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Beyond Wavelets and JPEG2000

Beyond Wavelets and JPEG2000. Tony Lin Peking University, Beijing, China Dec. 17, 2004. Outline. Wavelets and JPEG2000: A brief review Beyond wavelets and JPEG2000 My exploration Directional wavelet construction Adaptive wavelet selection Inter-subband transform Outlook. References.

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Beyond Wavelets and JPEG2000

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  1. Beyond Wavelets and JPEG2000 Tony Lin Peking University, Beijing, China Dec. 17, 2004

  2. Outline • Wavelets and JPEG2000: A brief review • Beyond wavelets and JPEG2000 • My exploration • Directional wavelet construction • Adaptive wavelet selection • Inter-subband transform • Outlook

  3. References • Classical books on wavelets and subband • I. Daubechies, "Ten lectures on wavelets," 1992. • P. P. Vaidyanathan, "Multirate systems and filter banks," 1992. • C. K. Chui, An Introduction to Wavelets, 1992. • Y. Meyer, “Wavelets: Algorithms and Applications,” 1993. • Vetterli and J. Kovacevic, "Wavelets and subband coding," 1995. • G. Strang and T. Nguyen, "Wavelet and filter banks," 1996. • C. K. Chui, Wavelets: A mathematical tool for signal analysis, 1997. • C. S. Burrus, R. A. Gopinath, and H. Guo, "Introduction to wavelets and wavelet transforms: A primer," 1998. • S. Mallat, "A wavelet tour of signal processing," second edition, 1998.

  4. References • Beyond • David Donoho, “Beyond Wavelets,” ten lectures, 2000. • Book: G. Welland ed., Beyond wavelets, 2003. • Martin Vetterli, "Wavelets, approximation and compression: Beyond JPEG2000," San Diego, Aug. 2003. • Martin Vetterli, "Fourier, wavelets and beyond: the search for good bases for images," Singapore, Oct. 2004. • M. N. Do, "Beyond wavelets: Directional multiresolution image representation," 2003.

  5. References • Beyond (cont.) • David Donoho, "Data compression and harmonic analysis," IEEE Trans. Info Theory, 1998. • Martin Vetterli, "Wavelets, approximation, and compression," IEEE Sig. Proc. Mag., Sept. 2001. • E. L. Pennec, S. Mallat, "Sparse geometric image representations with bandelets," July 2003.

  6. References • JPEG2000 • Book: D. Taubman & M. Marcellin, “JPEG2000: Image compression fundamentals, standards and pratice,” 2002. • D. Taubman, “High performance scalable image compression with EBCOT,” IEEE Trans. Image Proc., 2000. • Jin Li, “Image compression: mechanics of JPEG 2000,” 2001. • M. Adams, “The JPEG-2000 still image compression standard,” 2002.

  7. Main Contributors • Wavelets (Mathematics) • Daubechies, Mallat, Meyer, Donoho, Strang, Sweldens, … • Subband (EE) • Vaidyanathan, Vetterli, … • Image Compression (EE) • Shapiro (EZW), Said&Pearlman (SPIHT), Taubman (EBCOT), Jin Li (R-D optimization)

  8. Part I: Wavelets and JPEG2000: A brief review "Who controls the past, ran the Party slogan, controls the future; who controls the present, controls the past." -- George Orwell, 1984.

  9. Wavelets • Then dulcet music swelled • Concordant with the life-strings of the soul; • It throbbed in sweet and languid beatings there, • Catching new life from transitory death; • Like the vague sighings of a wind at even • That wakes the wavelets of the slumbering sea... • ---Percy Bysshe Shelley • Queen Mab: A Philosophical Poem, with Notes, published by the author, London, 1813. This is given by The Oxford English Dictionary as one of the earliest instances of the word "wavelet". For an instance in current poetry in this generic sense, see Breath, by Natascha Bruckner. • http://www.math.uiowa.edu/~jorgen/shelleyquotesource.html

  10. Wavelets = Wave + lets • Pure Mathematics • Algebra • Geometry • Analysis (mainly studying functions and operators) • Fourier, Harmonic, Wavelets

  11. Why Wavelets Work? • Wavelet functions are those functions such that their integer translate and two-scale dilations, i.e., f(2mx-n) for all integer m and n form a Riesz basis for the space of all square integrable functions ( L2(R) ). • Such functions provide a good basis for approximating signal and images. • -- From Ming-Jun Lai’s homepage • Notes: • Simple: Just do translation and dilations for f(x) • Complete: Riesz basis for L2(R)

  12. Basis: Tools to Divide and Conquer the Function Spaces • From rainbows to spectras • The following picture is from Vetterli’s ICIP04 talk

  13. Subband vs. Wavelets • Wavelets allow the use of powerful mathematical theory in function analysis, so that many function properties can be studied and used. • The values in DWT are fine-scale scaling function coefficients, rather than samples of some function. This specifies that the underlying continuous-valued functions are transformed. • Wavelets involve both spatial and frequency considerations. • G. Davis and A. Nosratinia, "Wavelet-Based Image Coding: An Overview", 1998.

  14. Regularity, or Vanishing Moments • From Vetterli’s SPIE’03 Talk

  15. Orthogonal vs. Biorthogonal-- B. Usevitch, "A turorial on modern lossy wavelet image compression: foundations of JPEG 2000," IEEE Trans. Sig. Proc. Mag., 2001. • Orthogonal: • Energy conservation: simplifies the designing wavelet-based image coder • Drawback: Coefficient expansion (e.g., 8 (input) + 4 (filter) = 12 (output) ). Worse for Multiple DWTs. • Biorthogonal CDF 9/7 filter: • Nearly orthogonal • Solve the “coefficient expansion” problem. • Symmetric extensions of the input data • Filters are symmetric or antisymmetric

  16. DWT Implementation: Convolution vs. Lifting • Daubechies and Sweldens, “Factoring wavelet transforms into lifting steps”, J. Fourier Anal. Appl., 1998.

  17. Forward and Inverse Lifting- From Jin Li’s Talk

  18. Operation flow of JPEG2000

  19. Secret 1 for the coding efficiency of JPEG2000: -- Multiple levels of DWT LL block • Only a small portion of coefficients are needed to coded. • Why 5-level decomposition? Because further decomposition can not improve the performance, since the LL block has been very small. • Divide and Conquer Five DWT decompositions of Barbara image

  20. Secret 2 for the coding efficiency of JPEG2000: -- EBCOT: Fractional bitplane coding and Multiple contexts to implement a high performance arithmetic coder • Divide and Conquer • Bitplane coding • Three passes for each bitplane: Significance, refinement, cleanup • Different contexts: Sig (LL+LH, HL, HH), Sign, Ref

  21. Part II: Beyond wavelets and JPEG2000 "My dream is to solve problems, with or without wavelets" -- Bruno Torresani, 1995

  22. Fourier vs. Wavelets

  23. The failure of Wavelets in 2-D

  24. Wavelets vs. New Scheme

  25. Curvelets: Breakthrough by Candes and Donoho, 1999

  26. Continuous Ridgelet Transform Translation Rotation Dilation

  27. Orthonormal Ridgelets

  28. Curvelets: Combining wavelets and ridgelets

  29. Curvelet Transform: An Example

  30. Second Generation of Curvelets: Without Ridgelets, 2002 Dilation Rotation Translation

  31. The Frequency-Domain Definition of Curvelets

  32. Beamlets

  33. Wedgelets

  34. Contourlets by M. Do and M. Vetterli

  35. Contourlet Transform

  36. Contourlet Transform (Cont.)

  37. Bandelets by E. Pennec & S. Mallat 2003 • Using separable wavelet basis, if no geometric flow • Using modified orthogonal wavelets in the flow direction, called bandelets • Quad-tree segmentation

  38. Example 1

  39. Example 2

  40. Compression Performance • Bandelets compared with CDF97 • Implemented with a scalar quantization and an adaptive arithmetic coder • No comparison with JPEG2000

  41. Curved Wavelet Transform-- D. Wang, ICIP’04

  42. Example

  43. Compression Performance

  44. Part III: My exploration: 1.Directional wavelet construction2. Adaptive wavelet selection3. Inter-subband transform "There have been too many pictures of Lena, and too many bad wavelet sessions at meetings." -- M. Vetterli, 1995. "If you steal from one author, it's plagiarism; if you steal from many, it's research" -- Wilson Mizner, 1953.

  45. Directional wavelet construction • Find a 2-D wavelet function such that their translations, dilations, and rotations form a basis for the space of all square integrable functions ( L2(R) ). • Build new multiresolution theory • Build fast algorithms to do multiscale transforms • How ? • If succeed, it would be similar to the curvelets by Candes.

  46. Adaptive Wavelet Selection • Different wavelets have different support lengths, vanishing moments, and smoothness • Longer and smoother wavelets for smooth image regions • Shorter and more rugged wavelets for edge regions • Adaptively select the best wavelet basis

  47. = matting? +

  48. Shortcomings • Difficult to find a measure to evaluate which wavelet basis is better • Big overhead • Segmentation information • The wavelet basis used in each segments • Solutions

  49. Further Transforms in Wavelet Domain • Curvelets, Contourlets, and Bandelets are new basis to approximate the ideal transform • Wavelets are far from the ideal basis, but they are on the midway • Further transforms in the wavelet domain can be benefited by the existing good properties offered by DWT

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