Digital Image Processing: Image Enhancement in the Frequency Domain

CSC447: Digital Image
Processing
Chapter 4:
Prof. Dr. Mostafa Gadal-Haqq M. Mostafa
Computer Science Department
Faculty of Computer & Information Sciences
AIN SHAMS UNIVERSITY

Foundation
 Fourier Theorem:
Any function that
periodically repeat itself
can be represented by the
some of sines and/or
cosines of different
frequencies, each
multiplied by a different
coefficient.
)cossin()(
0
xbxaxf ii
n
i
ii   
2CSC447: Digital Image Processing Prof. Dr. Mostafa GadalHaqq.

The Discrete Fourier Transform (DFT)
 1-D Fourier Transform:
 The Fourier transform, F(u), of a discrete 1-
D function, f(x); x = 0, 1, 2, …, M-1, is:
 Where u= 0, 1, 2, …, M-1
 1-D Inverse Fourier Transform:





1
0
/2
)(
1
)(
M
x
Muxj
exf
M
uF 




1
0
/2
)()(
M
u
Muxj
euFxf 

 F(u) is called the frequency component of the Fourier
Transform, and its domain (the values of u) is called the
frequency domain, because u determines the frequency
of the components of the transform:
 Since F(u) is complex quantity It is convenient to
express it in polar form
 |F(u)| is called the magnitude, and (u) is the phase
 The Power Spectrum P(u) = |F(u)|2
)](/)([tan(u)and,)]()([|)(|where,
|)(|)(
1-2/122
)(
uRuIuIuRuF
euFuF uj





 The Fourier transform, F(u,v), of a discrete
2-D function (MxN), f(x,y) is:
 Where u= 0,1,2, …,M-1, and v = 0,1,2, …, N-1
 2-D Inverse Fourier Transform:







1
0
1
0
)//(2
),(
1
),(
M
x
N
y
NvyMuxj
eyxf
MN
vuF 
 






1
0
1
0
)//(2
),(),(
M
u
N
v
NvyMuxj
evuFyxf 

 the Fourier spectrum , phase angle, andpower
spectrum , are defined as before:
),(),(|),(|
and)],,(/),([tanv)(u,
,)],(),([|),(|where,
|),(|),(
222
1-
2/122
),(
vuIvuRvuFP(u,v)
vuRvuI
vuIvuRvuF
evuFvuF vuj







 Properties of the Fourier Transform:

Filtering in the Frequency Domain
 Basic Operations

 2-D Fourier Transform

 Notch filter

 Ideal Low-pass Filter (ILPF)
cutoff
frequency

 How to find the cutoff frequency for a ILPF?
 Find the circle that enclose a certain amount of the
power spectrum of the image:
 Where P(u,v) is the Power spectrum at frequencies
(u,v) the. Then , a circle of radius r enclose a 
percentage of the power, where

 Distribution of the power spectrum:

 Filtering with power cutoff

 Butterworth Low-pass Filter

 Gaussian Low-pass Filter
 Where D(u,v) id the distance from the origin

 Gaussian Low-pass Filter

 Gaussian Low-pass and High-pass filters:

 Gaussian Low-pass filters:

 Ideal High-pass filters:

 Gaussian High-pass filters:

 High-pass filters:

 Ideal Band-Pass Filter

 The Laplacian filters:

 Gaussian High-pass filters:

 Homomorphic filters:

HW3
 4.9 and 4.12

Digital Image Processing: Image Enhancement in the Frequency Domain

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