computer vision – enhancement(part iii)
DESCRIPTION
Computer Vision – Enhancement(Part III). Hanyang University Jong-Il Park. The Fourier transform. Definition 1-D Fourier transform 2-D Fourier transform. Fourier series. 1- D case. M-point spectrum. 2 D Fourier series. 2-D case is periodic : period = 1 - PowerPoint PPT PresentationTRANSCRIPT
Computer Vision – Computer Vision – Enhancement(Part III)Enhancement(Part III)
Hanyang University
Jong-Il Park
Department of Computer Science and Engineering, Hanyang University
dxuxjxfuF )2exp()()(
duuxjuFxf )2exp()()(
dydxvyuxjyxfvuF
))(2exp(),(),(
dvduyvxujvuFyxf
))(2exp(),(),(
The Fourier transformThe Fourier transform
Definition 1-D Fourier transform
2-D Fourier transform
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1-D case
n
unujnxuX 5.05.0),2exp()()(
5.0
5.0)2exp()()( dunujuXnx
Fourier seriesFourier series
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M-point spectrumM-point spectrum
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22D Fourier seriesD Fourier series
2-D case
is periodic : period = 1
Sufficient condition for existence of
m n
vunvmujnmxvuX 5.0,5.0),)(2exp(),(),(
5.0
5.0
5.0
5.0 1 ))(2exp(),(),( dudvnvmujvuXnmx
),( vuX
,2,1,0,),,(),( lklvkuXvuX
|))(2exp(),(||),(|
m n
nvmujnmxvuX
m nm n
nmxnvmujnmx |),(||))(2exp(||),(|
),( vuX
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original 256x256 lena
Centered andnormalized spectrum(log-scale)
Eg. 2D Fourier transformEg. 2D Fourier transform
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Filtering in Frequency DomainFiltering in Frequency Domain
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Unitary TransformsUnitary Transforms
Unitary Transformation for 1-Dim. Sequence Series representation of
Basis vectors : Energy conservation :
}10),({ Nnnu
1
0
10),(),()(N
n
NknunkakvAuv
)matrixunitary ( where *1 TAA
1
0
** 10),(),()(N
n
NnkvnkanuvAu
TNnnka }10),,({ * *ka
22 |||||||| uvAuv
)|||||)(||)(||||| ( 21
0
2*1
0
22 uuuAuAuv **
N
n
TTTN
k
nukv
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Unitary Transformation for 2-D Sequence Definition :
Basis images : Separable Unitary Transforms:
1,0),,(),(),(1
0
1
0,
NlknmanmulkvN
m
N
nlk
1,0),,(),(),(1
0
1
0
*,
NnmnmalkvnmuN
k
N
llk
)},({ *, nma lk
22D Unitary TransformationD Unitary Transformation
)()(),(, nbmanma lklk
Tl
N
m
N
nk nanmumalkv AUAV
)(),()(),(1
0
1
0
**1
0
1
0
* )(),()(),( VAAUT
l
N
k
N
lk nalkvmanmu
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NNNN
NnmwwlkvN
nmu
NlkwwnmuN
lkv
T
T
T
TT
N
k
N
l
lnN
kmN
N
m
N
n
lnN
kmN
2
*
*
1
0
1
0
1
0
1
0
log2 DFT D-1 2separable.2
where
and
notation spacevector
since , .1
1,0,),(1
),(
1,0,),(1
),(
is DFTunitary D2
O
FFFF
vuuv
VFFU
FFFUFFFUV
F
FF
2-2-D DFTD DFT
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1,0,,
~,~
,~
,~
,~
1,,0
1,0,,,~
1,,0
1,0,,,
~
1,,,,
n theoremconvolutiocircular D2 .4
1,0,,,
real is if symmetry, e3.conjugat
1
*
MnmlkYDFTnmy
lkUlkHlkY
MnmN
Lnmnmunmu
MnmN
Nnmnmhnmh
LNMnmunmhnmy
NlklNkNvlkv
u(m,n)
NNLLMM
N+L-1
M
MN+L-1
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1,,0, ,),(11
),(1
0
1
0
NlkWnmfN
WN
lkFN
m
N
n
lnN
kmN
1,,0 ,),(11
),(1
0
1
0
Nm,nWlkFN
WN
nmfN
k
N
l
lnN
kmN
SeparabilitySeparability
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Transform OperationsTransform Operations
mask zonal ,;,,,
filteringlinear dgeneralize
image enhanced :
operationt enhancemen :,,then
image ed transform:,
imageinput :,
11
lkglkvlkglkv
lkvflkv
lkv
nmu
T
T
AVAU
AUAV
U
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Centered SpectrumCentered Spectrum
)2/,2/()1)(,( . NvMuFyxf nsFourierTrayx
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Generalized Linear FilteringGeneralized Linear Filtering
Generalized Linear Filtering
maskzonallkglkvlkglkv :),(),,(),(),(
Unitarytransform
TAUA
),( nmu ),( lkv Pointoperation
)(f
),( lkv Inversetransform
11 ][ TAVA
),( nmu
HPF
BPF
LPF
Zonal masks forOrthogonal(DCT, DHT etc) transforms
BPF
LPF
HPF
BPF
LPF
BPF
LPF
BPF
LPF
Zonal masks for DFT
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Eg. Filtering - DFTEg. Filtering - DFT
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Eg. Filtering - LPF and HPFEg. Filtering - LPF and HPF
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Eg. Filtering - HPF + DC Eg. Filtering - HPF + DC
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Correspondence between Spatial Domain Correspondence between Spatial Domain and Frequency Domainand Frequency Domain
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Ideal LPFIdeal LPF
NOT practical because of “ringing”
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RingingRinging
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Illustration of RingingIllustration of Ringing
convolution
Ideal LPF
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Butterworth LPFButterworth LPF
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Ringing in BLPFRinging in BLPF
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Eg. 2Eg. 2ndnd order Butterworth LPF order Butterworth LPF
A good compromise between Effective LPFand Acceptable ringing
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Gaussian LPF(GLPF)Gaussian LPF(GLPF)
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Eg. GLPFEg. GLPF
No ringing!
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Application of GLPF(1)Application of GLPF(1)
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Application of GLPF(2)Application of GLPF(2)
Soft and pleasing
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Homomorphic FilteringHomomorphic Filtering
Homomorphic Filtering f(x, y) = i(x, y) • r(x, y)
i(x,y) : - illumination component
- responsible for the dynamic range
- low freq. Components
r(x,y) : - reflectance component
- responsible for local contrast
- high frequency component
enhancement based on the image model
- reduce the illumination components
- enhance the reflectance components
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Transform OperationsTransform Operations
Homomorphic System
note
log LinearSystem exp
log exp
HP
LP
g(x, y)f(x, y)
<1
>1
yxrFyxiFyxfF
yxryxiyxf
,,,
,,,
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Eg. Homomorphic filtering(1)Eg. Homomorphic filtering(1)
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Eg. Homomorphic filtering(2)Eg. Homomorphic filtering(2)