Digital Image Processing

H.R. Pourreza

Digital Image Processing

بسمه تعالي

H.R. Pourreza

Image Enhancement in the Frequency Domain

(Chapter 4)

H.R. Pourreza

• Basic understanding of the frequency domain, Fourier series, and Fourier transform. • Image enhancement in the frequency domain.

Objective

H.R. Pourreza

Any function that periodically repeats itself can be expressedas the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficients.This sum is called a Fourierseries.

Fourier Series

H.R. Pourreza

00

0

00

0

00

0

00

00

00

1000

sin)(2

cos)(2

)(1

sincos)(

Tt

tn

Tt

tn

Tt

t

nnn

dttnwtgT

b

dttnwtgT

a

dttgT

a

tnwbtnwaatg

Fourier Series

H.R. Pourreza

A function that is not periodic but the area under its curve is finite can be expressed as the integral of sines and/or cosines multiplied by a weighing function. The formulation in this case is Fourier transform.

Fourier Transform

H.R. Pourreza

dueuFxf

dxexfuF

uxj

uxj

2

2

)()(

)()(

Continuous One-Dimensional Fourier Transform and Its Inverse

Where 1j

• (u) is the frequency variable.• F(u) is composed of an infinite sum of

sine and cosine terms and…• Each value of u determines the

frequency of its corresponding sine-cosine pair.

H.R. Pourreza

210211

)(x

xx

2 sinc )( uuF

0.5- 0.5x

2

4

u

Continuous One-Dimensional Fourier Transform and Its InverseExampleFind the Fourier transform of a gate function (t) defined by

H.R. Pourreza

Continuous One-Dimensional Fourier Transform and Its Inverse

H.R. Pourreza

Discrete One-Dimensional Fourier Transform and Its Inverse

• A continuous function f(x) is discretized into a sequence:

)}]1[(),...,2(),(),({ 0000 xNxfxxfxxfxf

by taking N or M samples x units apart.

H.R. Pourreza


• Where x assumes the discrete values (0,1,2,3,…,M-1) then

)()( 0 xxxfxf

• The sequence {f(0),f(1),f(2),…f(M-1)} denotes any M uniformly spaced samples from a corresponding continuous function.

H.R. Pourreza

1

0

1

0

2

2sin2cos)(1)(

)(1)(

M

x

M

x

Mxuj

Mxuj

Mxuxf

MuF

exfM

uF

1

0

2)()(

M

u

xMuj

euFxf

u =[0,1,2, …, M-1]

x =[0,1,2, …, M-1]


H.R. Pourreza


• The values u = 0, 1, 2, …, M-1 correspond to samples of the continuous transform at values 0, u, 2u, …, (M-1)u.

i.e. F(u) represents F(uu), where:

u 1

Mx• Each term of the FT (F(u) for every u) is

composed of the sum of all values of f(x)

H.R. Pourreza


• The Fourier transform of a real function is generally complex and we use polar coordinates:

)()(tan)(

)]()([)(

)()(

)()()(

1

2/122

)(

uRuIu

uIuRuF

euFuF

ujIuRuFuj

H.R. Pourreza

|F(u)| (magnitude function) is the Fourier spectrum of f(x) and (u) its phase angle.

• The square of the spectrum

)()()()( 222 uIuRuFuP

is referred to as the Power Spectrum of f(x) (spectral density).


H.R. Pourreza

• Fourier spectrum: 2/122 ),(),(),( vuIvuRvuF

• Phase:

),(),(tan),( 1

vuRvuIvu

• Power spectrum:

),(),(),(),( 222 vuIvuRvuFvuP

Discrete 2-Dimensional Fourier Transform

H.R. Pourreza


H.R. Pourreza 1 second

Fmax = 100 HzWhat is the sampling rate (Nyquist)?What is the time resolution?What is the frequency resolution?What if the sampling rate is higher than the Nyquist sampling rate?What if we take samples for twoseconds with the Nyquist samplingrate?

Time and Frequency Resolution and Sampling

H.R. Pourreza

2/122

1

0

1

0

)(2

1

0

1

0

)(2

),(),(),(

),(),(

),(1),(

vuIvuRvuF

evuFyxf

eyxfMN

vuF

M

u

N

v

yNvx

Muj

M

x

N

y

Nyv

Mxuj

Fourier Spectrum

Discrete Two-Dimensional Fourier Transform and Its Inverse

H.R. Pourreza

1

0

1

0),(1)0,0(

M

x

N

yyxf

MNF

F(0,0) is the average intensity of an image


H.R. Pourreza

Use Matlab to generate the above figures


H.R. Pourreza

),(),(

)()(

then)()(

If

00

)(2

0

00

0

vvuuFeyxf

Getg

Gtg

Nyv

Mxuj

tj

Frequency Shifting Property of the Fourier Transform

H.R. Pourreza

Frequency Shifting Property of the Fourier Transform

H.R. Pourreza

function Normalized_DFT = Img_DFT(img)img=double(img); % So mathematical operations can be conducted on % the image pixels.[R,C]=size(img);for r = 1:R % To phase shift the image so the DFT will be for c=1:C % centered on the display monitor phased_img(r,c)=(img(r,c))*(-1)^((r-1)+(c-1)); endendfourier_img = fft2(phased_img); %Discrete Fourier Transformmag_fourier_img = abs(fourier_img ); % Magnitude of DFTLog_mag_fourier_img = log10(mag_fourier_img +1);Max = max(max(Log_mag_fourier_img ));Normalized_DFT = (Log_ mag_fourier_img )*(255/Max);imshow(uint8(Normalized_DFT))

Basic Filtering in the Frequency Domain using Matlab

H.R. Pourreza

1. Multiply the input image by (-1)x+y to center the transform2. Compute F(u,v), the DFT of the image from (1)3. Multiply F(u,v) by a filter function H(u,v)4. Compute the inverse DFT of the result in (3)5. Obtain the real part of the result in (4)6. Multiply the result in (5) by (-1)x+y

Basic Filtering in the Frequency Domain

H.R. Pourreza

Filtering out the DC Frequency Component

H.R. Pourreza

1)2/,2/(),( if 0

),(NMvu

vuHotherwise

Notch Filter

Filtering out the DC Frequency Component

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Low Pass Filter attenuate high frequencies while “passing” low frequencies.

High Pass Filter attenuate low frequencies while “passing” high frequencies.

Low-pass and High-pass Filters

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Low-pass and High-pass Filters

H.R. Pourreza

2/122

0

0

)2/()2/(),(

),( if 0),( if 1

),(

NvMuvuD

DvuDDvuD

vuH

Smoothing Frequency Domain, Ideal Low-pass Filters

H.R. Pourreza

u vT

M

u

N

vT

PvuF

vuFP

/),(100

),(1

0

1

0

2

Total Power

The remained percentage power after filtration


H.R. Pourreza

fc =5 = 92%

fc =15 = 94.6%

fc =30 = 96.4%

fc =80 = 98%

fc =230 = 99.5%


H.R. Pourreza

Cause of Ringing

H.R. Pourreza

Implement in Matlab the Ideal low-pass filter in the following equation.

1. You must give the user the ability to specify the cutoff frequency D0

2. Calculate the The remained percentage power after filtration.

3. Display the image after filtration4. Use Elaine image to test your program

2/122

0

0

)2/()2/(),(

),( if 0),( if 1

),(

NvMuvuD

DvuDDvuD

vuH

Project #5, Ideal Low-pass Filter

H.R. Pourreza

nDvuDvuH 2

0/),(11),(

Smoothing Frequency Domain, Butterworth Low-pass Filters

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Radii= 5

Radii= 15 Radii=

30

Radii= 80

Radii= 230

Butterworth Low-pass Filter: n=2

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H.R. Pourreza

20

2 2/),(),( DvuDevuH

Smoothing Frequency Domain, Gaussian Low-pass Filters

H.R. Pourreza

Radii= 5

Radii= 15 Radii=

30

Radii= 80

Radii= 230

Gaussian Low-pass


H.R. Pourreza


H.R. Pourreza

Hhp(u,v) = 1 - Hlp(u,v)

Sharpening Frequency Domain Filters

H.R. Pourreza

Sharpening Frequency Domain Filters

H.R. Pourreza

),( if 1

),( if 0),(

0

0

DvuDDvuD

vuH

Sharpening Frequency Domain, Ideal High-pass Filters

H.R. Pourreza

nvuDDvuH 2

0 ),(/11),(

Sharpening Frequency Domain, Butterworth High-pass Filters

H.R. Pourreza

20

2 2/),(1),( DvuDevuH

Sharpening Frequency Domain, Gaussian High-pass Filters

H.R. Pourreza

Homomorphic Filtering

H.R. Pourreza

g t k t h tM

k t m h mm

M

( ) ( ) * ( ) ( ) ( )

1

0

1

h(t) g(t)k(t) g(t) = k(t)*h(t)G(f) = K(f)H(f)

-1

1

t

h(t)

123

t

k(t)

1 2 3

* is a convolution operator and not multiplication

Convolution

H.R. Pourreza

gM

k m h mm

M

( ) ( ) ( )01

00

1

m

k(-m)

-1

1

m

h(m)

g(t)

Convolution

H.R. Pourreza

-1

1

m

h(m)

Convolution

m

k(1-m)

g(t)

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-1

1

m

h(m)

Convolution

m

k(2-m)

g(t)

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-1

1

m

h(m)

Convolution

m

k(3-m)

g(t)

H.R. Pourreza

-1

1

m

h(m)

Convolution

m

k(4-m)

g(t)

H.R. Pourreza

-1

1

m

h(m)

Convolution

m

k(5-m)

g(t)

H.R. Pourreza

-1

1

m

h(m)

Convolution

m

k(6-m)

g(t)

H.R. Pourreza

f x y h x yMN

f m n h x m y nn

N

m

M

( , ) ( , ) ( , ) ( , )

10

1

0

1

2-Dimensions Convolution

H.R. Pourreza

g t k t h tM

k t m h mm

M

( ) ( ) ( ) ( ) ( )

1

0

1

h(t) g(t)k(t)g(t) = k(t)h(t)G(f) = K(f)* . H(f)

-1

1 1

2

t t

k(t)h(t)

g(t)

2

Correlation

H.R. Pourreza

Correlation

H.R. Pourreza

Convolution

Digital Image Processing

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