2D Linear Systems 2D Fourier Transform and its...

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E. Fatemizadeh, Sharif University of Technology, 2012

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Digital Image Processing

Filtering in the Frequency Domain

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• 2D Linear Systems

• 2D Fourier Transform and its Properties

• The Basics of Filtering in Frequency Domain

• Image Smoothing

• Image Sharpening

• Selective Filtering

• Implementation Tips

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• General Definition:

System

H ,f x y ,g x y

, ,g x y H f x y

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• 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

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• Unit Impulse Function (Pinhole):

, 0,0,

0 , 0,0

, 1

x yx y

x y

x y dxdy

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• 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

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• 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

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• 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

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• 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

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• Aliasing in Images:

• See Figures 4.16-4.22 for practical examples.

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• 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

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• Phase Significance

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• Symmetry Properties

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• Definition and Properties (1)

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• Definition and Properties (2)

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• DFT Pairs (1)

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• DFT Pairs (2)

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• FFT Shift, Centering and Symmetry

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• Fourier Transform Centering

– fftshift in Matlab

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• Example

Without Shift

With Shift

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• Translation

• Rotation

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• Phase Changes:

Original Translated Rotated

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

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• 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

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• Frequency Domain Manipulation

– Set F(0,0) to zero

– Clip negative value

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• Lowpass, Highpass, Highboost

– No shift

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• Zero-Padding Effect:

– Blurring With Gaussian

Original No Zero-Padding Zero-Padding

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• Inherent Periodicity of DFT and Zero-Padding

– With (Right) and Without (Left) Padding

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• Zero-Padding Side effect

• Read Pg. 260-262!

– Ideal Filter (TL)

– Time Domain (BL)

– Zero Padding (TR)

– Ringing Effect (BR)

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• Effect of small changes in phase

– Why we prefer zero-phase filters?

0.5j FIDFT F e 0.25j FIDFT F e

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• 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

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• 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

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• Example (1)

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• Example (2)

– An image and its spectrum

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• Filtering in Spatial and Frequency Domain

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• Image Smoothing in Frequency Domain

– Ideal Lowpass Filter

– Butterworth Lowpass Filter

– Gaussian Lowpass Filter

– More Examples

2 2,D u v u v

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• Ideal Lowpass Filter

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• Test Pattern and Energy Circles

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

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• Ideal Lowpass Filter

– 10, 30, 60, 160, and 460 (Radius)

– Smoothing

– Blurring

– Ringing Effect

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• Origin of Ringing Effect

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• Butterworth Lowpass Filter:

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• Butterworth Lowpass Filter:

– Order (2) Same radius

– Smoothing

– Blurring

– Less Ringing Effect

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• Less Ringing Effect of Butterworh Filter:

– Order 1,2 ,5, and 20

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• Gaussian Lowpass Filter:

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• Gaussian Lowpass Filter:

– Smoothing

– Blurring

– No Ringing Effect!

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• Low Resolution Images Repairing

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

– Original, D0=100, and D0=80

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• Remove Unwanted Pattern:

– Remove Horizontal Lines (Imaging System Deficiency)

– Large Recognizable Features

– Original, D0=50, and D0=20

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• Image Sharpening in Frequency Domain

– Ideal Highpass Filter

– Butterworth Highpass Filter

– Gaussian Highpass Filter

– More Examples

, 1 ,HP LPH u v H u v

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• Highpass Filters

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• Spatial Representation of Highpass Filters

– Ringing (Ideal, Butterworth, and Gaussian)

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• Ideal Highpass Filter

D0=30 D0=60 D0=160

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• Butterworth (n=2) Highpass Filters

D0=30 D0=60 D0=160

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• Gaussian Highpass Filter

D0=30 D0=60 D0=160

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• 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

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• 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

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• Example (Laplacian):

– Similar But Not identical to Spatial Domain

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• 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

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• Example: GHPF, D0=40

HFE, k1=0.5, k2=0.25

Histogram EQ.

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• 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

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• Illumination-Reflection Control Filter

2 2

0,, 1

c D u v D

H L LH u v e

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• Example: Whole Body PET Scan Enhancement

0

0.25

2

1

80

L

H

c

D

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• Selective Filtering:

– Bandpass

– Band Reject

– Notch

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• Bandreject and Bandpass Filter

– HBP(u, v)= 1- HBR(u, v)

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• Notch Filters:

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

– HNP(u, v)= 1- HNR(u, v)

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• Example (1)

– Spot in Frequency Domain

Multiplied Spectrum

Spectrum

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• Example (2)

– Vertical sin Pattern

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• Example (2) – Cont.

– Extract Vertical sin Pattern

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• Matlab Command

– fft2, ifft2, fftshift, ifftshift

– freqz2, fspecial

2D Linear Systems 2D Fourier Transform and its...

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