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

2 CSC447: 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

MuxjexfM

uF

1

0

/2)()(M

u

MuxjeuFxf



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

NvyMuxjeyxfMN

vuF

1

0

1

0

)//(2),(),(M

u

N

v

NvyMuxjevuFyxf



the Fourier spectrum , phase angle, andpower

spectrum , are defined as before:

),(),(|),(|

and )],,(/),([ tan v)(u,

,)],(),([|),(| where,

|),(|),(

222

1-

2/122

),(

vuIvuRvuF P(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



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:



Gaussian Low-pass filters:



Ideal High-pass filters:









Gaussian High-pass filters:



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

Education