1. IMAGE ANALYSIS CHAPTER 8 Fourier Filters A. Dermanis

2. 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

3. 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

4. gij = hi–n,j–mfnm ++   n = –m = – Discrete convolution theorem {fij} {Fuv} DFT convolution multiplication Guv = Huv Fuv inverse DFT {gij} {Guv} A. Dermanis

5. Circular Filters High Pass Low Pass 1 0 1 0 1 1 A. Dermanis

6. 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