DIGITAL IMAGE PROCESSING

18CS741
[As per Choice Based Credit System (CBCS) scheme]
(Effective from the academic year 2018 -2019)
SEMESTER – VII
MODULE 3
Notes

Prepared By
Athmaranjan K
Associate Professor
Dept. of Information Science & Eng.
Srinivas Institute of Technology, Mangaluru

Athmaranjan K Dept of ISE

DIGITAL IMAGE PROCESSING 18CS741 Module 3

MODULE 3
SYLLABUS
Image Enhancement In Frequency Domain: Introduction, Fourier Transform, Discrete Fourier Transform
(DFT), properties of DFT, Discrete Cosine Transform (DCT), Image filtering in frequency domain.

Textbook 1: Rafael C G., Woods R E. and Eddins S L, Digital Image Processing, Prentice Hall, 2nd edition,
2008
Textbook 1: Ch.4.1, 4.2

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FOURIER SERIES:
Any function that periodically repeats itself can be expressed as the sum of sines and/or cosines of different
frequencies, each multiplied by a different coefficient.
FOURIER TRANSFORM:
Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral
of sines and/or cosines multiplied by a weighting function.
The frequency domain refers to the plane of the two dimensional discrete Fourier transform of an image. The
purpose of the Fourier transform is to represent a signal as a linear combination of sinusoidal signals of
various frequencies.

The function at the bottom is the sum of the four functions above it.
PRELIMINARY CONCEPTS
A complex number, C, is defined as: C = R + j I where R and I are real numbers, and j is an imaginary
number equal to the square of -1; that is j equal to square root of -1.
Here, R denotes the real part of the complex number and I its imaginary part.
The conjugate of a complex number C, denoted is defined as C* and is defined as
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*
C = R – jI
POLAR REPRESENTATION:
C = |C| (cos ɵ + j sin ɵ)

is the magnitude

ɵ is the angle between vector and the real axis

EULER’S FORMULA

For example, the polar representation of the complex number 1 +2j is:

FOURIER SERIES:
A function (t) of a continuous variable t that is periodic with period, T, can be expressed as the sum of sines
and cosines multiplied by appropriate coefficients. This sum, known as a Fourier series, has the form:

IMPULSES AND THEIR SIFTING PROPERTY

Impulse: is a distribution or a generalised function. Sifting: to separate
A unit impulse of a continuous variable t located at t=0, is defined as:

and is constrained also to satisfy the identity:

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DIGITAL IMAGE PROCESSING 18CS741 Module 3

An impulse has the so called sifting property with respect to integration,

The Fourier Transform of Functions of One Continuous Variable

The Fourier transform of a continuous function (t) of a continuous variable, t, denoted F(f(t)), is defined by
the equation

Where μ is also a continuous variable we denote this fact explicitly by writing the Fourier transform

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CONVOLUTION
The convolution of two functions, f(t) and h(t), of one continuous variable t is denoted by,

Where minus sign accounts for flipping, t is the displacement and τ is a dummy variable that is integrated
out.
SAMPLING AND THE FOURIER TRANSFORM OF SAMPLED FUNCTIONS
Continuous functions have to be converted into a sequence of discrete values before they can be processed in
a computer. This is accomplished by using sampling and quantization
Consider a continues function, f(t), that we wish to sample at uniform intervals (∆T) of the independent
variable t. We assume that the function extends from -∞ to ∞ with respect to t.

THE DISCRETE FOURIER TRANSFORM (DFT) OF ONE VARIABLE

***************Obtain the equation for one dimensional Discrete Fourier Transform and its inverse
from the continuous transform of sampled function of one variable.

Obtaining the DFT from the continuous transform of a sampled function , From the definition of
Fourier transform, we have,

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DIGITAL IMAGE PROCESSING 18CS741 Module 3
Where sampled function is given by:

By substituting this in above equation we get;

Suppose that we want to obtain M equally spaced samples of Taken over the period

This is accomplished by taking the samples at the following frequencies.

Substituting this result for μ into the above equation and letting Fm denote the result yields

The inverse Fourier transform is given by,

Corresponding to discrete variables m and n in two dimensions, we use the notation x and y for image
coordinate variables and u and v for frequency variables, Then the equations becomes

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THE 2-D DISCRETE FOURIER TRANSFORM (DFT)

The 2D-discrete Fourier Transfer (DFT) of a digital image f(x, y) of size M x N is given by:

The function F(u) must be evaluated for values of the discrete variables u and v in the ranges;
u = 0, 1, 2, ……., M- 1 and v = 0, 1, 2, ……… , N - 1.

THE 2-D INVERSE DISCRETE FOURIER TRANSFORM (IDFT)

In order to get the original digital image f(x, y) from Frequency domain (Fourier Transform), we need to
apply Inverse Fourier Transform function.
The 2D-Inverse discrete Fourier Transfer (IDFT) of a Fourier Transform F(u, v) is given by f(x, y) of size M
x N is given by:

The function f(x, y) must be evaluated for values of the discrete variables x and y in the ranges;
x = 0, 1, 2, ……., M- 1 and y = 0, 1, 2, ……… , N - 1.

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Problem:
1. *******Compute the DFT of the sequence f(x) = {1, 0, 0, 1}
Answer:
DFT of 1-D function f(x) is given by:

Where M = the number of samples given = 4

= f(0) + f(1) e-j2Πu /4 + f(2) e-jΠu + f(3) e-j3Πu /2

= 1 + 0 + 0 + e-j3Πu /2
Take different values of u:
When u = 0;
F(0) = 1 + 1 = 2

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When u = 1;

F(1) = 1 + e-j3Πu /2 = 1 + cos(3Π/2) - j sin(3Π/2) = 1 + 0 –j (-1) = 1 + j

When u = 2;
F(2) = 1 + e-j3Π u/2 =1 + e-j6Π /2 = 1 + cos(3Π) - j sin(3Π) = 1 =1 = 0
When u = 3;
F(3) = 1 + e-j3Π u/2 =1 + e-j9Π /2 = 1 -j
Therefore F(u) = { 2, 1 + j, 0, 1 – j }
KERNEL DFT
It is the impulse response of the filter or the frequency response of the filter.
To find N point Kernel matrix we have to use the following rule:
n
WNn = e-i2Π/N
WN0 = e-i2Π/N 0 = 1
For different values of n we can find out WNn . N = the sample size (number of rows/columns)
OR
By using unit circle method:
How to find 4 Point DFT Kernel Matrix?
0 1 2 3
0 W40 W40 W40 W40
1 W40 W41 W42 W43
2 W40 W42 W44 W46
3 W40 W43 W46 W49

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0
By referring above unit circle, W4 = 1, W41 =-j, W42 = -1, W43 4
= +j, W4 = 1, W46= 9
-1 and W4 = -j
The 4 Point DFT Kernel Matrix:

1 1 1 1
1 -j -1 j
1 -1 1 -1
1 j -1 -j

NOTE:
1. By using Kernel matrix 1-D DFT is given by F(u) = Kernel Matrix x Image f(x, y)
2. By using Kernel matrix 1-D IDFT is given by X(n) = 1/N [Kernel Matrix x X(u)]
3. By using Kernel matrix 2D-DFT is given by F(u, v) = Kernel Matrix x Image f(x, y) x Kernel MatrixT
4.
Problem 2:
********Find the DFT of the following sequence using matrix and verify whether DFT works correctly for
X = {1, 2, 8, 9}
Answer:
Using matrix (N point kernel) DFT of 1-D function X is given by:
F(u) = Kernel Matrix x Image X
Here N = 4, so we have to use 4 point Kernel matrix which is given by:
1 1 1 1
1 -j -1 j
1 -1 1 -1
1 j -1 -j
DFT of X(n) is = X(u)
1 1 1 1 1 20
1 -j -1 j x 2 = -7 + j7
1 -1 1 -1 8 -2
1 j -1 -j 9 -7 - j7

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To verify the result we have to use IDFT function
X(n) = 1/N [ (Kernel Matrix)-1 x X(u)]

X(0) 1 1 1 1 20
X(1) = 1/4 1 j -1 -j x -7 + j7
X(2) 1 -1 1 -1 -2
X(3) 1 -j -1 j -7 - j7

X(0) 4 1
X(1) = 1/4 x 8 = 2
X(2) 32 8
X(3) 36 9

Therefore the input sequence X = {1, 2, 8, 9 }

NOTE:
To perform DFT Kernel Matrix-1 we have to take the inverse of each element of Kernel Matrix
That means inverse of 1 = 1/1 = 1
-1 = -1/1 = -1

( Multiplying and dividing Nr and Dr by j)

DFT Kernel Matrix-1 =

1 1 1 1
1 j -1 -j
1 -1 1 -1
1 -j -1 j

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DIGITAL IMAGE PROCESSING 18CS741 Module 3
*************Discuss about two dimensional DFT and its inverse
Discrete Fourier Transform (DFT) is an important digital image processing tool which is used to decompose
an image into sine and cosine components. DFT help us to convert images in spatial domain to Frequency
domain. Transformation of images using DFT may isolate critical components of image pattern so that they
are directly accessible for analysis.
The 2D-discrete Fourier Transfer (DFT) of a digital image f(x, y) of size M x N is given by:

The function F(u) must be evaluated for values of the discrete variables u and v in the ranges;
u = 0, 1, 2, ……., M- 1 and v = 0, 1, 2, ……… , N - 1.
The definition of the Fourier transform uses a complex exponential. In consequence, the DFT of an image is
possibly complex, so it can be represented as amplitude (modulus) and phase (argument) of the DFT. The
amplitude and phase represent the distribution of energy in the frequency plane. The low frequencies are
located in the center of the image, and the high frequencies near the boundaries.
We define the magnitude as:

is called Fourier (Frequency) spectrum and

is the Phase angle and power spectrum as:

Multiplying an image f(x, y) of size M x N by (-1)x+y will shift the origin of F(u, v) to frequency coordinates
(M/2, N/2), which is the center of the M x N area (Frequency rectangle) occupied by the 2D-DFT.
The value of DFT at (u, v) = (0, 0)

whichh is the average of f(x,y).

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DIGITAL IMAGE PROCESSING 18CS741 Module 3
If f(x, y) is an image, the value of the DFT at the origin is equal to the average gray level of the image,
because both frequencies are 0 at the origin.
The 2-D Inverse Discrete Fourier Transform (IDFT)
In order to get the original digital image f(x, y) from Frequency domain (Fourier Transform), we need to
apply Inverse Fourier Transform function.
The 2D-Inverse discrete Fourier Transfer (IDFT) of a Fourier Transform F(u, v) is given by f(x, y) of size M
x N is given by:

The function f(x, y) must be evaluated for values of the discrete variables x and y in the ranges;
x = 0, 1, 2, ……., M- 1 and y = 0, 1, 2, ……… , N - 1.
PROPERTIES OF 2D-DISCRETE FOURIER TRANSFORM (DFT)
****************Explain any four properties of two-dimensional Discrete Fourier Transform
The various properties of 2D- DFT is:
1. LINEARITY
2. TRANSLATION
3. ROTATION
4. SEPARABILITY
5. PERIODICITY
6. CONVOLUTION
7. CORRELATION
8. SYMMETRY PROPERTIES
9. TRANSLATION TO CENTER OF THE FREQUENCY RECTANGLE (M/2, N/2)
LINEARITY:
The DFT of a weighted sum of two 2D- images is equal to the weighted sum of their individual Fourier
transforms.
Let us consider two images f1(x, y) and f2(x, y) are multiplied by two scaling factors a and b respectively.
The DFT [a. f1(x, y) + b. f2(x, y)] = a F1 (u, v) + b. F2 (u, v)
Proof:

DFT [a. f1(x, y) + b. f2(x, y)] =

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DFT [a. f1(x, y) + b. f2(x, y)] = a F1(u, v) + b F2(u, v)

TRANSLATION
2D- DFT of a shifted (in x direction / y direction/ both) image in spatial domain is unaltered except for a
linearly varying Phase factor. That means magnitude is same only phase changes will be there.
Suppose an image f (x, y) is shifted in x direction; the shifted version of the image is f (x-x0, y) and its DFT
is given by:

Introduce the term x = x - x0 +x0 and substitute this in above exponent, we get

DFT ( f (x-x0, y) ) =

ROTATION:

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SEPARABILITY
2D-DFT of an image can be computed by computing 1D-DFT transforms along the rows (columns) of the
image, followed by 1D- DFT transforms along the columns (rows) of the result
We know that:
The 2D-discrete Fourier Transfer (DFT) of a digital image f(x, y) of size M x N is given by:

Let us perform 1D- DFT along the rows that can be represented as:

Now let us perform 1D- DFT along the columns of the above resultant DFT; we get 2D DFT

= F (u, v)
PERIODICITY
2-D Fourier transform and its inverse are infinitely periodic in the u and v directions; that is,

Any integral multiple of period (M, N) will not change the spectrum F (u, v) then we say that image is
periodic.
CONVOLUTION
The 2D- DFT of convolution of two images in spatial domain is equal to multiplication of two images in
frequency domain and vice versa.

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Let us consider the two images f(x, y) and h(x, y) in spatial domain then the DFT of convolution of these two
images:

CORRELATION
Let F(u, v) and H(u, v) denote the DFTs of f(x, y) and h(x, y) then the correlation between these two is given
by:

SYMMETRY PROPERTIES
Fourier transform of a real function, (x, y), is conjugate symmetric

Also

TRANSLATION TO CENTER OF THE FREQUENCY RECTANGLE (M/2, N/2)

Multiplying an image f(x, y) of size M x N by (-1)x+y will shift the origin of F(u, v) to frequency coordinates
(M/2, N/2), which is the center of the M x N area (Frequency rectangle) occupied by the 2D-DFT

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2D-CONVOLUTION THEOREM OF DISCRETE FOURIER TRANSFORM
Explain 2D-convolution theorem of DFT frequency domain Filtering.
The 2D Convolution of two images f(x, y) and h(x, y) is given by:

The 2D- Convolution theorem is given by the expression:

and, conversely

DFT of the product F(u, v) H(u, v) yields the 2-D spatial convolution of f and h.

2D Convolution theorem states that DFT of convolution of two images f(x, y) and h(x, y) is equal to the
multiplication of those two images in Frequency domain and vice versa.
Apply Discrete Fourier Transform for the following image:
2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
Answer:
The 2D- DFT for the above image is given by:
4 Point DFT Kernel Matrix x Image f(x, y) x DFT Kernel MatrixT

4 Point-DFT Kernel Matrix Image Kernel MatrixT

1 1 1 1 2 0 0 0 1 1 1 1
1 -j -1 j x 0 0 0 0 x 1 -j -1 j
1 -1 1 -1 0 0 0 0 1 -1 1 -1
1 j -1 -j 0 0 0 0 1 j -1 -j

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2 0 0 0 1 1 1 1
2 0 0 0 x 1 -j -1 j
2 0 0 0 1 -1 1 -1
2 0 0 0 1 j -1 -j
2D DFT of the given image is:
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
DISCRETE COSINE TRANSFORM (DCT)
• DCT is similar to DFT, transforms an image from spatial domain to Frequency domain
• The difference between these two is in terms of basis function used by each transform.
• DFT uses set of harmonically related complex exponential functions, while DCT uses only (Real
valued) cosine functions.
Define and explain 1D and 2D discrete cosine transform.
The discrete cosine transform (DCT) represents an image as a sum of sinusoids of varying magnitudes and
frequencies. DCTs are widely used for applications such as encoding, decoding, video, audio, multiplexing,
control signals, signaling, and analog-to-digital conversion. DCTs are also commonly used for high-
definition television (HDTV) encoder/decoder chips.
 DCT uses only (Real valued) cosine functions.
 The discrete cosine transform (DCT) helps to separate the image into parts (or spectral sub-bands) of
differing importance (with respect to the image's visual quality).
 DCT is real term and it is orthogonal.
 DCT is often used in JPEG Image compression application.
The most common DCT definition of a 1-D sequence of length N is:

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The Inverse DCT is defined as:

The DCT definition of a 2-D Image of length N x N is:

DCT definition of a 1-D sequence of length N using N X N Cosine matrix:

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C(u) = (N point Cosine Matrix) x f(x)
DCT definition of a 2-D image of length N using N X N Cosine matrix:
C(u, v) = (N point Cosine Matrix) x f(x, y) x Cosine MatrixT

4 x 4 Cosine Matrix is:

0.5 0.5 0.5 0.5
0.653 0.2705 -.2705 -0.653
0.5 -0.5 -0.5 0.5
0.2705 -0.653 0.653 -.2705
FILTERING IN THE FREQUENCY DOMAIN
The slowest varying frequency component (u=0, v= 0) corresponds to the average gray level of an image. As
we move away from the origin of the transform, the low frequencies correspond to the slowly varying
components of an image. In an image of a room, for example, these might correspond to smooth gray-level
variations on the walls and door. As we move further away from the origin, The higher frequencies begin to
correspond to faster and faster gray level changes in the image. These are the edges of objects and other
components of an image characterized by abrupt changes in gray level, such as noise.
*******Explain the steps involved in Image filtering in Frequency domain.

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DIGITAL IMAGE PROCESSING 18CS741 Module 3
1. Given an input image f(x, y) of size M x N obtain the padding parameters P and Q by typically
selecting P =2M and Q = 2N
2. Form a padded image, fp(x, y) of size P x Q by appending the necessary number of zeros to f(x, y).
3. Multiply fp(x, y) by (-1)x+y to centre its transform.
4. Compute the DFT F(u, v) of the image from step 3.
5. Generate a real symmetric filter function H(u, v), of size P x Q with centre at coordinates (P/2, Q/2).
Form the product G(u, v) = H(u, v) F(u, v) using array multiplication; that is G(i, k) = H(i, k) F(i, k)
6. Obtain the processed image:

7. Obtain the final processed result, g(x, y), by extracting the M x N region from the top, left quadrant of
gp(x, y).
IMAGE SMOOTHING IN FREQUENCY DOMAIN
*******Explain image smoothing in frequency domain.
Noise characterized by sharp transitions in image intensity. Such transitions contribute significantly to high
frequency components of Fourier transform. So we need to remove these high frequency components in
order to obtain all blurred image or smoothen the image. This can be done with the help of filters.
To attenuate all high frequency components and to pass only low frequency components we use Low pass
filters. There are 3 types of low-pass filters are used for smoothing purpose;
1. Ideal Low Pass Filters
2. Butterworth LPF
3. Gaussian LPF
IDEAL LOW PASS FILTERS:
A 2-D low-pass filter that passes without attenuation all frequencies within a circle of radius D0 from the
origin and “cuts off” all frequencies outside this circle is called an ideal low-pass filter (ILPF

where D0 is a positive constant and D(u, v) is the distance between a point (u, v) in the frequency domain and
the center of the frequency rectangle; that is

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Ideal Low Pass Filter 3-D view and 2-D view and line graph
Drawback: Blurred edges
BUTTERWORTH LOW-PASS FILTERS
The transfer function of a Butterworth low-pass filter (BLPF) of order n, and with cut-off frequency at a
distance D0 from the origin, is defined as:

Unlike the ILPF, the BLPF transfer function does not have a sharp discontinuity that gives a clear cut-off
between passed and filtered frequencies.
It is useful in defining the edges in an image.

(a) perspective plot of a Butterworth lowpass-filter transfer function. (b) Filter displayed as an image.
(c)Filter radial cross sections of order 1 through 4.
GAUSSIAN LOWPASS FILTERS
The transfer function is smooth, like Butterworth filter and it is given by

D is the distance from the center of the frequency rectangle, where D0 is the cut-off frequency, When D(u, v)
= D0 the GLPF is down to 0.607 of its maximum value.
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Advantage: No ringing artifacts

IMAGE SHARPENING USING FREQUENCY DOMAIN FILTERS
Explain image sharpening in frequency domain
Image sharpening can be achieved in the frequency domain by high-pass filtering, which attenuates the low-
frequency components without disturbing high-frequency information in the Fourier transform
A high-pass filter is obtained from a given low-pass filter using the equation

IDEAL HIGH-PASS FILTERS

BUTTERWORTH HIGH-PASS FILTERS

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Butter-worth high-pass filter to behave smoother than IHPFs.

GAUSSIAN HIGH-PASS FILTERS:

As expected, the results obtained are more gradual than with the previous two filters

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QUESTION BANK
MODULE 3
1 Obtain the equation for one dimensional Discrete Fourier Transform and its inverse from the 10

continuous transform of sampled function of one variable

2 Discuss about two dimensional DFT and its inverse 8

3 Explain any four properties of two-dimensional Discrete Fourier Transform 8

4 Explain 2D-convolution theorem of DFT frequency domain Filtering. 8

5 Define and explain 1D and 2D discrete cosine transform. 8

6 Explain the steps involved in Image filtering in Frequency domain. 6

7 Explain image smoothing and sharpening in frequency domain. 8

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