chapter 4 image enhancement in the frequency domain
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![Page 1: Chapter 4 Image Enhancement in the Frequency Domain](https://reader035.vdocuments.site/reader035/viewer/2022081501/56649d3a5503460f94a1509d/html5/thumbnails/1.jpg)
Chapter 4
Image Enhancement in the Frequency Domain
![Page 2: Chapter 4 Image Enhancement in the Frequency Domain](https://reader035.vdocuments.site/reader035/viewer/2022081501/56649d3a5503460f94a1509d/html5/thumbnails/2.jpg)
Fourier Transform1-D Fourier Transform1-D Discrete Fourier Transform (DFT)MagnitudePhasePower spectrum
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2D DFT Definition:
1
0
)//(21
0
1
0
)//(21
0
),(),(
),(1
),(
N
v
NvyMuxjM
u
N
y
NvyMuxjM
x
evuFyxf
eyxfMN
vuF
1
0
1
0
),(1
)0,0(M
x
N
y
yxfMN
F
),(),(
),(*),(
vuFvuF
vuFvuF
if f(x,y) is real
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Centered Fourier Spectrum
It can be shown that:
)2/,2/()1)(,( NvMuFyxf yx
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Example
SEM Image
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Filtering in the Frequency Domain
1. Multiply the input image by (-1)^x+y to center the transform
2. Compute F(u,v), the DFT of input3. Multiply F(u,v) by a filter H(u,v)4. Computer the inverse DFT of 35. Obtain the real part of 46. Multiply the result in 5 by (-
1)^(x+y)
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Fourier Domain Filtering
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Some Basic FiltersNotch filter:
otherwise 1
N/2)(M/2,v)(u, if 0),( vuH
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Lowpass and Highpass Filters
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Convolution TheoremDefinition
Theorem
Need to define the discrete version of impulse function to prove these results.
1
0
1
0
),(),(1
),(),(M
m
N
n
nymxhnmfMN
yxhyxf
),(),(),(),(
),(),(),(),(
vuHvuFyxhyxf
vuHvuFyxhyxf
),(),(),( 00
1
0
1
000 yxAsyyxxAyxs
M
x
N
y
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Gaussian Filters
Difference of Gaussians (DoG)
222
22
2
2/
2)(
)(x
u
Aexh
AeuH
22
221
2 2/2/)( uu BeAeuH
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Illustration
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Smoothing FiltersIdeal lowpass filtersButterworth lowpass filtersGaussian lowpass filters
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Ideal Lowpass Filters
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Example
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Ringing Effect
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Butterworth Lowpass FiltersDefinition:
nDvuDvuH 2
0/),(1
1),(
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Example
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Ringing Effect
![Page 20: Chapter 4 Image Enhancement in the Frequency Domain](https://reader035.vdocuments.site/reader035/viewer/2022081501/56649d3a5503460f94a1509d/html5/thumbnails/20.jpg)
Gaussian Lowpass FiltersDefinition:
22 2/),(),( vuDevuH
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Example
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More example
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Sharpening FiltersHigh-pass filtersIn general,Ideal highpass filterButterworth highpass filter:
Gaussian highpass filters
),(1),( vuHvuH lphp
nvuDDvuH
20 )],(/[1
1),(
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Relationship between Lowpass and Highpass Filters
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Spatial Domain Representation
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Ideal Highpass Example
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Butterworth Highpass Example
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Gaussian Highpass Example
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Laplacian in the Frequency Domain
It can be shown that:
Therefore,
)()()(
uFjudx
xfd nn
n
),()()],([ 222 vuFvuyxf
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Illustration
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Other FiltersUnsharp masking: High-boost filtering:High-frequency emphasis filtering:
),(),(),( yxfyxfyxf lphp
),(),(),( yxfyxAfyxf lphp
),(),( vubHavuH hphfe
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Homomorphic Filtering
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Example
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DFT: Implementation Issues
RotationPeriodicity and conjugate symmetrySeparabilityNeed for paddingCircular convolutionFFT
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Properties of 2D FT (1)
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Properties of 2D FT (2)
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FT Pairs