ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20111

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• General Definition:

SystemH ,f x y ,g x y

, ,g x y H f x y

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20112

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Linearity

• Shift Invariant or Spatially Invariant:

• Causality: We do not care about it• Stability as before.

1 2 1 2, , , ,H af x y bf x y aH f x y bH f x y

0 0 0 0, , , ,g x y H f x y g x x y y H f x x y y

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20113

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Unit Impulse Function (Pinhole):

0 , 0,0,

, 0,0

, 1

x yx y

x y

x y dxdy

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20114

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Point Spread Function (Impulse Response):

• Linear Shift Invariant Systems:

0 0 0 0, ; , ,H x y x y H x x y y

0 0 0 0 0 0, ; , , ,

, ,

H x y x y H x x y y H x x y y

H x y H x y

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20115

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Convolution/Correlation Integral:

• Convolution/Corrlelation Summation:

, , , ,

, , , ,

f x y h x y f s t h x s y t dsdt

f x y h x y f s t h x s y t dsdt

1 1

0 0

1 1

0 0

, , , ,

, , , ,

M N

p q

M N

p q

f m n h m n f p q h m p n q

f m n h m n f p q h m p n q

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20116

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Discrete Fourier Transform:

1 1

0 0

1 1

0 0

1, , exp 2

1, , exp 21

M N

m n

M N

u v

um vnF u v f m n jM N

um vnf m n F

N

jN

M

u vM

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20117

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20118

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 20119

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201110

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

10

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201111

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Fourier Transform Centering– fftshift in Matlab

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201112

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Example

Without Shift

With Shift

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E. Fatemizadeh, Sharif University of Technology, 201113

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Phase Importance in Digital Image Processing

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E. Fatemizadeh, Sharif University of Technology, 201114

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Phase Importance in Digital Image Processing

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201115

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

Double Rect

Double Sinc

• A Sample DFT:

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201116

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Fourier Spectrum

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201117

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Image Processing in Frequency Domain:

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201118

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Notch Filter Effect on DC Level

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E. Fatemizadeh, Sharif University of Technology, 201119

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Lowpass and Highpass effects

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E. Fatemizadeh, Sharif University of Technology, 201120

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Highboost (Highpass + Original)

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E. Fatemizadeh, Sharif University of Technology, 201121

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Ideal Low Pass Filtering:

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E. Fatemizadeh, Sharif University of Technology, 201122

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• An Image and its Spectrum (Circle Energy)!

92.0%

94.6%

96.4%

98.0%

99.5%

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201123

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

Blurring Effect of ILPF

(Back to BLPF)

(Back to GLPF)

-8%

-5.4% -3.6%

-2% -0.5%

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201124

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Origin of Ringing Effect

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E. Fatemizadeh, Sharif University of Technology, 201125

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Butterworth Low Pass Filtering

2

0

1,,

1

H u vD u v

D

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201126

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

BLPF Blurring Effect (ILPF)

(Back to GLPF)

D0=5

D0=15 D0=30

D0=80D0=230

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201127

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

BLPF(1) BLPF(2) BLPF(5) BLPF(20)

• Butterworth LPF Spatial Representation:

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E. Fatemizadeh, Sharif University of Technology, 201128

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Gaussian Low Pass Filter

2

20

,, exp

2D u v

H u vD

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201129

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

GLPF Blurring Effect (ILPF)

(BLPF)

D0=5

D0=15 D0=30

D0=80D0=230

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201130

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

IHPF

BHPF

GHPF

• Highpass filter

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E. Fatemizadeh, Sharif University of Technology, 201131

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

IHPF BHPF GHPF

• Ringing!

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201132

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

IHPF

Ringing

D0=15 D0=30 D0=80

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201133

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

BHPF

Ringing

D0=15 D0=30 D0=80

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201134

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

GHPF

Ringing

D0=15 D0=30 D0=80

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201135

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

Laplacian in (u,v) Image view

IDFT of –(u2+v2)

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201136

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

Original Laplacian

EnhancedLaplacian

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E. Fatemizadeh, Sharif University of Technology, 201137

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

Original Laplacian

A=2 A=2.7

, 1 ,hb hpH u v A H u v

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201138

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Combinational Processes:

Original BHPF

HFE

, ,hfe hpH u v a bH u v

HE

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201139

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Implementation‐Practical Consideration:

Ordinary ViewShifted View

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201140

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Zero Padding:– Convolution of N‐points signal and M‐Points signals will results a M+N‐1 points signal

– In Frequency Domain we need zero padding before Fourier transform.

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201141

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

ee.sharif.edu/~miap

E. Fatemizadeh, Sharif University of Technology, 201142

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Zero Padding:

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E. Fatemizadeh, Sharif University of Technology, 201143

Medical Image Analysis and Processing

Image Enhancement in Frequency Domain

• Matlab Command– fft2, ifft2, fftshift, ifftshift– freqz2, – fspecial (average, disk , gaussian, laplacian, log, motion, prewitt, sobel, unsharp)