1. Communication & Multimedia C. -H. Hong 2015/6/18 Contourlet Student: Chao-Hsiung Hong Advisor: Prof. Hsueh-Ming Hang

2. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Contourlet Transform Simulation Results Conclusion Reference

3. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Goal The failure of wavelet The inefficiency of wavelet Contourlet Transform Simulation Results Conclusion Reference

4. Communication & Multimedia C. -H. Hong 2015/6/18 Goal Sparse representation for typical image with smooth contours Action is at the edges!!!

5. Communication & Multimedia C. -H. Hong 2015/6/18 The failure of wavelet 1-D: Wavelets are well adapted to singularities 2-D: Separable wavelets are only well adapted to point- singularity However, in line- and curve-singularities …

6. Communication & Multimedia C. -H. Hong 2015/6/18 The inefficiency of wavelet Wavelet: fails to recognize that boundary is smooth New: require challenging non-separable constructions

7. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Contourlet Transform Laplacian Pyramid Directional Filter Bank Pyramid Directional Filter Banks Simulation Results Conclusion Reference

8. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(1) Multiscale decomposition

9. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(2) Multiscale subspaces generated by the Laplacian pyramid

10. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(3) Avoid frequency scrambling by downsampling the lowpass channel only Sample rate:4/3 (wavelet:1)

11. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(4) H and G correspond to (↓M)H and G(↑M) c = Hx, p = Gc, and d = x-p = x-GHx = (I-GH) x

12. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(5) If H and G are biorthogonal with respect to the sampling lattice M, HG = I GHd = GH(x-GHx) = GHx-GHx = 0

13. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(6) p = GHx computes the projection of x onto V d = x-p and Hd = H(x-GHx) = 0, so d is a projection of x ontp W and perpendicular to, eliminate the error that is parallel to V

14. Communication & Multimedia C. -H. Hong 2015/6/18 Laplacian Pyramid(7)

15. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(1) Division of 2-D spectrum into fine slices using iterated tree structured filter banks

16. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(2) Diamond shape filter, or fan filter The black region represents ideal frequency supports of the filters Q: quincunx sampling lattice

17. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(3) X Y X’ Y’

18. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(4) X Y ↓Q 0 ↑Q0↑Q0 X’ Y’ X’ Y’

19. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(5) X Y ↓Q 1 ↑Q1↑Q1 X’ Y’ X’ Y’

20. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(6) ↓M H(ω) H(M T ω) ↓M ↓Q 0 A(ω)B(ω) Time domain : upsampled by Q 0 (multiplied by Q 0 ) Frequency domain :

21. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(7) 0 1 2 3

22. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(8) Quincunx filter banks with resampling operations that are used in the DFB starting from the third level

23. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(9) X Y ↓R 0 ↑R0↑R0 X’ Y’ X’ Y’

24. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(10) Time domain : upsampled by R 0 (multiplied by R 0 ) Frequency domain : ↓R 0 ↓Q 0 ↓P 0 A(ω) B(ω)

25. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(11) X Y ↓Q 0 ↓R 0 ↓Q 0 X’ Y’ X’ Y’

26. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(12) 7 6 5 4 3 2 1 0

27. Communication & Multimedia C. -H. Hong 2015/6/18 Directional Filter Bank(13) Impulse response of 32 equivalent filters for the first half channels of a 6-levels DFB use the Haar filters.

28. Communication & Multimedia C. -H. Hong 2015/6/18 Pyramid Directional Filter Banks The number of directional frequency partition is decreased from the higher frequency bands to the lower frequency bands

29. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Contourlet Transform Simulation Results Conclusion Reference

30. Communication & Multimedia C. -H. Hong 2015/6/18 Simulation Results(1) 0 1 1 2 2 3 34 4 5 5 6 6 7 78 8 16 9 9 10 11 12 13 14 15 16 10 012 34 5 6 78 15161314 1112 9

31. Communication & Multimedia C. -H. Hong 2015/6/18 Simulation Results(2) 1 1 2 2 3 34 4 5 5 6 67 7 8 8 9 9 10 11 12 0 4 0 1 2 3 5 6 78 91011

32. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Contourlet Transform Simulation Results Conclusion Reference

33. Communication & Multimedia C. -H. Hong 2015/6/18 Conclusion Offer sparse representation for piecewise smooth images Provide multi-scale and multi-direction decomposition Small redundancy

34. Communication & Multimedia C. -H. Hong 2015/6/18 Outline Introduction Contourlet Transform Simulation Results Conclusion Reference

35. Communication & Multimedia C. -H. Hong 2015/6/18 Reference M. N. Do, “ Directional Multiresolution Image Representations ”, Ph.D. Thesis, Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November 2001

36. Communication & Multimedia C. -H. Hong 2015/6/18 Thank you for your attention! Any questions?