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   -> Volume 7, Issue 10

Thesis: Shift-Invariant Adaptive Wavelet Decompositions
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Israel Cohen (

PostPosted: Tue Sep 29, 1998 8:36 pm    
Subject: Thesis: Shift-Invariant Adaptive Wavelet Decompositions
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#9 Thesis: Shift-Invariant Adaptive Wavelet Decompositions

Title: Shift-Invariant Adaptive Wavelet Decompositions and Applications

D.Sc. Dissertation, Technion - Israel Institute of
Technology, Haifa, Israel, May~1998.

Author: Israel Cohen

Supervisors: Prof. Shalom Raz and Prof. David Malah


Adaptive representations in libraries of bases, including the
wavelet-packet and local trigonometric decompositions, are widely used
in various applications. A major drawback restricting their use,
particularly in statistical signal processing applications, such as
detection, identification or noise removal (denoising), is the lack of
shift-invariance. The expansion, as well as the information cost
measuring its suitability for a particular application, may be
significantly influenced by the alignment of the input signal with
respect to the basis functions. Furthermore, the time-frequency
tilings, produced by the best-basis expansions, do not generally
conform to standard time-frequency energy distributions.

The objective of this work is to develop a general approach for
achieving shift-invariance, enhanced time-frequency decompositions and
robust signal estimators using libraries of orthonormal bases. The
first problem we address is that of shift-invariant adaptive
decompositions in libraries of wavelet packet and local trigonometric
bases. We introduce shift-invariant decompositions that are
characterized by lower information cost, improved time-frequency
resolution, and for a prescribed data set yield more stable cost
functions. The computational complexities are investigated, and
efficient procedures for their control at the expense of the attained
information cost are presented.

A second issue, closely related to the problem of shift-invariance, is
that of adaptive decompositions of time-frequency distributions and
removal of interference terms associated with bilinear
distributions. We show that utilizing the shift-invariant
decompositions, various useful properties relevant to time-frequency
analysis, including high energy concentration and suppressed
interference terms, can be achieved simultaneously in the Wigner
domain. Instead of smoothing, which broadens the energy distribution
of signal components, we propose cross-term manipulations that are
adapted to the local distribution of the signal. The properties of the
resultant modified Wigner distribution are investigated, and its
distinctive applicability to resolving multicomponent signals is

The final topic concerns the problem of translation-invariant
denoising, using the Minimum Description Length (MDL) criterion. We
define a collection of signal models based on an extended library of
orthonormal bases, and apply the MDL principle to derive an
approximate additive cost function. The description length of the
noisy observed data is then minimized by optimizing the expansion-tree
associated with the best-basis algorithm and thresholding the
resulting coefficients. We show that the signal estimator can be
efficiently combined with the modified Wigner distribution, yielnding
robust time-frequency representations. The proposed methods are
compared to alternative existing methods, and their superiority is
demonstrated by synthetic and real data examples.

Chapter 1 Introduction
Chapter 2 Shift-Invariant Wavelet Packet Decompositions
Chapter 3 Shift-Invariant Trigonometric Decompositions
Chapter 4 Adaptive Time-Frequency Distributions
Chapter 5 Translation-Invariant Denoising
Chapter 6 Conclusion

The thesis is available at:

Dr. Israel Cohen
Yale University Tel: +(203) 432-1287
Department of Computer Science Fax: +(203) 432-0593
P.O. Box 208285
New Haven, CT 06520-8285
All times are GMT + 1 Hour
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