CHAPTER 8CHAPTER 8

Fourier FiltersFourier Filters

IMAGE ANALYSISIMAGE ANALYSIS

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Continuous and discrete Fourier transform in two dimensions Continuous and discrete Fourier transform in two dimensions

f(x,y) = F(u,v) ei 2

(ux+vy) dxdy

– –

+ +Continuous inverse Fourier transform

F(u,v) = f(x,y) e–i 2

(ux+vy) dxdy

– –

+ +Continuous Fourier transform

f(x,y) F(u,v)

Fuv = fnm eNM

–i 2 ( + ) un vmN M

n=1 m=1

N M1

Discrete Fourier transform

fij Fuv

Discrete inverse Fourier transform

fnm = Fuv ei 2 ( + ) un vm

N M u=1 v=1

N M

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{gij} = {hij} {fij} Guv = Huv Fuv

G(u) = F(u) H(u)

gij = hi–n,j–m fnm = hnm fi–n,j–m n = – m = –

+ +

n = – m = –

+ +

Discrete convolution theorem

Continuous convolution theorem

g(x) = h( – x) f() d f(x) h(x)–

+

f(x) F()

g(x) G()

h(x) H()

fij Fuv

gij Guv

hij Huv

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{gij} {Guv}

Guv = Huv Fuv

{ Fuv}

Discrete convolution theoremDiscrete convolution theorem

DFT

convolution

inverseDFT

multiplication

{fij}

gij = hi–n,j–m fnm n = – m = –

+ +

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Circular FiltersCircular Filters

Low PassLow Pass High PassHigh Pass

1

1

1

1

0

0

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Original Fourier transform

After circular low-pass filter, R = 100 After circular low-pass filter, R = 75 After circular low-pass filter, R = 50

After circular high-pass filter, R = 50

An example of Fourier filtering with circular filters An example of Fourier filtering with circular filters

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