Filtering in the Frequency Domain
Image Processing
CSE 166
Lecture 7
Announcements
• Assignment 3 will be released today
– Due Apr 27, 11:59 PM
• Reading
– Chapter 4: Filtering in the Frequency Domain
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Overview: Image processing in the frequency domain
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Image in spatial domain
f(x,y)
Image in spatial domain
g(x,y)
Fouriertransform
Image in frequency domain
F(u,v)
Inverse Fourier
transform
Image in frequency domain
G(u,v)
Frequency domain processing
Jean-Baptiste Joseph Fourier1768-1830
2D continuous Fourier transform
• (Forward) Fourier transform
• Inverse Fourier transform
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2D continuous Fourier transform
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1D
2D
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Example: box function
Unit discrete impulse
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1D
2D
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Impulse train
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1D
2D
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Fourier transform of sampled functionand extracting one period
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1D
2D
Over-sampled Under-sampled
Recovered Imperfect recoverydue to
interference
Aliasing
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1D
2D
Aliasing
Original
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Aliasing in real images
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AliasingOriginal No aliasing
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2D Discrete Fourier Transform
2D discrete Fourier transform (DFT)
• (Forward) Fourier transform
• Inverse Fourier transform
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Centering the DFT
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1D
2D
In MATLAB, use fftshift and ifftshift
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Centering the DFT
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Original
DFT(look at corners)
Shifted DFTLog of
shifted DFT
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DFT magnitude of geometrically transformed images
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Translated
Rotatedabout center
Same magnitude as original
(invariant to translation)
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DFT phase of geometrically transformed images
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TranslatedRotated
about centerOriginal
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Contributions of magnitude and phaseto image formation
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Phase
IDFT: Phase only
(zero magnitude)
IDFT: Magnitude
only (zero phase)
IDFT: Boy magnitude
and rectangle phase
IDFT: Rectangle magnitude
and boy phase 17
2D convolution theorem
• 2D discrete (circular) convolution
• 2D convolution theorem
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Filtering using convolution theorem
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Filtering in spatial domain
using convolution
expectedresult
Filtering in frequencydomain
using productwithout
zero-padding
wraparounderror
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Filtering using convolution theorem
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Filtering in frequencydomain
using productwith
zero-padding
no wraparounderror
Gaussian lowpass filter in
frequency domain
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Fourier transform
Product
Inverse Fourier transform
Zero padding
Filtering using convolution theorem
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Filtering in spatial
domain using
convolution
Filtering in frequencydomain
usingproduct
Identical results
DFT
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Filtering in the frequency domain
• Ideal lowpass filter (LPF)
– Frequency domain
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Filtering in the frequency domain
• Ideal lowpass filter (LPF)
– Spatial domain
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H(u,v) h(x,y)
Filtering in the frequency domain
• Gaussian lowpass filter (LPF)
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Filtering in the frequency domain
• Butterworth lowpass filter (LPF)
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Gaussian LPF
Filtering in the frequency domain
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Ideal LPF Butterworth LPF
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Example: character recognition
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Gaussian LPFjoins broken characters
Highpass filter (HPF)Frequency domain
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Ideal HPF
Gaussian HPF
Butterworth HPF
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Butterworth HPF
Highpass filter (HPF)Spatial domain
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Ideal HPF Gaussian HPF
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Filtering in the frequency domain
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Ideal HPF Gaussian HPF Butterworth HPF
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H (u)
u u
x x
H (u)
h (x)
1
16–– 3
h (x)
2 12 12 1
2 1 8 2 1
2 12 12 1
0 2 1 0
2 1 4 2 1
0 2 1 0
1 2 1
2
1
9–– 3
4 2
1 2 1
1 1 1
1 1 1
1 1 1
Filtering in the frequency domain
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1D
Lowpass filter Sharpening filter31
Frequencydomain
Spatialdomain
Filtering in the frequency domain
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2D
32
Lowpass filter Highpass filter Offset highpass filter
Bandreject filters
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ButterworthIdeal Gaussian
Filtering in the frequency domain
• Sharpening filter
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Overview: Image processing in the frequency domain
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Image in spatial domain
f(x,y)
Image in spatial domain
g(x,y)
Fouriertransform
Image in frequency domain
F(u,v)
Inverse Fourier
transform
Image in frequency domain
G(u,v)
Frequency domain processing
Jean-Baptiste Joseph Fourier1768-1830
Next Lecture
• Image restoration
• Reading
– Chapter 5: Image Restoration and Reconstruction
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