1. Communication & Multimedia C. -H. Hong 2015/6/12 Contourlet Student: Chao-Hsiung Hong Advisor: Prof. Hsueh-Ming Hang

2. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Contourlet Transform Simulation Results Conclusion Reference

3. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Goal The failure of wavelet The inefficiency of wavelet Curvelet Transform Contourlet Transform Simulation Results Conclusion Reference

4. Communication & Multimedia C. -H. Hong 2015/6/12 Goal Sparse representation for typical image with smooth contours Action is at the edges!!!

5. Communication & Multimedia C. -H. Hong 2015/6/12 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 2015/6/12 The inefficiency of wavelet Wavelet: fails to recognize that boundary is smooth New: require challenging non-separable constructions

7. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Key idea Ridgelet Decomposition Non-linear approximation Problem Contourlet Transform Simulation Results Conclusion Reference

8. Communication & Multimedia C. -H. Hong 2015/6/12 Key Idea Optimal representation for function in R 2 with curved singularities Anisotropy scaling relation for curves: width ≈ length 2

9. Communication & Multimedia C. -H. Hong 2015/6/12 Ridgelet(1)

10. Communication & Multimedia C. -H. Hong 2015/6/12 Ridgelet(2) Ridgelet functions ψ a, b,θ (x 1, x 2 ) = a -1/2 ψ((x 1 cos(θ)+ x 2 sin(θ) – b)/a) x 1 cos(θ)+ x 2 sin(θ) = constant, oriented at angel θ Essentially localized in the corona | ω | in [2 a, 2 a+1 ] and around the angel θin the frequency domain Wavelet functions ψ a, b, (x) = a -1/2 ψ((x – b)/a) ψ a1, b1,a2,b2 (x) = ψ a1, b1, (x 1 )ψ a2, b2, (x 2 )

11. Communication & Multimedia C. -H. Hong 2015/6/12 Decomposition Segments of smooth curves would look straight in smooth windows → can be captured efficiently by a local ridgelet transform Window’s size and subband frequency are coordinated → width ≈ length 2

12. Communication & Multimedia C. -H. Hong 2015/6/12 Non-Linear Approximation Along a smooth boundary, at the scale 2 -j Wavelet: coefficient number ≈ O(2 j ) Curvelet: coefficient number ≈ O(2 j/2 ) Keep nonzero coefficient up to level J Wavelet: error ≈ O(2 -J ) Curvelet: error ≈ O(2 -2J )

13. Communication & Multimedia C. -H. Hong 2015/6/12 Problem(1) Translates it into discrete world Block-based transform: have blocking effects and overlapping windows to increase redundancy Polar coordinate Group the nearby coefficients since their locations are locally correlated due to the smoothness of the discontinuity curve Gather the nearby basis functions at the same scale into linear structure

14. Communication & Multimedia C. -H. Hong 2015/6/12 Problem(2) Multiscale and directional decomposition Multiscale decomposition: capture point discontinuities Directional decomposition: link point discontinuities into linear structures

15. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Contourlet Transform Multiscale decomposition Directional decomposition Pyramid Directional Filter Banks Basis Functions Simulation Results Conclusion Reference

16. Communication & Multimedia C. -H. Hong 2015/6/12 Multiscale Decomposition(1) Laplacian pyramid (avoid frequency scrambling)

17. Communication & Multimedia C. -H. Hong 2015/6/12 Multiscale Decomposition(2) Multiscale subspaces generated by the Laplacian pyramid

18. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(1) Directional Filter Bank Division of 2-D spectrum into fine slices Use quincunx FB ’ s, modulation, and shearing Test: zone plate image decomposed by d DFB with 4 levels that leads to 16 subbands

19. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(2): Sampling in Multiple Dimensions Quincunx sampling lattice Downsample by 2 Rotate 45 degree

20. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(3): Quincunx Filter Bank Diamond shape filter, or fan filter The black region represents ideal frequency supports of the filters Q: quincunx sampling lattice

21. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(4): Directional Filter Bank At each level QFB ’ s with fan filters are used The first two levels of DFB

22. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(5): 2 Level Directional Filter Bank

23. Communication & Multimedia C. -H. Hong 2015/6/12 Directional Decomposition(8): 3 Level Directional Filter Bank

24. Communication & Multimedia C. -H. Hong 2015/6/12 Pyramid Directional Filter Banks The number of directional frequency partition is decreased from the higher frequency bands to the lower frequency bands

25. Communication & Multimedia C. -H. Hong 2015/6/12 Basis Functions

26. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Contourlet Transform Simulation Results Conclusion Reference

27. Communication & Multimedia C. -H. Hong 2015/6/12 Simulation Results

28. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Contourlet Transform Simulation Results Conclusion Reference

29. Communication & Multimedia C. -H. Hong 2015/6/12 Conclusion Offer sparse representation for piecewise smooth images Small redundancy Energy compactness

30. Communication & Multimedia C. -H. Hong 2015/6/12 Outline Introduction Curvelet Transform Contourlet Transform Simulation Results Conclusion Reference

31. Communication & Multimedia C. -H. Hong 2015/6/12 Reference M. N. Do and Martin Vetterli, “ The Finite Ridgelet Transform for Image Representation ”, IEEE Transactions on Image Processing, vol. 12, no. 1, Jan. 2003. M. N. Do, “ Directional Multiresolution Image Representations ”, Ph.D. Thesis, Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November 2001

32. Communication & Multimedia C. -H. Hong 2015/6/12 Thank you for your attention! Any questions?