2D Linear Systems 2D Fourier Transform and its …ee.sharif.edu/~dip/Files/ImageEnhancementInFrequencyDomain(For... · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 2012

1

Digital Image Processing

Filtering in the Frequency Domain

1

• 2D Linear Systems

• 2D Fourier Transform and its Properties

• The Basics of Filtering in Frequency Domain

• Image Smoothing

• Image Sharpening

• Selective Filtering

• Implementation Tips

ee.sharif.edu/~dip


2



2

• General Definition:

System

H ,f x y ,g x y

, ,g x y H f x y

ee.sharif.edu/~dip


3



3

• Linearity

• Shift Invariant or Spatially Invariant:

• Causality: Same as before

– Don’t worry about it!

• Stability: Same 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/~dip


4



4

• Unit Impulse Function (Pinhole):

, 0,0,

0 , 0,0

, 1

x yx y

x y

x y dxdy

ee.sharif.edu/~dip


5



5

• 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

PSD of Hubble Telescope

ee.sharif.edu/~dip


6



6

• Convolution/Correlation Integral:

• Convolution/Correlation 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/~dip


7



7

• The 2-D Continuous Fourier Transform

2

2

, ,

, ,

j ux vy

j ux vy

F u v f x y e dxdy

f x y F u v e dudv

x y

Arect rect ATZsinc uT sinc vZZ T

ee.sharif.edu/~dip


8



8

• 2-D Sampling and Sampling Theorem

– 2-D impulse train:

– Error Free Reconstruction:

,m m

x m x y n y

max max

1 1,

2 2x y

u v

ee.sharif.edu/~dip


9



9

• Aliasing in Images:

• See Figures 4.16-4.22 for practical examples.

ee.sharif.edu/~dip


10



10

• 2-D Discrete Fourier Transform

• Spatial and Frequency Intervals

1 1

0 0

1 1

0 0

1, , exp 2

1, , ex

1

p 2

M N

m n

M N

u v

um vnF u v f m n j

M N

um vnf m n F u v j

M NMN

1 1,u v

M x N y

ee.sharif.edu/~dip


11



11

• Phase Significance

ee.sharif.edu/~dip


12



12


ee.sharif.edu/~dip


13



13


ee.sharif.edu/~dip


14



14

• Symmetry Properties

ee.sharif.edu/~dip


15



15

• Definition and Properties (1)

ee.sharif.edu/~dip


16



16

• Definition and Properties (2)

ee.sharif.edu/~dip


17



17

• DFT Pairs (1)

ee.sharif.edu/~dip


18



18

• DFT Pairs (2)

ee.sharif.edu/~dip


19



19

• FFT Shift, Centering and Symmetry

ee.sharif.edu/~dip


20



20

• Fourier Transform Centering

– fftshift in Matlab

ee.sharif.edu/~dip


21



21

• Example

Without Shift

With Shift

ee.sharif.edu/~dip


22



22

• Translation

• Rotation

ee.sharif.edu/~dip


23



23

• Phase Changes:

Original Translated Rotated

ee.sharif.edu/~dip


24



24

• Line Pattern in Spatial and Frequency Domain – Strong ±45˚ edge in Spatial Strong ±45˚ edge in Frequency

ee.sharif.edu/~dip


25



25

• Frequency Domain Filtering Fundamentals:

– Convolution Theorem:

– Zero Padding:

– Zero-Padding is necessary to avoid Wraparound error. • Circular vs. Linear Convolution

, , , ,f m n h m n F u v H u v

1 1M N P Q M P N Qf h g

ee.sharif.edu/~dip


26



26

• Frequency Domain Manipulation

– Set F(0,0) to zero

– Clip negative value

ee.sharif.edu/~dip


27



27

• Lowpass, Highpass, Highboost

– No shift

ee.sharif.edu/~dip


28



28

• Zero-Padding Effect:

– Blurring With Gaussian

Original No Zero-Padding Zero-Padding

ee.sharif.edu/~dip


29



29

• Inherent Periodicity of DFT and Zero-Padding

– With (Right) and Without (Left) Padding

ee.sharif.edu/~dip


30



30

• Zero-Padding Side effect

• Read Pg. 260-262!

– Ideal Filter (TL)

– Time Domain (BL)

– Zero Padding (TR)

– Ringing Effect (BR)

ee.sharif.edu/~dip


31



31

• Effect of small changes in phase

– Why we prefer zero-phase filters?

0.5j FIDFT F e 0.25j FIDFT F e

ee.sharif.edu/~dip


32



32

• Steps for Frequency Domain Filtering a) Original

b) Padding

c) Multiply by (-1)x+y

d) FFT

e) GLP (Centered)

f) Multiply © and (e)

g) Multiply IFFTReal by (-1)x+y

h) Cropping

ee.sharif.edu/~dip


33



33

• Spatial-Frequency Correspondences

• The most used Filter (Gaussian)

, ,h x y H u v

2 2

2 2 2 22 222

2

12

2

x yu v

e e

ee.sharif.edu/~dip


34



34

• Example (1)

ee.sharif.edu/~dip


35



35

• Example (2)

– An image and its spectrum

ee.sharif.edu/~dip


36



36

• Filtering in Spatial and Frequency Domain

ee.sharif.edu/~dip


37



37

• Image Smoothing in Frequency Domain

– Ideal Lowpass Filter

– Butterworth Lowpass Filter

– Gaussian Lowpass Filter

– More Examples

2 2,D u v u v

ee.sharif.edu/~dip


38



38

• Ideal Lowpass Filter

ee.sharif.edu/~dip


39



39

• Test Pattern and Energy Circles

460 (99.2%) 160 (97.8%) 60 (95.7%) 30 (93.1%) 10 (87.0%)

ee.sharif.edu/~dip


40



40

• Ideal Lowpass Filter

– 10, 30, 60, 160, and 460 (Radius)

– Smoothing

– Blurring

– Ringing Effect

ee.sharif.edu/~dip


41



41

• Origin of Ringing Effect

ee.sharif.edu/~dip


42



42

• Butterworth Lowpass Filter:

ee.sharif.edu/~dip


43



43

• Butterworth Lowpass Filter:

– Order (2) Same radius

– Smoothing

– Blurring

– Less Ringing Effect

ee.sharif.edu/~dip


44



44

• Less Ringing Effect of Butterworh Filter:

– Order 1,2 ,5, and 20

ee.sharif.edu/~dip


45



45

• Gaussian Lowpass Filter:

ee.sharif.edu/~dip


46



46

• Gaussian Lowpass Filter:

– Smoothing

– Blurring

– No Ringing Effect!

ee.sharif.edu/~dip


47



47

• Low Resolution Images Repairing

ee.sharif.edu/~dip


48



48

• Pre-print Processing (Smooth and soft-Looking)

– Original, D0=100, and D0=80

ee.sharif.edu/~dip


49



49

• Remove Unwanted Pattern:

– Remove Horizontal Lines (Imaging System Deficiency)

– Large Recognizable Features

– Original, D0=50, and D0=20

ee.sharif.edu/~dip


50



50

• Image Sharpening in Frequency Domain

– Ideal Highpass Filter

– Butterworth Highpass Filter

– Gaussian Highpass Filter

– More Examples

, 1 ,HP LPH u v H u v

ee.sharif.edu/~dip


51



51

• Highpass Filters

ee.sharif.edu/~dip


52



52

• Spatial Representation of Highpass Filters

– Ringing (Ideal, Butterworth, and Gaussian)

ee.sharif.edu/~dip


53



53

• Ideal Highpass Filter

D0=30 D0=60 D0=160

ee.sharif.edu/~dip


54



54

• Butterworth (n=2) Highpass Filters

D0=30 D0=60 D0=160

ee.sharif.edu/~dip


55



55

• Gaussian Highpass Filter

D0=30 D0=60 D0=160

ee.sharif.edu/~dip


56



56

• Thumb Print Processing:

– Original (Left)

– Butterworth Highpass Filter with n=4, D0=50 (Middle)

– Thresholding (Right) • Setting Negative Value to Black and Positive value to White

ee.sharif.edu/~dip


57



57

• Laplacian in Frequency Domain:

• Image Laplacian:

• Enhanced Image:

2 2 2 2 2, 4 4 ,H u v u v D u v

2 1, , ,f x y H u v F u v

2

1

1 2 2

, , ,

, , ,

1 4 , ,

g x y f x y c f x y

F u v H u v F u v

D u v F u v

ee.sharif.edu/~dip


58



58

• Example (Laplacian):

– Similar But Not identical to Spatial Domain

ee.sharif.edu/~dip


59



59

• Unsharp Masking, Highboost, High-Frequency Emphasis:

• Unsharp Masking (K=1) and Highboost Filters(K>1):

• High Frequency Emphasing:

1

, , ,

, , ,

mask LP

LP LP

g x y f x y f x y

f x y H u v F u v

1

, , ,

, 1 1 , ,

mask

LP

g x y f x y kg x y

g x y k H u v F u v

1 2

1

1

, ,1 ,

,, ,

HP

HP

g x y F u v

g x

kH u v

k k H uy F u vv

ee.sharif.edu/~dip


60



60

• Example: GHPF, D0=40

HFE, k1=0.5, k2=0.25

Histogram EQ.

ee.sharif.edu/~dip


61



61

• Homomorphic Filtering:

• Linear Process is not Possible:

• Summary of Steps

, , ,f x y i x y r x y

ln , ln , ln ,f x y i x y r x y

ee.sharif.edu/~dip


62



62

• Illumination-Reflection Control Filter

2 2

0,, 1

c D u v D

H L LH u v e

ee.sharif.edu/~dip


63



63

• Example: Whole Body PET Scan Enhancement

0

0.25

2

1

80

L

H

c

D

ee.sharif.edu/~dip


64



64

• Selective Filtering:

– Bandpass

– Band Reject

– Notch

ee.sharif.edu/~dip


65



65

• Bandreject and Bandpass Filter

– HBP(u, v)= 1- HBR(u, v)

ee.sharif.edu/~dip


66



66

• Notch Filters:

– Pass/Reject predefined, both (u0, v0) and (-u0, -v0)

– HNP(u, v)= 1- HNR(u, v)

ee.sharif.edu/~dip


67



67

• Example (1)

– Spot in Frequency Domain

Multiplied Spectrum

Spectrum

ee.sharif.edu/~dip


68



68

• Example (2)

– Vertical sin Pattern

ee.sharif.edu/~dip


69



69

• Example (2) – Cont.

– Extract Vertical sin Pattern

ee.sharif.edu/~dip


70



70

• Matlab Command

– fft2, ifft2, fftshift, ifftshift

– freqz2, fspecial

Documents

2D Linear Systems 2D Fourier Transform and its …ee.sharif.edu/~dip/Files/ImageEnhancementInFrequencyDomain(For... · ee.sharif.edu/~dip . E. Fatemizadeh, Sharif University of Technology,