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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