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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
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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), Said&Pearlman (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/shelleyquotesource.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 Lai’s 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 Vetterli’s 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 Vetterli’s SPIE’03 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 Li’s Talk
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Operation flow of JPEG2000
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Secret 1 for the coding efficiency of JPEG2000: -- Multiple levels of DWT 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
LL block
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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 (LL+LH, 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
TranslationRotationDilation
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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, ICIP’04
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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 VC+J2000
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 (30:1), 35.70dB
Sorting
434 bytes (30:1), 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