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Hello, welcome to the video lecture series on digital image processing. During our last
few lectures, we have talked about various image enhancement techniques.
So, we have talked about image enhancement techniques both in the spatial domain as well
as in the frequency domain. So, among spatial domain techniques, we have talked about the
point processing techniques and we have also talked about the mask processing techniques
and in frequency domain; we have talked about ideal and butter worth low pass filters, we
have talked about ideal and butter worth high pass filters, we have talked about Gaussian
filters and we have also talked about homomorphic filters and we have said that when we are
filtering an image in the frequency domain using a low pass filter, if the low pass filter
is an ideal low pass filter; in that case, there is a ringing effect in the output of
the image.
The ringing effect is reduced by using the butter worth filter because of smooth transition
which is given by the butter worth filter from low frequency region to the high frequency
region. However, even in the butter worth filter if we use a butter worth filter of
order more than 1 that is if I use a butter worth filter of order 2 or order 3 and so
on; in such cases also, the butter worth filter leads to the ringing effect.
However, we have discussed that if we use Gaussian filters, then Gaussian filters do
not lead to ringing effect at all. Same is the situation in case of the high pass filters
where the high pass filters try to enhance the high frequency components or detailed
contents of an image and it suppresses the low frequency components and that is the reason
that the output of a high pass filter we have seen that if there is any smooth region in
the image, the smooth region is almost appearing as black in the processed image.
Homomorphic filter as we have discussed is a very very interesting filter. It tries to
enhance the reflectance component in an image and it tries to suppress the contribution
of the intensity component of the image or the effect of the illumination of the same
object and by using this, we have seen some interesting result that even in areas of very
low illumination where the areas is not illuminated properly while taking the images, even in
such areas, some details of the image, we have been able to extent.
Now, in today’s lecture or in a number of lectures starting from today, we will talk
about image restoration techniques. So, we will talk about image restoration techniques
and we will see what is the difference between image enhancement and image restoration. We
will talk about image formation process and the degradation model involved in it and we
will see the degradation model and the degradation operation in continuous functions and how
it can be formulated in the discrete domain.
Now, when we have talked about the image enhancement, particularly using a low pass filter or using
smoothing masks in the special domain; we have seen that one of the effect of using
a low pass filter or the effect of using a smoothing mask in the special domain is that
the noise content of the image gets reduced.
The simple reason is the noise content leads to high frequency components in the displayed
image. So, I if can remove or reduce the high frequency components that also leads to reduction
of the noise. Now, this type of reduction of the noise is also a sort of restoration.
But these are not usually termed as restoration. Rather a process which tries to recover or
which tries to restore an image which has been degraded by some knowledge of a degradation
method which has degraded the image; this is an operation which is known as image restoration.
So, in case of image restoration, the image degradation model is very very important.
So, we have to find out what is the phenomena or what is the model which has degraded the
image and once that model, the degradation model is known; then we have to apply the
inverse process to recover or restore the desired image.
So, this is the difference between an image enhancement or simple noise filtering in terms
of image enhancement and image restoration. That is in case of image enhancement or simple
noise filtering, we do not make use of any of the degradation model or we do not bother
about what is the process which is degrading the image. Whereas in case of image restoration,
we will talk about the degradation model, we will try to estimate the model that has
degraded the image and using that model; we apply the inverse process and try to restore
the image.
So, the degradation modeling is very very important in case of image restoration and
when we try to restore an image, in most of the cases, we define some goodness criteria.
So, using this goodness criteria, we can find out an optimally restored image which more
or less which is almost same as the original image and we will see later that image restoration
operations can be applied as in case of image enhancement both in the frequency domain as
well as in the spatial domain.
So, first of all, let us see that what is the image degradation model that we will consider
in our subsequent lectures. So, let us see the image degradation model first.
So here, we assume that our input image is image f (x, y). It is a 2 dimensional function
as before and we assume that this f (x, y), the input image f (x, y) is degraded by a
degradation function H. So, we will put it like this that we have a degradation function
H which operates on the input image f (x, y).
Then, the output of this degradation function is added to an additive noise. So here, we
add a noise term which we represent by say eta (x, y) which is added to the degradation
output and this finally gives us the output image g (x, y). So, this g (x, y) is the degraded
image which we want to recover. So, from this g (x, y), we want to recover the input image,
the original input image f (x, y) using the image restoration techniques.
So, for recovering this f (x, y), what we have to do is we have to perform some filtering
operation and we will see later that this filters, they are actually derived using the
knowledge of the degradation function that is H and output of the filters is our restored
image and let us put it as f hat (x, y) and we put it as f hat (x, y) because in most
of the cases, we are unable to restore the image exactly. That means it is very difficult
to get the exact image f (x, y) rather by using the goodness criteria that we have just
mentioned; what we can do is we can get an approximation of the original image f (x,
y). So, that is this reconstructed image f hat (x, y) which is an approximation of the
original image f (x, y).
So, the blocks from here to here that is upto obtaining g (x, y), this is actually the process
of degradation; so you will find that in the degradation, we first have a degradation function
H which operates on the input image f a f (x, y), then the output of this degradation
function block that is added with an additive noise which in this particular case we have
represented as eta (x, y) and this degradation function output added to this additive noise
that is what is the degraded image that we actually absorb and this degraded image is
filtered by using the restoration filters. So, this filters that we use they are actually
restoration filters.
So, this g (x, y) is passed through the restoration filters where we get the filter output as
the reconstructed image f hat (x, y) and as we have just said that this f hat (x, y) is
an approximation of the original image f (x, y). So, this particular block which represents
an operation this is a restoration operation and as we have said that the process we call
as image restoration in that, the knowledge of the degradation model is very very essential.
So, one of the fundamental task, one of the very important task in the restoration process
is to estimate the degradation model of the degradation model which has degraded the input
image and later on we will see various techniques of how to estimate the degradation model.
That is how to estimate the degradation function H and we will see in a short while from now
that this particular operation that is the conversion from f (x, y) to g (x, y), this
can be represented in special domain as g (x, y) is equal to h (x, y) convolution with
f (x, y) plus the noise eta (x, y).
So, this is the operation which is done in the spatial domain and the corresponding operation
in frequency domain will be represented by G (u, v) is equal to H (u, v) into F (u, v)
plus N (u, v) where H (u, v) is the Fourier transformation of H (x, y), F (u, v) is the
Fourier transformation of the input image f (x, y), N (u, v) is the Fourier transform
of the additive noise eta (x, y) and G (u, v) is the Fourier transform of the degraded
image G (x, y).
And, this operation is the frequency domain operation and the equivalent operation in
the spatial domain is the other one and here you see that in the special domain, we have
represented this operation as the convolution operation and we had said earlier that a convolution
in the special domain is equivalent to multiplication in the frequency domain. So, that is what
the second term that is G (u, v) is equal to H (u, v) into F (u, v) plus N (u, v).
So here, the convolution in the spatial domain is replaced by the multiplication in the frequency
domain. So, these 2 are very very important expressions and we will make use of these
expressions subsequently more or less throughout our discussion on image restoration process.
Now, before we proceed further, let us try to recapitulate some of the definitions. So,
first we will look at some of the definitions that will be used throughout our discussion
on image restoration. So here, what we have is we have a degraded image g (x, y) which
now let us represent it is like this H of f (x, y) plus eta (x, y) where in this particular
case, we assume that this H is the degradation operator which operates on the input image
f (x, y) and that when added with the additive noise eta (x, y) gives us the degraded image
g (x, y).
Now here, if we assume or for the time being if we neglect the term eta (x, y) or we said
eta (x, y) equal to 0 for the time being for simplicity of our analysis, then what we get
is g (x, y) is equal to H in f (x, y) and as we said that here this H, we assume that
this is the degradation operator.
Now, the first term that we will define in our case is what is known as linearity. So,
what do you mean by the linearity or we say that this degradation operator H is a linear
operator.
So, for defining linearity, we know that if we have 2 functions say f1 (x, y) and f2 (x,
y); then we say that if H [k1 f1 (x, y) plus some constant k2 f2(x, y)], this is equal
to k1 H [f1 (x, y)] plus k2 H [f2 (x, y)]. So, if for these 2 functions f1 (x, y) and
f2 (x, y) and for these 2 constants k1 and k2, this particular relation is true that
is H [k1 f1 (x, y) plus k2 f2 (x, y)] is equal to k1 H [f1 (x, y)] plus k2 H [f2 (x, y)]
if this relation is true, then the operator H is said to be a linear operator.
And, we know very well from our linear system theory that this is nothing but the famous
super position theorem. So, this is what is known as the super position theorem and as
per our definition of a linear system, we know already that the super position theorem
must hold true if the system is a linear system. Now, using this same equation if I said say
k1 is equal to k2 is equal to 1, then the same equation leads to H [f1 (x, y) plus f2
(x, y)] this is nothing but H [f1 (x, y) plus H f2 (x, y)].
Simply, we have replaced k1and k2 by 1 and this is what is known as additivity property.
So, the additivity property simply says that the response of the system to the sum of 2
inputs is same as the sum of their individual responses. So here, we have 2 inputs f1 (x,
y) and f2 (x, y).
So, if I take the summation of f1 (x, y) and f f2 (x, y) and then allow H to operate on
it, then whatever result we will get that will same as when H operates on f1 and f2
individually and we take the sum of those individual responses and this 2 must be equal
to true for a linear system and this is what is known as the additivity property. So, this
is what is the additivity property in this particular case.
Now here, again if i assume that f 2 (x, y) is equal to 0. So, this gives H of k1 f1 (x,
y) should be equal to k1 H [f1 (x, y)] and this is the property which is known as homogeneity
property. So, these are the different properties of a linear system and the system is also
called position invariant if certain properties hold.
So, the system will be position invariant or location invariant if H [f (x minus alpha, y minus beta)] is
same as g of x minus alpha, y minus beta. So, in this case obviously, what we have assumed
is g (x, y) is equal to H [f (x, y)]. So, when this is true that g (x, y) is equal to
H [f (x, y)], then this particular operator H will be called to be position invariant
if H (x minus alpha, y minus beta) is equal to g (x minus alpha, y minus beta) and that
should be true for any function f (x, y) and any value of alpha, beta.
So, this position invariant property this simply says that the response at any point
in the image, the response of H at any point in the image should solely depend upon the
value of the pixel at that particular point and the response will not depend upon the
position of the point in the image and that is what is given by this particular expression
that is H [f (x minus alpha y, minus beta)] equal to g (x minus alpha, y minus beta).
Now given these definitions, let us see that what will be the degradation model for what
will be the degradation model in case of continuous functions.
So, to look at the degradation model in case of continuous functions; we make use of an
old mathematical expression where we have seen that if I take a delta function say delta
(x, y) and the definition of delta (x, y) we have seen earlier that this is equal to
1 if x equal to 0 and y equal to 0 and this is equal to 0 otherwise.
So, this is the definition of a delta function that we have already used and we can use a
shifted version of this delta function. That is delta x minus x0 and y minus y0 will be
equal to 1 if x equal to x0 and y equal to y0 and it will be 0 otherwise. So, this is
the definition of a delta function.
Now, earlier we have seen that if we have an image say f (x, y) or a 2 dimensional function
f (x, y), then multiply this with delta x minus x0, y minus y0 and integrate this product
over the interval minus infinity to infinity. Then the result of the integral will be simply
equal to f (x0, y0).
So, this says that if I multiply a 2 dimensional function f (x, y) with the delta function
delta x minus x0, y minus y0 and integrate the product over the interval minus infinity
to infinity, then the result will be simply the value of the 2 dimensional function f
(x, y) at location (x0, y0).
So, by slightly modifying this particular expression, we can have an equivalent expression
which is given by I can formulate the 2 dimensional function f (x, y) as a similar integral operation
and in this case, I will take f (alpha, beta) delta (x minus alpha, y minus beta) d alpha
d beta and take the integral from minus infinity to infinity.
So, we find that we have an equivalent mathematical expression which is equivalent to just the
earlier expression that we have said and in this case, we can formulate f (x, y) the 2
dimensional function f (x, y) in terms of the value of the function at a particular
point alpha beta and in terms of the delta function delta (x minus alpha, y minus beta).
Now, for the time being if we consider say the noise term eta (x, y) is equal to 0 for
simplicity, then we can write the degraded image g (x, y), we have seen earlier that
g (x, y) we have written as H f (x, y) plus eta (x, y); so for the time being, we are
assuming that this additive noise term eta (x, y) is 0 or it is negligible, then the
degraded image g (x, y) can now be written in the form H of… I replace this f (x, y)
by this integral term. So, this will be simply H of double integral f (alpha, beta) delta
(x minus alpha, y minus beta) d alpha d beta where the integral has to be taken from minus
infinity to infinity.
So, I can write, I can get an expression of the degraded image g (x, y) in terms of this
integral definition of the function f (x, y) which is operated by the degradation operator
H. Now, once I get this kind of expression, now if I apply the linearity and additivity
property of the linear system; then this particular expression gets converted to g (x, y) is equal
to… I can take this double summation outside, it becomes H of f (alpha, beta) delta (x minus
alpha, y minus beta) d alpha d beta, take the integral from minus infinity to infinity
and this is what we have obtained by applying the linearity and additivity property to this
earlier expression of this degraded image.
Now, here you find that this term f (alpha, beta), this is independent of the variables
x and y. So, because the term f (alpha, beta) is independent of the variables x and y, the
same expression can now be rewritten in a slightly different form.
So, that form give us that g (x, y) can now be written as same double integral. We take
f (alpha, beta) outside the scope of the operator H. So, this simply becomes f (alpha, beta),
then H delta (x minus alpha, y minus beta) d alpha d beta. Take the integral over minus
infinity to infinity.
Now, this particular term H of delta (x minus alpha, y minus beta), we can write this as
h (x, alpha, y, beta) and this is nothing but what is known as the impulse response
of H. So, this is what is known as the impulse response. That is the response of the operator
H when the input is an impulse given in the form delta (x minus alpha, y minus beta) and
in case of optics, this impulse response is popularly known as point spread function or
PSF.
So, using this impulse response, now the same g (x, y), we can write as double integral
again f of (alpha, beta) h (x, alpha, y, beta) d alpha d beta, integral from minus infinity
to infinity and this is what is popularly known as super position integral of first
kind. Now, this particular expression is very very important. It simply says that if the
impulse response of the operator H is known, then it is possible to find out the response
of this operator H to any arbitrary input f (alpha, beta).
So, that is what has been done here that using the knowledge of this impulse response h (x,
alpha, y, beta), we have been able to find out the response of this system to an input
f (alpha, beta) and this impulse response is the one which uniquely or completely characterizes
a particular system. So, given any system, if we know what is the impulse response of
the system, then we can find out what will be the response of that system to any other
arbitrary function.
Now, in addition to this, if the function H, this operator H is position invariant;
so we use H to be position invariant, so if H is position invariant, then obviously H
[delta x minus alpha, y minus beta)] as per of our definition of position invariance will
be same as h (x minus alpha, y minus beta). This is as per the definition of position
invariance of a system.
Now, using this position invariance property, now we can write g (x, y) that is the degraded
image as simply double integral f (alpha, beta) into h (x minus alpha, y minus beta)
d alpha d beta, take the integral from minus infinity to infinity. And, if you look at
this particular expression, you will find that this expression is not is nothing but
the convolution operation. This is nothing but the convolution operation of the 2 functions
f (x, y) and h (x, y) and that is what we said that when we have drawn our degradation
model, we have said that input image f (x, y) is actually convolved by the degradation
process that is H (x, y). So, this is nothing but that convolution operation.
And now, if I take, you will find that earlier we have considered this noise term eta (x,
y) to be equal to 0. So now, if I consider this noise term eta (x, y), then our degradation
function or the degradation model becomes simply g (x, y) is equal to f (alpha, beta)
h (x minus alpha, y minus beta) d alpha d beta, take the integral from minus infinity
to infinity plus the noise term eta (x, y).
So, this is the general image degradation model and you will find that here we have
assumed that the degradation function H is linear and position invariant and it is very
important to note that many of the degradation operations which we encounter in reality can
be approximated by such linear space invariant or linear position invariant models.
The advantage is once a degradation model can be approximated by a linear position invariant
model, then the inter mathematical tool of linear system theory can be used to find out
the solution for such image restoration process. That means we can use all those tools of linear
system theory to estimate what will be the restored image f (x, y) from a given degraded
image g (x, y) provided, we know we have some knowledge of the degradation function that
is H (x, y) and we have some knowledge of what is the noise function eta (x, y).
Now, this formulation that we have done till now, this formulation is for the continuous
case and as we have said many times that in order to use this mathematical operation for
our digital image processing techniques, we have to find out a discrete formulation of
this mathematical model. So, let us see that how we can have an equivalent discrete formulation
of this particular degradation model.
So, to get a discrete formulation, firstly we will consider; so we have to get a discrete
formulation. So, to obtain this discrete formulation, for simplicity, initially we will assume the
cases in 1 dimension and later on this we will extend to 2 dimensional cases for digital
image processing operations. Again for simplicity, initially, we will neglect the contribution
of the noise term that is eta (x, y).
So, in case of 1 dimension as we have done in case of in the continuous signal; we have
2 signals f (x) and another one is h (x). So, we have said that f (x) is the input signal
and h (x) tells us that what is the degradation function. So, f (x) is the input function
and h is the h (x) is the degradation function. For discretization of the same formulation,
what we have to do is we have to uniformly sample these 2 functions f (x) and h (x) and
we assume that f (x) is uniformly sampled to give an array of dimension A and h (x)
is uniformly sampled to give an array of dimension B.
That means for f (x) in the discrete case, x varies from 0, 1 to A minus 1 and h (x)
for h (x), x varies from 0, 1 to b minus 1. Then what we will do, we will add additional
0s to this f (x) and b (x) to make both of them of the same dimension and dimension equal
to say capital M.
So, we make both of them to be of dimension capital M by adding additional number of 0s
and we assume that both f (x) and h (x) after addition of this 0 terms and making both of
them to be of dimension M, they become periodic with a periodicity capital M. So, once we
have done this, now the same convolution operation that we have done in case of our continuous
case, now can also be written in case of discrete case.
So, in discrete case, the convolution operation, we will write in this manner. So, after converting
both f (x) and h (x) into arrays of dimension M, this new arrays that we will get, we represent
it by fe (x) that is f extended x as we have extended it and h we represent by he (x) that
is the extended version of h (x).
And now, in discrete domain, the convolution function can be written as ge (x) is equal
to summation fe (m) he (x minus m) where this m will be varying from 0 to capital M minus
1 and x we will assume values from 0 to capital M minus 1. So, this is the discrete formulation
of the convolution equation that we have obtained in case of continuous signal cases.
Now, if you analyze this convolution expression, you will find that this convolution expression
can be written in the form of a matrix, matrix operation. So we can have the matrix form.
In matrix form, these equations will be like this - g equal to some matrix H times f where
the function f or array f will be simply fe (0), fe (1), this way upto fe (capital M minus
1) and function g similarly will be ge (0), ge (1), so like this it will be ge (M capital
M minus 1).
So, you recollect, you just recollect that fe and ge, these are the names which are given
to the sample versions of the functions f (x) and g (x) after extending the functions
by addition of addition by adding additional number of 0’s to make them of dimension
capital M.
And, in this particular case, the matrix h will have the matrix h will be of dimension
capital M by capital M. But the elements of H will be like this - he (0), he (minus 1),
continue like this, it will be he (minus M plus 1), here it will be he (1), he (0), it
will be he (minus capital M plus 2) and if we continue like this, it will be he (capital
M minus 1), he (capital M minus 2), like this it will be he (0). So, this is the form of
the matrix capital H which is the degradation matrix in this particular case.
And here, you find that that elements of this degradation matrix capital H are actually
generated from the degradation function he (x). Now, remember that we have assumed that
our he (x), this function is actually periodic. This is which we have assumed with periodicity
of capital M. So, if this function is periodic with periodicity capital M that means he x
plus capital M that will be same as he of x.
So, by using this periodicity assumption, now this particular degradation matrix H can
be written in a different form where this matrix H will now be represented as he (0),
he (capital M minus 1), he (capital M minus 2) upto he (1). The second row will be he
(1), he (0), he (capital M minus 1) and this will be he (2). Third row will be he (2),
he (1), he (0) like this it will be he (3) and the last row continue in the same manner
will be he (capital M minus 1), he (capital M minus 2), he (capital M minus 3) and the
last term will be equal to he (0).
Now, if you analyze this particular matrix, you will find that this degradation matrix
capital H has a very very interesting property. That means the first property is different
rows of this matrix are actually generated by rotation to the right of the previous term.
So, here if you look at the second row; you will find that this second row is actually
generated by rotating the first row to the right. Similarly, third row is generated by
rotating the second row to right by 1.
So, this is so in this particular matrix, the different rows are actually generated
by rotating the previous row to the right. So, this is called circulant matrix because
different rows are generated by a circular rotation and the circularity in this particular
matrix is also complete in the sense that if I rotate this last row to right, what I
get is the first row of the matrix. So, this kind of matrix is known as a circulant matrix.
So here, I find we find that in case of discrete formulation, the discrete formulation is also
a convolution operation and here in the matrix equation of the degradation model, the degradation
matrix H that we obtain that is actually a circulant matrix. Now, let us extend the concept
of this discrete formulation from 1 dimension to 2 dimensions.
So, let us see what we get in case of 2 dimensional functions that is in case of 2 dimensional
images. So, in case of 2 dimension, we have the input function or the image function which
is given by f (x, y) and we have the degradation function which is given by h (x, y) and we
assume that this if f (x, y) is sampled to an array of dimension capital A by capital
B and say h (x, y) is sampled to an array of dimension say capital C by capital D.
Now, as we have done in 1 dimensional case that is the functions f (x) and h (x) are
actually extended by using by putting additional number of 0’s to make both of them of same
size say capital N; in the same manner, here we add additional number of 0’s to both
this f (x, y) and h (x, y) to get the extended functions fe (x, y) and he (x, y) to make
both of them of dimension say capital M by capital N and we also assume that this fe
(x, y) and he (x, y), they are periodic and in x dimension, the periodicity will be of
period capital M and in y dimension, the periodicity will be of period capital N.
Now, following similar procedure, we can obtain a convolution expression in 2 dimensions which
is given by ge (x, y) which is nothing but fe (m, n) he (x minus m, y minus n) where
n varies from 0 to capital N minus 1 and m varies from 0 to capital M minus 1.
And, if I write this convolution expression in the form of a matrix and incorporating
the noise term eta (x, y), I will get a matrix equation which is of the form g equal to Hf
plus n where this matrix where this vector f is a vector of dimension capital M into
N which is obtained by concatenating different rows of the 2 dimensional function f (x, y)
that is the first N number of elements of this vector f will be the elements of the
first row of matrix f (x, y).
Similarly, we also obtain this particular vector n by concatenation of the rows of the
matrix eta (x, y) and this particular degradation matrix h (x) in this case will be of dimension
M into N by M into N and this matrix H will have a very very interesting form. This matrix
H can now be represented as H0 H M minus 1 like this upto H1. The second row can be H1
H0 upto H2 and the last row is H M minus 1 H M minus 2 like this we have H0 where each
of these terms Hj is a matrix, so each of this Hj is actually a matrix of dimension
N by N where this Hj is generated from the j’th row of the degradation function H (x,
y).
That is this Hj we can write this matrix Hj in the form he (j, 0) he (j, N minus 1) like
this upto he (j, 1). Second row will be he (j, 1) he (j, 0) this way he (j, 2) and if
I continue like this the last row will be he (j, N minus 1) he (j, N minus 2) like this
if I continue, the last element will be he (j, 0). So, you find that this matrix Hj which
is actually a component of the degradation matrix capital H is a circulant matrix that
we have defined earlier and using this block matrix, the degradation matrix H is also have
been subscripted in the form of a circulant matrix. So, this matrix H in this particular
case is what is known as a block circulant matrix. So, this is what is called a block
circulant matrix.
So, in case of 2 dimensional function that is in case of a digital image, we have seen
that the degradation model can simply be represented by this expression g equal to H into f plus
n where this vector f is a vector of dimension m into n and the degradation matrix H which
is of dimension m into n by m into n is actually a block circulant matrix where for each block,
the matrix is obtained from the j’th row of the degradation function H (x, y).
So, in our next lecture, we will see what will be the applications of this particular
degradation model to restore an image from its degraded version.
So now, let us see some of the questions of this particular lecture. So, the first question
is what is the difference between image enhancement and image restoration? Second question is
what is a linear position invariant system? Third question, what is homogeneity property?
Fourth, what is a circulant matrix? What is a block circulant matrix? Why does the degradation
matrix H become circulant?
Thank you.
Hello, welcome to the video lecture series on digital image processing. In the last class,
we have started discussion on image restoration.
We have said that there are certain cases where image restoration is necessary in the
sense that in many cases, while capturing the image or while acquiring the image, some
distortions appear in the image. For example, if you want to capture a moving object with
a camera; in that case, because of the movement of the camera, it is possible that the image
that is captured will be blurred which is known as motion blurring.
There are many other situations, say for example if the camera in not properly focused, then
also the image that you get is a distorted image. So, in such situations, what we have
to go for is restoration of the image or recovery of the original image from the distorted image.
Now, regarding this in the last class, we had talked about what is image restoration
technique. In previous classes, we have talked about image filtering that is if the image
is contaminated with noise. Then, we have talked about various types of filters both
in spatial domain as well as in frequency domain to remove that noise and we just mentioned
in our last class that this kind of noise removal is also a sort of restoration because
there also we are trying to recover the original image from a noisy image.
But conventionally, this kind of simple filtering is not known as restoration. But what is by
restoration what I what we mean is that if we know a degradation model by which the image
has been degraded and on that degradation model on the degraded image some noise has
been added. So, recovery or restoration of the original image from a degraded image using
the acquired knowledge of the degradation function of the model using which the image
has been degraded; so, that kind of recovery is normally known as restoration process.
So, this is the basic difference between restoration and image filtering or image enhancement.
Then, we have seen an image formation process where the degradation is involved and we have
talked about the degradation model in continuous functions as well as its discrete formulation.
So, in today’s lecture, we will talk about the …..