chapter 4 image enhancement in the frequency domain
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Chapter 4
Image Enhancement in the Frequency Domain
Fourier Transform1-D Fourier Transform1-D Discrete Fourier Transform (DFT)MagnitudePhasePower spectrum
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
Centered Fourier Spectrum
It can be shown that:
)2/,2/()1)(,( NvMuFyxf yx
Example
SEM Image
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)
Fourier Domain Filtering
Some Basic FiltersNotch filter:
otherwise 1
N/2)(M/2,v)(u, if 0),( vuH
Lowpass and Highpass Filters
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
Gaussian Filters
Difference of Gaussians (DoG)
222
22
2
2/
2)(
)(x
u
Aexh
AeuH
22
221
2 2/2/)( uu BeAeuH
Illustration
Smoothing FiltersIdeal lowpass filtersButterworth lowpass filtersGaussian lowpass filters
Ideal Lowpass Filters
Example
Ringing Effect
Butterworth Lowpass FiltersDefinition:
nDvuDvuH 2
0/),(1
1),(
Example
Ringing Effect
Gaussian Lowpass FiltersDefinition:
22 2/),(),( vuDevuH
Example
More example
Sharpening FiltersHigh-pass filtersIn general,Ideal highpass filterButterworth highpass filter:
Gaussian highpass filters
),(1),( vuHvuH lphp
nvuDDvuH
20 )],(/[1
1),(
Relationship between Lowpass and Highpass Filters
Spatial Domain Representation
Ideal Highpass Example
Butterworth Highpass Example
Gaussian Highpass Example
Laplacian in the Frequency Domain
It can be shown that:
Therefore,
)()()(
uFjudx
xfd nn
n
),()()],([ 222 vuFvuyxf
Illustration
Other FiltersUnsharp masking: High-boost filtering:High-frequency emphasis filtering:
),(),(),( yxfyxfyxf lphp
),(),(),( yxfyxAfyxf lphp
),(),( vubHavuH hphfe
Homomorphic Filtering
Example
DFT: Implementation Issues
RotationPeriodicity and conjugate symmetrySeparabilityNeed for paddingCircular convolutionFFT
Properties of 2D FT (1)
Properties of 2D FT (2)
FT Pairs
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