration problem is now reduced to simple matrix
manipulations. Unfortunately this is not the case. The problem with the matrix
formulation is that the H matrix is very large MN*MN and that its inverse is firstly
very sensitive to noise and secondly does not necessarily exist.
One way to deal with these issues is to base optimality of restoration on a measure
of smoothness, such as the second derivative of the image, e.g. the Laplacian.
subject to the constraint

42. Constrained Least Squares Filtering
The Frequency domain solution to this optimization problem is given by
Where ϒ a parameter that must be adjusted so that the constraint is satisfied,
and P(u,v) is the Fourier transform of the Laplace operator

43. Optimal Selection for Gamma