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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

Beyond Wavelets and JPEG2000

Tony Lin

Peking University, Beijing, China

Dec. 17, 2004


Outline
Outline

  • Wavelets and JPEG2000: A brief review

  • Beyond wavelets and JPEG2000

  • My exploration

    • Directional wavelet construction

    • Adaptive wavelet selection

    • Inter-subband transform

  • Outlook


References
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.


References1
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.


References2
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.


References3
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.


Main contributors
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)


Part i wavelets and jpeg2000 a brief review

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.


Wavelets
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


Wavelets wave lets
Wavelets = Wave + lets

  • Pure Mathematics

    • Algebra

    • Geometry

    • Analysis (mainly studying functions and operators)

      • Fourier, Harmonic, Wavelets


Why wavelets work
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)


Basis tools to divide and conquer the function spaces
Basis: Tools to Divide and Conquer the Function Spaces

  • From rainbows to spectras

  • The following picture is from Vetterli’s ICIP04 talk


Subband vs wavelets
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.


Regularity or vanishing moments
Regularity, or Vanishing Moments

  • From Vetterli’s SPIE’03 Talk


Beyond wavelets and jpeg2000

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


Dwt implementation convolution vs lifting
DWT Implementation: Convolution vs. Lifting

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


Forward and inverse lifting from jin li s talk
Forward and Inverse Lifting- From Jin Li’s Talk



Secret 1 for the coding efficiency of jpeg2000 multiple levels of dwt
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


Beyond wavelets and jpeg2000

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


Part ii beyond wavelets and jpeg2000

Part II: Beyond wavelets and JPEG2000

"My dream is to solve problems, with or without wavelets"

-- Bruno Torresani, 1995





Curvelets breakthrough by candes and donoho 1999
Curvelets: Breakthrough by Candes and Donoho, 1999


Continuous ridgelet transform
Continuous Ridgelet Transform

Translation

Rotation

Dilation



Curvelets combining wavelets and ridgelets
Curvelets: Combining wavelets and ridgelets



Second generation of curvelets without ridgelets 2002
Second Generation of Curvelets: Without Ridgelets, 2002

Dilation

Rotation

Translation








Bandelets by e pennec s mallat 2003
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




Compression performance
Compression Performance

  • Bandelets compared with CDF97

  • Implemented with a scalar quantization and an adaptive arithmetic coder

  • No comparison with JPEG2000


Curved wavelet transform d wang icip 04
Curved Wavelet Transform-- D. Wang, ICIP’04




Beyond wavelets and jpeg2000

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.


Directional wavelet construction
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.


Adaptive wavelet selection
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


Beyond wavelets and jpeg2000

= matting?

+


Shortcomings
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


Further transforms in wavelet domain
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


Inter subband transform
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 ?


Pca for the three subbands lh hl hh
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




Shortcomings1
Shortcomings

  • Spherical approximation

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


Sorting based on edge directions
Sorting based on edge directions

  • Edge-detection in three subbands

  • Rearrange the coefficients based on edge directions

  • We obtain compact energy !

DWT

Subband Sorting


Example2
Example

DWT

443 bytes (30:1), 35.70dB

Sorting

434 bytes (30:1), 35.49dB

Saving several cleanup passes


Part iv outlook

Part IV: Outlook

"Predicting is hard, especially about the future."

-- Victor Borge, quoted by Philip Kotler.


Wish lists for next generation basis
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


Beyond wavelets and jpeg2000
Over

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

  • Questions