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Computer Vision – Enhancement(Part III)

Computer Vision – Enhancement(Part III). Hanyang University Jong-Il Park. The Fourier transform. Definition 1-D Fourier transform 2-D Fourier transform. Fourier series. 1- D case. M-point spectrum. 2 D Fourier series. 2-D case is periodic : period = 1

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Computer Vision – Enhancement(Part III)

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  1. Computer Vision – Enhancement(Part III) Hanyang University Jong-Il Park

  2. The Fourier transform • Definition • 1-D Fourier transform • 2-D Fourier transform

  3. Fourier series • 1-D case

  4. M-point spectrum

  5. 2D Fourier series • 2-D case • is periodic : period = 1 • Sufficient condition for existence of

  6. Eg. 2D Fourier transform original 256x256 lena Centered and normalized spectrum (log-scale)

  7. Filtering in Frequency Domain

  8. Unitary Transforms • Unitary Transformation for 1-Dim. Sequence • Series representation of • Basis vectors : • Energy conservation :

  9. 2D Unitary Transformation • Unitary Transformation for 2-D Sequence • Definition : • Basis images : • Separable Unitary Transforms:

  10. 2-D DFT

  11. Separability

  12. Transform Operations

  13. Centered Spectrum

  14. Unitary transform Point operation Inverse transform LPF LPF BPF BPF LPF HPF BPF HPF BPF BPF LPF LPF Generalized Linear Filtering • Generalized Linear Filtering Zonal masks for Orthogonal(DCT, DHT etc) transforms Zonal masks for DFT

  15. Eg. Filtering - DFT

  16. Eg. Filtering - LPF and HPF

  17. Eg. Filtering - HPF + DC

  18. Correspondence between Spatial Domain and Frequency Domain

  19. Ideal LPF  NOT practical because of “ringing”

  20. Ringing

  21. convolution Illustration of Ringing Ideal LPF

  22. Butterworth LPF

  23. Ringing in BLPF

  24. Eg. 2nd order Butterworth LPF A good compromise between Effective LPF and Acceptable ringing

  25. Gaussian LPF(GLPF)

  26. Eg. GLPF No ringing!

  27. Application of GLPF(1)

  28. Application of GLPF(2) Soft and pleasing

  29. Homomorphic 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

  30. Linear System log exp LP exp log <1 f(x, y) HP g(x, y) >1 Transform Operations • Homomorphic System note

  31. Eg. Homomorphic filtering(1)

  32. Eg. Homomorphic filtering(2)

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