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   -> Volume 2, Issue 9

Thesis available
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Marc Goldburg, Stanford University

PostPosted: Fri Nov 29, 2002 3:31 pm    
Subject: Thesis available
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Thesis available


Title: Applications of Wavelets to Quantization and Random Process

Author: Marc Goldburg
Date: May, 1993
Advisor: Thomas Kailath
Institution: Stanford University
Availability: 1) University Microfilms, or
2) via anonymous ftp to
'pub/marcg/' and
'pub/marcg/' are 'compress'ed PostScript
files suitable for one and two-sided printing, respectively.

Despite its short history, the wavelet transform has found application in a
remarkable diversity of disciplines: Mathematics, Physics, Numerical
Analysis, Signal Processing and others. In this thesis, we examine the
utility of this transform for three different signal processing
applications: the representation of continuously indexed random processes;
transform vector quantization systems; and partial representations and
subband coding of discretely indexed random processes.

For continuously indexed random processes, we show that orthogonal wavelet
series representations converge in $L^2(R)$ with probability one and in
mean-square for all finite energy random processes. With compactly
supported wavelets and scaling functions, these same convergence modalities
hold for finite power random processes on any compact interval. Between
them, the finite power and finite energy random processes include all
physical random processes.

Next, we present a high-rate analysis of Transform Vector Quantization
(TVQ) systems subject to a rate or index entropy constraint. This analysis
provides expressions for the best achievable distortion and optimal
sub-vector rate or index entropy assignments for a given source, orthogonal
transform, and sub-vector partitioning scheme. Bounds are provided for the
performance penalty incurred by TVQ relative to unstructured vector
quantization. These analytical results are especially useful in the design
of TVQ systems; they permit the relative performance of competing
transforms to be assessed using only the source statistics. If the class
of candidate transforms has a parametric description, our analytical
results lead to a tractable optimization problem whose solution is the
optimal (distortion minimizing) transform within the class. To demonstrate
this application, the general results are specialized to the case of a
Gaussian source and a transform cost function identified. The properties
of transforms minimizing this cost and a computationally efficient upper
bound on the cost are then presented. Using the Discrete Time Wavelet
Transform (DTWT) as our parametric family of transforms, we present design
examples of optimal TVQ systems and compare their performance to systems
based on the Discrete Cosine Transform.

The final topic addressed is the ability of a single DTWT step to
concentrate the energy in its input sequence into one of its two component
output subsequences. This issue relates, for example, to maximization of
the coding gain afforded by scalar subband coding systems employing the
DTWT. We present bounds on the concentration capability of the transform,
and discuss the design of optimal (maximally concentrating) DTWT's. For
any wide-sense stationary process, we show that the optimal DTWT with
length four filters will always be one of two fixed choices, one of which
corresponds to Daubechies' $D_2$ scaling function.
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