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

2. Department of Computer Science and Engineering, Hanyang University The Fourier transform Definition  1-D Fourier transform  2-D Fourier transform

3. Department of Computer Science and Engineering, Hanyang University 1-D case Fourier series

4. Department of Computer Science and Engineering, Hanyang University M-point spectrum

5. Department of Computer Science and Engineering, Hanyang University 2D Fourier series 2-D case  is periodic : period = 1  Sufficient condition for existence of

6. Department of Computer Science and Engineering, Hanyang University original 256x256 lena Centered and normalized spectrum (log-scale) Eg. 2D Fourier transform

7. Department of Computer Science and Engineering, Hanyang University Filtering in Frequency Domain

8. Department of Computer Science and Engineering, Hanyang University Unitary Transforms Unitary Transformation for 1-Dim. Sequence  Series representation of  Basis vectors :  Energy conservation :

9. Department of Computer Science and Engineering, Hanyang University Unitary Transformation for 2-D Sequence  Definition :  Basis images :  Separable Unitary Transforms: 2D Unitary Transformation

10. Department of Computer Science and Engineering, Hanyang University 2-D DFT

11. Department of Computer Science and Engineering, Hanyang University

12. Department of Computer Science and Engineering, Hanyang University Separability

13. Department of Computer Science and Engineering, Hanyang University Transform Operations

14. Department of Computer Science and Engineering, Hanyang University Centered Spectrum

15. Department of Computer Science and Engineering, Hanyang University Generalized Linear Filtering Unitary transform Point operation Inverse transform HPF BPF LPF Zonal masks for Orthogonal(DCT, DHT etc) transforms BPF LPF HPF BPF LPF BPF LPF BPF LPF Zonal masks for DFT

16. Department of Computer Science and Engineering, Hanyang University Eg. Filtering - DFT

17. Department of Computer Science and Engineering, Hanyang University Eg. Filtering - LPF and HPF

18. Department of Computer Science and Engineering, Hanyang University Eg. Filtering - HPF + DC

19. Department of Computer Science and Engineering, Hanyang University Correspondence between Spatial Domain and Frequency Domain

20. Department of Computer Science and Engineering, Hanyang University Ideal LPF  NOT practical because of “ringing”

21. Department of Computer Science and Engineering, Hanyang University Ringing

22. Department of Computer Science and Engineering, Hanyang University Illustration of Ringing convolution Ideal LPF

23. Department of Computer Science and Engineering, Hanyang University Butterworth LPF

24. Department of Computer Science and Engineering, Hanyang University Ringing in BLPF

25. Department of Computer Science and Engineering, Hanyang University Eg. 2 nd order Butterworth LPF A good compromise between Effective LPF and Acceptable ringing

26. Department of Computer Science and Engineering, Hanyang University Gaussian LPF(GLPF)

27. Department of Computer Science and Engineering, Hanyang University Eg. GLPF No ringing!

28. Department of Computer Science and Engineering, Hanyang University Application of GLPF(1)

29. Department of Computer Science and Engineering, Hanyang University Application of GLPF(2) Soft and pleasing

30. Department of Computer Science and Engineering, Hanyang University 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. Department of Computer Science and Engineering, Hanyang University Transform Operations Homomorphic System note log Linear System exp log exp HP LP g(x, y) f(x, y)  <1  >1

32. Department of Computer Science and Engineering, Hanyang University Eg. Homomorphic filtering(1)

33. Department of Computer Science and Engineering, Hanyang University Eg. Homomorphic filtering(2)