2d linear systems 2d fourier transform and its...
TRANSCRIPT
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Aliasing in Images:
• See Figures 4.16-4.22 for practical examples.
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Phase Significance
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Digital Image Processing
Filtering in the Frequency Domain
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• Phase Significance
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Digital Image Processing
Filtering in the Frequency Domain
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• Phase Significance
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Digital Image Processing
Filtering in the Frequency Domain
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• Symmetry Properties
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Digital Image Processing
Filtering in the Frequency Domain
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• Definition and Properties (1)
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Digital Image Processing
Filtering in the Frequency Domain
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• Definition and Properties (2)
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Digital Image Processing
Filtering in the Frequency Domain
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• DFT Pairs (1)
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Digital Image Processing
Filtering in the Frequency Domain
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• DFT Pairs (2)
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Digital Image Processing
Filtering in the Frequency Domain
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• FFT Shift, Centering and Symmetry
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Digital Image Processing
Filtering in the Frequency Domain
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• Fourier Transform Centering
– fftshift in Matlab
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Digital Image Processing
Filtering in the Frequency Domain
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• Example
Without Shift
With Shift
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Digital Image Processing
Filtering in the Frequency Domain
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• Translation
• Rotation
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Digital Image Processing
Filtering in the Frequency Domain
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• Phase Changes:
Original Translated Rotated
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Frequency Domain Manipulation
– Set F(0,0) to zero
– Clip negative value
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Digital Image Processing
Filtering in the Frequency Domain
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• Lowpass, Highpass, Highboost
– No shift
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Digital Image Processing
Filtering in the Frequency Domain
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• Zero-Padding Effect:
– Blurring With Gaussian
Original No Zero-Padding Zero-Padding
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Digital Image Processing
Filtering in the Frequency Domain
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• Inherent Periodicity of DFT and Zero-Padding
– With (Right) and Without (Left) Padding
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Example (1)
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Digital Image Processing
Filtering in the Frequency Domain
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• Example (2)
– An image and its spectrum
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Digital Image Processing
Filtering in the Frequency Domain
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• Filtering in Spatial and Frequency Domain
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Digital Image Processing
Filtering in the 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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Digital Image Processing
Filtering in the Frequency Domain
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• Ideal Lowpass Filter
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Ideal Lowpass Filter
– 10, 30, 60, 160, and 460 (Radius)
– Smoothing
– Blurring
– Ringing Effect
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Digital Image Processing
Filtering in the Frequency Domain
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• Origin of Ringing Effect
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Digital Image Processing
Filtering in the Frequency Domain
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• Butterworth Lowpass Filter:
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Digital Image Processing
Filtering in the Frequency Domain
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• Butterworth Lowpass Filter:
– Order (2) Same radius
– Smoothing
– Blurring
– Less Ringing Effect
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Digital Image Processing
Filtering in the Frequency Domain
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• Less Ringing Effect of Butterworh Filter:
– Order 1,2 ,5, and 20
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Digital Image Processing
Filtering in the Frequency Domain
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• Gaussian Lowpass Filter:
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Digital Image Processing
Filtering in the Frequency Domain
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• Gaussian Lowpass Filter:
– Smoothing
– Blurring
– No Ringing Effect!
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Digital Image Processing
Filtering in the Frequency Domain
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• Low Resolution Images Repairing
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Digital Image Processing
Filtering in the Frequency Domain
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• Pre-print Processing (Smooth and soft-Looking)
– Original, D0=100, and D0=80
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Highpass Filters
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Digital Image Processing
Filtering in the Frequency Domain
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• Spatial Representation of Highpass Filters
– Ringing (Ideal, Butterworth, and Gaussian)
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Digital Image Processing
Filtering in the Frequency Domain
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• Ideal Highpass Filter
D0=30 D0=60 D0=160
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Digital Image Processing
Filtering in the Frequency Domain
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• Butterworth (n=2) Highpass Filters
D0=30 D0=60 D0=160
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Digital Image Processing
Filtering in the Frequency Domain
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• Gaussian Highpass Filter
D0=30 D0=60 D0=160
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Example (Laplacian):
– Similar But Not identical to Spatial Domain
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Digital Image Processing
Filtering in the Frequency 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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Digital Image Processing
Filtering in the Frequency Domain
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• Example: GHPF, D0=40
HFE, k1=0.5, k2=0.25
Histogram EQ.
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Example: Whole Body PET Scan Enhancement
0
0.25
2
1
80
L
H
c
D
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Digital Image Processing
Filtering in the Frequency Domain
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• Selective Filtering:
– Bandpass
– Band Reject
– Notch
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Digital Image Processing
Filtering in the Frequency Domain
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• Bandreject and Bandpass Filter
– HBP(u, v)= 1- HBR(u, v)
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Digital Image Processing
Filtering in the Frequency Domain
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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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Digital Image Processing
Filtering in the Frequency Domain
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• Example (1)
– Spot in Frequency Domain
Multiplied Spectrum
Spectrum
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Digital Image Processing
Filtering in the Frequency Domain
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• Example (2)
– Vertical sin Pattern
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Digital Image Processing
Filtering in the Frequency Domain
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• Example (2) – Cont.
– Extract Vertical sin Pattern
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Digital Image Processing
Filtering in the Frequency Domain
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• Matlab Command
– fft2, ifft2, fftshift, ifftshift
– freqz2, fspecial