digital image processing unit-6
TRANSCRIPT
Page 1 of 4
LCE/7.5.1/RC 01
TEACHING NOTES
Department: ELECTRONICS & COMMUNICATION ENGINEERING
Unit: VI Date:
Topic name: Introduction to Degradation Model No. of marks allotted by JNTUK:
Books referred: 01. Digital Image Processing by R C Gonzalez and R E Woods
02. www.wikipedia.org
03. www.google.com
Introduction:
The purpose of image restoration is to compensate for or undo defects which degrade an
image. Degradation comes in many forms such as motion blur, noise, and camera misfocus. In cases
like motion blur, it is possible to come up with a very good estimate of the actual blurring function
and undo the blur to restore the original image. In cases where the image is corrupted by noise, the
best we may hope to do is to compensate for the degradation it caused. In this project, we will
introduce and implement several of the methods used in the image processing world to restore
images.
Degradation model:
The block diagram for our general degradation model is,
Where g is the corrupted image obtained by passing the original image f through a low pass
filter (blurring function) b and adding noise to it. We present four different ways of restoring the
image.
Inverse Filter:
In this method we look at an image assuming a known blurring function. We will see that
restoration is good when noise is not present and not so good when it is.
Weiner Filtering:
In this section we implement image restoration using wiener filtering, which provides us
with the optimal trade-off between de-noising and inverse filtering. We will see that the result is in
general better than with straight inverse filtering.
Wavelet Restoration:
We implement three wavelet based algorithms to restore the image.
Blind De-convolution:
In this method, we assume nothing about the image. We do not have any information about
the blurring function or on the additive noise. We will see that restoring an image when we know
nothing about it is very hard.
Faculty/Date: HOD/Date:
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Page 2 of 4
LCE/7.5.1/RC 01
TEACHING NOTES
Department: ELECTRONICS & COMMUNICATION ENGINEERING
Unit: VI Date:
Topic name: Inverse Filtering No. of marks allotted by JNTUK:
Books referred: 01. Digital Image Processing by R C Gonzalez and R E Woods
02. www.wikipedia.org
03. www.google.com
Inverse Filtering:
If we know of or can create a good model of the blurring function that corrupted an image
the quickest and easiest way to restore that is by inverse filtering. Unfortunately, since the inverse
filter is a form of high pass filter, inverse filtering responds very badly to any noise that is present in
the image because noise tends to be high frequency. In this section, we explore two methods of
inverse filtering – a thresholding method and an iterative method.
Method-1 (Thresholding):
We can model a blurred image by,
Where f is the original image, b is some kind of a low pass filter and g is our blurred image.
So to get back the original image, we would just have to convolve our blurred function with some
kind of a high pass filter,
In the ideal case, we would just invert all the elements of B to get a high pass filter.
However, notice that a lot of the elements in B have values either at zero or very close to it. Inverting
these elements would give us either infinities or some extremely high values. In order to avoid these
values, we will need to set some sort of a threshold on the inverted element. So instead of making a
full inverse out of B, we can an almost full inverse by
So the higher we set γ, the closer H is to the full inverse filter.
Method-2 (Iterative procedure):
The idea behind the iterative procedure is to make some initial guess of f based on g and to
update that guess after every iteration. The procedure is,
Where f₀ is an initial guess based on g. if our fk is a good guess, eventually fk convolved with
b will be close to g. when that happens the second term in the fk+1 equation will disappear and fk and
fk+1 will converge. λ is our convergence factor and it lets us determine how fast fk and fk+1 converge.
If we take both of the above equations to the frequency domain, we get
Faculty/Date: HOD/Date:
www.jntuworld.com
Department: ELECTRONICS & COMMUNICATION
Unit: VI
Topic name: Least Mean Square Filt
Books referred: 01. Digital Image Processing by R C Gonzalez and R E Woods
02. www.wikipedia.org
03. www.google.com
Solving for Fk recursively, we get
So if (1-λB(ω1,ω2))k+1
goes to zero as k goes to infinity, we would get the result as obtained by
the inverse filter. In general, this method will not give t
can be less sensitive to noise in some cases.
Least Mean Square Filters:
Least mean squares (LMS) algorithms are used in adaptive filters to find the filter coefficients
that relate to producing the least mean squares of the error signal (
and the actual signal). It is a stochastic gradient descent
based on the error at the current time. It was invented in 1960 by Stanford University professor
Bernard Widrow and his first Ph.D. student, Ted Hoff.
Most linear adaptive filtering problems can be formulated using the block diagram above.
That is, an unknown system is to be identified and the adaptive filter attempts to adapt the filter to
make it as close as possible to , while using only observable
and h(n) are not directly observable. Its solution is closely related to the Wiener filter.
The idea behind LMS filters is to use the method of steepest descent to find a coefficient
vector h(n) which minimizes a cost
Faculty/Date:
Page 3 of 4
LCE/7.5.1/RC 01
TEACHING NOTES
OMMUNICATION ENGINEERING
Date:
ters No. of marks allotted by JNTUK
: 01. Digital Image Processing by R C Gonzalez and R E Woods
www.wikipedia.org
www.google.com
recursively, we get
to zero as k goes to infinity, we would get the result as obtained by
the inverse filter. In general, this method will not give the exact same results as inverse filtering
can be less sensitive to noise in some cases.
) algorithms are used in adaptive filters to find the filter coefficients
that relate to producing the least mean squares of the error signal (difference between the desired
). It is a stochastic gradient descent method in that the filter is only adapted
based on the error at the current time. It was invented in 1960 by Stanford University professor
Bernard Widrow and his first Ph.D. student, Ted Hoff.
Most linear adaptive filtering problems can be formulated using the block diagram above.
That is, an unknown system is to be identified and the adaptive filter attempts to adapt the filter to
make it as close as possible to , while using only observable signals x(n), d(n) and e(n); but y(n), v(n)
and h(n) are not directly observable. Its solution is closely related to the Wiener filter.
The idea behind LMS filters is to use the method of steepest descent to find a coefficient
function. We start the discussion by defining the cost function as
HOD/Date:
LCE/7.5.1/RC 01
LCE/7.5.1/RC 01
No. of marks allotted by JNTUK:
to zero as k goes to infinity, we would get the result as obtained by
he exact same results as inverse filtering, but
) algorithms are used in adaptive filters to find the filter coefficients
difference between the desired
method in that the filter is only adapted
based on the error at the current time. It was invented in 1960 by Stanford University professor
Most linear adaptive filtering problems can be formulated using the block diagram above.
That is, an unknown system is to be identified and the adaptive filter attempts to adapt the filter to
signals x(n), d(n) and e(n); but y(n), v(n)
The idea behind LMS filters is to use the method of steepest descent to find a coefficient
function. We start the discussion by defining the cost function as
LCE/7.5.1/RC 01
www.jntuworld.com
Department: ELECTRONICS & COMMUNICATION
Unit: VI
Topic name: Constrained Least Squares Restoration
Books referred: 01. Digital Image Processing by R C Gonzalez and R E Woods
02. www.wikipedia.org
03. www.google.com
Where e(n) is defined in the block diagram section of the general adaptive filter and E{.}
denotes the expected value. Applying the steepest descent method means to take the partial
derivatives with respect to the individual entries of the filter coefficient vector
Where is the gradient operator
it follows,
Now, is a vector which points towards the steepest ascent of the cost function. To
find the minimum of the cost functio
To express that in mathematical terms
Where is the step size. That means we have found a sequential update algorithm which
minimizes the cost function. Unfortunately, this algorithm is not realizabl
.
Constrained Least Squares Restoration
Image restoration using the constrained
the image being processed. In addition, it only requires knowing the modulation transfer function of
the imaging system when applied to nuclear medicine images. Prompted by these observations, a
systematic evaluation of the effects of the form of the "coarseness function" [
method has been conducted. Nine C(f)'s are evaluated using an observer preference and a
normalized mean-squared error (NMSE
transfer functions and a wide range of count levels. The results of the subjective studies support
using the form of C(f) which has been most widely employed in previous studies, i.e., the form
designed to minimize the energy in the s
C(f) is generally found to be optimal by the mean
compared to: (1) no processing, (2) count
restoration. When evaluated using objective measurements of error and contrast, the CLS method is
found to be slightly inferior to the best method, Metz restoration. However, CLS restoration is found
to be equal to or better than the other methods when judged by
studies.
Faculty/Date:
Page 4 of 4
TEACHING NOTES
OMMUNICATION ENGINEERING
Date:
Constrained Least Squares Restoration No. of marks allotted by JNTUK
: 01. Digital Image Processing by R C Gonzalez and R E Woods
www.wikipedia.org
www.google.com
here e(n) is defined in the block diagram section of the general adaptive filter and E{.}
denotes the expected value. Applying the steepest descent method means to take the partial
atives with respect to the individual entries of the filter coefficient vector
operator with
is a vector which points towards the steepest ascent of the cost function. To
find the minimum of the cost function we need to take a step in the opposite direction of
To express that in mathematical terms
is the step size. That means we have found a sequential update algorithm which
minimizes the cost function. Unfortunately, this algorithm is not realizable until we
Constrained Least Squares Restoration:
Image restoration using the constrained least-squares (CLS) method theoretically adapts to
the image being processed. In addition, it only requires knowing the modulation transfer function of
the imaging system when applied to nuclear medicine images. Prompted by these observations, a
c evaluation of the effects of the form of the "coarseness function" [C (f)] used by the CLS
method has been conducted. Nine C(f)'s are evaluated using an observer preference and a
NMSE) criterion. This evaluation is conducted for three modulation
transfer functions and a wide range of count levels. The results of the subjective studies support
using the form of C(f) which has been most widely employed in previous studies, i.e., the form
designed to minimize the energy in the second derivative of the restored image. A different form of
C(f) is generally found to be optimal by the mean-squared error criterion. The CLS method is then
compared to: (1) no processing, (2) count-dependent smoothing, and (3) count-dependent Metz
ation. When evaluated using objective measurements of error and contrast, the CLS method is
found to be slightly inferior to the best method, Metz restoration. However, CLS restoration is found
to be equal to or better than the other methods when judged by the results of observer preference
HOD/Date:
ks allotted by JNTUK:
here e(n) is defined in the block diagram section of the general adaptive filter and E{.}
denotes the expected value. Applying the steepest descent method means to take the partial
and
is a vector which points towards the steepest ascent of the cost function. To
n we need to take a step in the opposite direction of .
is the step size. That means we have found a sequential update algorithm which
e until we know
) method theoretically adapts to
the image being processed. In addition, it only requires knowing the modulation transfer function of
the imaging system when applied to nuclear medicine images. Prompted by these observations, a
] used by the CLS
method has been conducted. Nine C(f)'s are evaluated using an observer preference and a
for three modulation
transfer functions and a wide range of count levels. The results of the subjective studies support
using the form of C(f) which has been most widely employed in previous studies, i.e., the form
econd derivative of the restored image. A different form of
squared error criterion. The CLS method is then
dependent Metz
ation. When evaluated using objective measurements of error and contrast, the CLS method is
found to be slightly inferior to the best method, Metz restoration. However, CLS restoration is found
the results of observer preference
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