1. Wavelet Transform 國立交通大學電子工程學系 陳奕安 2007.8.15

2. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

3. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

4. Fourier Transform Frequency domain: Fourier Transform (Joseph Fourier 1807 ) Cannot provide simultaneously time and frequency information.

5. Short Time Fourier Transform (STFT) Time-Frequency analysis: STFT (Dennis Gabor 1946) Windowed Fourier transform A function of time and frequency

6. Short Time Fourier Transform (STFT) Frequency and time resolutions are fixed: Narrow (Wide) window for poor freq. (time) resolution Via Narrow WindowVia Wide Window The two figures were from Robi Poliker, 1994

7. Continuous Wavelet Transform Width of the window is changed as the transform is computed for every spectral components. Altered resolutions are placed. Translation (The location of the window) Scale Mother Wavelet

8. Comparison of Transformations From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf

9. Wavelet Series Expansion Linear decomposition of a function: Basis orthogonal: Then the coefficients can be calculated by

10. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

11. Multiresolution Analysis Idea: If a set of signals can be represented by a weighted sum of φ(t-k), a larger set (including the original), can be represented by a weighted sum of φ (2t-k). Increase the size of the subspace changing the time scale of the scaling functions:

12. Multiresolution Analysis The spanned spaces are nested: Wavelets span the differences between spaces w i. Wavelets and scaling functions should be orthogonal: simple calculation of coefficients.

13. Multiresolution Analysis

14. Multiresolution Formulation. ( Scaling coefficients) ( Wavelet coefficients)

15. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

16. Discrete Wavelet Transform (DWT) Discrete Wavelet Transform Calculation: Using Multiresolution Analysis:

17. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

18. Fast Wavelet Transform Basic idea of Fast Wavelet Transform (Mallat’s herringbone algorithm): Pyramid algorithm provides an efficient calculation. DWT (direct and inverse) can be thought of as a filtering process. After filtering, half of the samples can be eliminated: subsample the signal by two. Subsampling: Scale is doubled. Filtering: Resolution is halved.

19. Fast Wavelet Transform (a)A two-stage or two-scale FWT analysis bank and (b)its frequency splitting characteristics.

20. Fast Wavelet Transform Inverse Fast Wavelet Transform

21. Fast Wavelet Transform A two-stage or two-scale FWT-1 synthesis bank.

22. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

23. Lifting Scheme The lifting scheme is an alternative method of computing the wavelet coefficients. Advantages of the lifting scheme: Requires less computation and less memory. Linear, nonlinear, and adaptive wavelet transform is feasible, and the resulting transform is invertible and reversible.

24. Lifting Scheme A spatial domain construction of bi-orthogonal wavelets, consists of the 4 operations: Split : s k (0) =x 2i (0), d k (0) =x 2i+1 (0) Predict : d k (r) = d k (r-1) –  p j (r) s k+j (r-1) Update : s k (r)= s k (r-1) +  u j (r) d k+j (r) Scaling : s k (R) =K 0 s k (R), d k (R) =K 1 d k (R)

25. Lifting Scheme A spatial domain construction of bi-orthogonal wavelets, consists of the 4 operations:

26. Lifting Scheme A spatial domain construction of bi-orthogonal wavelets, consists of the 4 operations:

27. Lifting Scheme Example: Conventional 5/3 filter C 0 = (4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[- 2]) )/8 C 1 = x[0]- (x[1]+x[-1])/2 Number of operations per pixel = 9+3 = 12

28. Lifting Scheme Example: (2,2) lifting scheme Prediction rule : interpolation : [1,1]/2 Update rule: preservation of average (moments) of the signal : [1,1]/4

29. Lifting Scheme Conventional 5/3 filter C 0 =(4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[-2]))/8 C 1 = x[0]- (x[1]+x[-1])/2 Number of operations per pixel = 9+3 = 12 The (2,2) lifting D[0] = x[0]- (x[1]+x[-1])/2 S[0] = x[0] + (D[0]+D[1])/4 Number of operations per pixel = 6

30. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

31. Beyond Wavelet Ridgelet Transform Curvelet Transform

32. Continuous Ridgelet Transform Ridgelet Transform (Candes, 1998): Ridgelet function: The function is constant along lines.Transverse to these ridges, it is a wavelet.

33. Continuous Ridgelet Transform The ridgelet coefficients of an object f are given by analysis of the Radon transform via:

34. The Curvelet Transform Decomposition of the original image into subbands. Spatial partitioning of each subband. Appling the ridgelet transform.

35. Beyond Wavelet A standard multiscale decomposition into octave bands, where the lowpass channel is subsampled while the highpass is not.

36. Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

37. [1] P. P. Vaidyanathan, "Multirate systems and filter banks,“pp.457-538 1992. [2] Howard L. Resnikoff, Raymond O. Wells, "Wavelet Analysis: The Scalable Structure of Information", Springer, 1998 [3] Martin Vetterli, "Wavelets, approximation and compression," IEEE Sig. Proc. Mag., Sept. 2001. [4] Sweldens W. "The lifting scheme: A custom-design construction of biorthogonal wavelets." Applied and Computational Harmonic Analysis, 1996,3(2):186~200. [5] E. L. Pennec, S. Mallat, "Sparse geometric image representations with bandelets," July 2003.

38. Reference [6] Candes, E. Ridgelets: theory and applications, Ph. D. thesis, Department of Statistics, Stanford University, 1998. [7] J.L. Starck, E.J. Candès and D.L. Donoho, The curvelet transform for image denoising, IEEE Transactions on Image Processing 11 (2002) (6), pp. 670–684.