Filtering in the Frequency Domaincseweb.ucsd.edu/classes/sp20/cse166-a/lec7.pdfFiltering in spatial...
Transcript of Filtering in the Frequency Domaincseweb.ucsd.edu/classes/sp20/cse166-a/lec7.pdfFiltering in spatial...
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Filtering in the Frequency Domain
Image Processing
CSE 166
Lecture 7
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Announcements
• Assignment 3 will be released today
– Due Apr 27, 11:59 PM
• Reading
– Chapter 4: Filtering in the Frequency Domain
CSE 166, Spring 2020 2
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Overview: Image processing in the frequency domain
CSE 166, Spring 2020 3
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
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2D continuous Fourier transform
• (Forward) Fourier transform
• Inverse Fourier transform
CSE 166, Spring 2020 4
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2D continuous Fourier transform
CSE 166, Spring 2020
1D
2D
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Example: box function
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Unit discrete impulse
CSE 166, Spring 2020
1D
2D
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Impulse train
CSE 166, Spring 2020
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
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Aliasing
CSE 166, Spring 2020
1D
2D
Aliasing
Original
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Aliasing in real images
CSE 166, Spring 2020
AliasingOriginal No aliasing
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2D Discrete Fourier Transform
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2D discrete Fourier transform (DFT)
• (Forward) Fourier transform
• Inverse Fourier transform
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Centering the DFT
CSE 166, Spring 2020
1D
2D
In MATLAB, use fftshift and ifftshift
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Centering the DFT
CSE 166, Spring 2020
Original
DFT(look at corners)
Shifted DFTLog of
shifted DFT
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DFT magnitude of geometrically transformed images
CSE 166, Spring 2020
Translated
Rotatedabout center
Same magnitude as original
(invariant to translation)
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DFT phase of geometrically transformed images
CSE 166, Spring 2020
TranslatedRotated
about centerOriginal
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Contributions of magnitude and phaseto image formation
CSE 166, Spring 2020
Phase
IDFT: Phase only
(zero magnitude)
IDFT: Magnitude
only (zero phase)
IDFT: Boy magnitude
and rectangle phase
IDFT: Rectangle magnitude
and boy phase 17
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2D convolution theorem
• 2D discrete (circular) convolution
• 2D convolution theorem
CSE 166, Spring 2020 18
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Filtering using convolution theorem
CSE 166, Spring 2020
Filtering in spatial domain
using convolution
expectedresult
Filtering in frequencydomain
using productwithout
zero-padding
wraparounderror
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Filtering using convolution theorem
CSE 166, Spring 2020
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
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Filtering using convolution theorem
CSE 166, Spring 2020
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
CSE 166, Spring 2020 23
H(u,v) h(x,y)
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Filtering in the frequency domain
• Gaussian lowpass filter (LPF)
CSE 166, Spring 2020 24
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Filtering in the frequency domain
• Butterworth lowpass filter (LPF)
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Gaussian LPF
Filtering in the frequency domain
CSE 166, Spring 2020
Ideal LPF Butterworth LPF
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Example: character recognition
CSE 166, Spring 2020 27
Gaussian LPFjoins broken characters
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Highpass filter (HPF)Frequency domain
CSE 166, Spring 2020
Ideal HPF
Gaussian HPF
Butterworth HPF
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Butterworth HPF
Highpass filter (HPF)Spatial domain
CSE 166, Spring 2020
Ideal HPF Gaussian HPF
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Filtering in the frequency domain
CSE 166, Spring 2020
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
CSE 166, Spring 2020
1D
Lowpass filter Sharpening filter31
Frequencydomain
Spatialdomain
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Filtering in the frequency domain
CSE 166, Spring 2020
2D
32
Lowpass filter Highpass filter Offset highpass filter
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Bandreject filters
CSE 166, Spring 2020 33
ButterworthIdeal Gaussian
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Filtering in the frequency domain
• Sharpening filter
CSE 166, Spring 2020 34
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Overview: Image processing in the frequency domain
CSE 166, Spring 2020 35
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
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Next Lecture
• Image restoration
• Reading
– Chapter 5: Image Restoration and Reconstruction
CSE 166, Spring 2020 36