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IMAGE ANALYSIS. CHAPTER 8. Fourier Filters. A. Dermanis. Continuous Fourier transform. f ( x , y ) F ( u , v ). + + . . Continuous inverse Fourier transform. F ( u , v ) = f ( x , y ) e – i 2 ( ux + vy ) dxdy. un vm. – i 2 ( + ). 1. – – .

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## CHAPTER 8

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**IMAGE ANALYSIS**CHAPTER 8 Fourier Filters A. Dermanis**Continuous Fouriertransform**f(x,y) F(u,v) + + Continuous inverse Fouriertransform F(u,v) = f(x,y) e–i2(ux+vy)dxdy un vm –i 2 (+ ) 1 – – N M Fuv = fnm e NM NM NM + + Discrete Fouriertransform f(x,y) = F(u,v) ei2(ux+vy)dxdy fij Fuv n=1 m=1 u=1 v=1 – – Discrete inverse Fouriertransform un vm i 2 (+ ) fnm = Fuv e N M Continuous and discrete Fouriertransform in two dimensions A. Dermanis**g(x) = h(–x) f() d f(x)h(x)**+ – gij = hi–n,j–mfnm = hnmfi–n,j–m ++ ++ n = –m = – n = –m = – f(x) F() g(x) G() h(x) H() Continuous convolution theorem G(u) = F(u)H(u) fij Fuv gij Guv hij Huv Discrete convolution theorem {gij} = {hij}{fij} Guv = HuvFuv A. Dermanis**gij = hi–n,j–mfnm**++ n = –m = – Discrete convolution theorem {fij} {Fuv} DFT convolution multiplication Guv = Huv Fuv inverse DFT {gij} {Guv} A. Dermanis**Circular Filters**High Pass Low Pass 1 0 1 0 1 1 A. Dermanis**An example of Fourier filtering with circular filters**Original Fourier transform After circular high-pass filter, R = 50 After circular low-pass filter, R = 100 After circular low-pass filter, R = 75 After circular low-pass filter, R = 50 A. Dermanis

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