Computer Vision – Enhancement(Part III)

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


1-D case

n

unujnxuX 5.05.0),2exp()()(

5.0

5.0)2exp()()( dunujuXnx

Fourier seriesFourier series


M-point spectrumM-point spectrum


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


original 256x256 lena

Centered andnormalized spectrum(log-scale)

Eg. 2D Fourier transformEg. 2D Fourier transform


Filtering in Frequency DomainFiltering in Frequency Domain


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


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


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


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


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


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


Centered SpectrumCentered Spectrum

)2/,2/()1)(,( . NvMuFyxf nsFourierTrayx


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


Eg. Filtering - DFTEg. Filtering - DFT


Eg. Filtering - LPF and HPFEg. Filtering - LPF and HPF


Eg. Filtering - HPF + DC Eg. Filtering - HPF + DC


Correspondence between Spatial Domain Correspondence between Spatial Domain and Frequency Domainand Frequency Domain


Ideal LPFIdeal LPF

NOT practical because of “ringing”


RingingRinging


Illustration of RingingIllustration of Ringing

convolution

Ideal LPF


Butterworth LPFButterworth LPF


Ringing in BLPFRinging in BLPF


Eg. 2Eg. 2ndnd order Butterworth LPF order Butterworth LPF

A good compromise between Effective LPFand Acceptable ringing


Gaussian LPF(GLPF)Gaussian LPF(GLPF)


Eg. GLPFEg. GLPF

No ringing!


Application of GLPF(1)Application of GLPF(1)


Application of GLPF(2)Application of GLPF(2)

Soft and pleasing


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


Transform OperationsTransform Operations

Homomorphic System

note

log LinearSystem exp

log exp

HP

LP

g(x, y)f(x, y)

<1

>1

yxrFyxiFyxfF

yxryxiyxf

,,,

,,,


Eg. Homomorphic filtering(1)Eg. Homomorphic filtering(1)


Eg. Homomorphic filtering(2)Eg. Homomorphic filtering(2)

Computer Vision – Enhancement(Part III)

Documents

Transcript of Computer Vision – Enhancement(Part III)