Computer Vision – Enhancement(Part III)

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

12. Separability

13. Transform Operations

14. Centered Spectrum

15. 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

16. Eg. Filtering - DFT

17. Eg. Filtering - LPF and HPF

18. Eg. Filtering - HPF + DC

19. Correspondence between Spatial Domain and Frequency Domain

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

21. Ringing

22. convolution Illustration of Ringing Ideal LPF

23. Butterworth LPF

24. Ringing in BLPF

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

26. Gaussian LPF(GLPF)

27. Eg. GLPF No ringing!

28. Application of GLPF(1)

29. Application of GLPF(2) Soft and pleasing

30. 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

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

32. Eg. Homomorphic filtering(1)

33. Eg. Homomorphic filtering(2)