Beyond Wavelets and JPEG2000

1
Beyond Wavelets and JPEG2000

• Tony Lin
• Peking University, Beijing, China
• Dec. 17, 2004

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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.

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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.

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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.

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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.

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Main Contributors

• Wavelets (Mathematics)
• Daubechies, Mallat, Meyer, Donoho, Strang,
  Sweldens,
• Subband (EE)
• Vaidyanathan, Vetterli,
• Image Compression (EE)
• Shapiro (EZW), SaidPearlman (SPIHT), Taubman
  (EBCOT), Jin Li (R-D optimization)

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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.
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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/shelleyquotesour
  ce.html

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Wavelets Wave lets

• Pure Mathematics
• Algebra
• Geometry
• Analysis (mainly studying functions and
  operators)
• Fourier, Harmonic, Wavelets

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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 Lais homepage
• Notes
• Simple Just do translation and dilations for
  f(x)
• Complete Riesz basis for L2(R)

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Basis Tools to Divide and Conquer the Function
Spaces

• From rainbows to spectras
• The following picture is from Vetterlis ICIP04
  talk

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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.

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Regularity, or Vanishing Moments

• From Vetterlis SPIE03 Talk

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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

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DWT Implementation Convolution vs. Lifting

• Daubechies and Sweldens, Factoring wavelet
  transforms into lifting steps, J. Fourier Anal.
  Appl., 1998.

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Forward and Inverse Lifting - From Jin Lis Talk
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(No Transcript)
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Operation flow of JPEG2000
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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
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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 (LLLH, HL, HH), Sign, Ref

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Part II Beyond wavelets and JPEG2000

• "My dream is to solve problems, with or without
  wavelets"
• -- Bruno Torresani, 1995

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Fourier vs. Wavelets
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The failure of Wavelets in 2-D
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Wavelets vs. New Scheme
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Curvelets Breakthrough by Candes and Donoho,
1999
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Continuous Ridgelet Transform
Translation
Rotation
Dilation
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Orthonormal Ridgelets
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Curvelets Combining wavelets and ridgelets
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Curvelet Transform An Example
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Second Generation of Curvelets Without
Ridgelets, 2002
Translation
Rotation
Dilation
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The Frequency-Domain Definition of Curvelets
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Beamlets
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Wedgelets
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Contourlets by M. Do and M. Vetterli
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Contourlet Transform
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Contourlet Transform (Cont.)
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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

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Example 1
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Example 2
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Compression Performance

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

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Curved Wavelet Transform-- D. Wang, ICIP04
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Example
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Compression Performance
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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.

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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.

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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

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matting ?

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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

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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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Inter-subband transform

• EBCOT or JPEG2000 uses neighbor coefficients to
  predict the current values
• EZW or SPIHT uses cross-scale correlations to do
  prediction
• Wavelet packets do further decomposition in each
  subband to reduce correlation
• How about the inter-subband transform that push
  the energy into the first or the second subbands ?

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PCA for the three subbands (LH, HL, HH)

• Programming with Matlab and VCJ2000 codec
• Found that the PCA transform matrix is very close
  to Identity matrix
• Sometimes it provide slightly better performance
  than JPEG2000, but it is not always

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Spherical Coordinate Transform
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Example
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Shortcomings

• Spherical approximation
• Hard to design the rate-distortion allocation for
  the two angular subbands, because they depend on
  the R subband

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Sorting based on edge directions

• Edge-detection in three subbands
• Rearrange the coefficients based on edge
  directions
• We obtain compact energy !

DWT
Subband Sorting
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Example
DWT 443 bytes (301), 35.70dB
Sorting 434 bytes (301), 35.49dB Saving several
cleanup passes
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Part IV Outlook

• "Predicting is hard, especially about the
  future."
• -- Victor Borge, quoted by Philip Kotler.

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Wish lists for next-generation basis

• Multiresolution or Multiscale
• Localization in both space and frequency
• Critical sampling no coefficient expansion
• Easily control the filter length, smoothness,
  vanishing moments, and symmetry
• Directionality
• Anisotropy spheres, ellipses, needles
• Adaptive basis

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Over

• There is a long way to go
  beyond wavelets and JPEG2000
• Questions