1. CHAPTER 8 Fourier Filters IMAGE ANALYSIS A. Dermanis

2. Continuous and discrete Fourier transform in two dimensions f(x,y) = F(u,v) e i 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) F uv = f nm e NM –i 2   ( + ) un vm N M    n=1 m=1 N M 1 Discrete Fourier transform f ij  F uv Discrete inverse Fourier transform f nm = F uv e i 2   ( + ) un vm N M    u=1 v=1 N M A. Dermanis

3. {g ij } = {h ij }  {f ij } G uv = H uv F uv G(u) = F(u) H(u) g ij = h i–n,j–m f nm = h nm f i–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(  ) f ij  F uv g ij  G uv h ij  H uv A. Dermanis

4. {g ij }{G uv } G uv = H uv F uv { F uv } Discrete convolution theorem DFT convolution inverse DFT multiplication {f ij } g ij = h i–n,j–m f nm  n = –  m = –  +  A. Dermanis

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

6. OriginalFourier transform After circular low-pass filter, R = 100After circular low-pass filter, R = 75After circular low-pass filter, R = 50 After circular high-pass filter, R = 50 An example of Fourier filtering with circular filters A. Dermanis