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   -> Volume 3, Issue 18


Thesis: Wavelet Methods for the Inversion of Certain Homogeneous
 
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Eric Kolaczyk (kolaczyk@galton.uchicago.edu)
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PostPosted: Mon Dec 02, 2002 1:28 pm    
Subject: Thesis: Wavelet Methods for the Inversion of Certain Homogeneous
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Thesis: Wavelet Methods for the Inversion of Certain Homogeneous
Linear Operators in the Presence of Noisy Data


Author: Eric D. Kolaczyk
Advisor: David L. Donoho
Title: Wavelet Methods for the Inversion of Certain Homogeneous
Linear Operators in the Presence of Noisy Data
Institution: Department of Statistics, Stanford University

This thesis may be obtained in hard form by writing to

Attn: Department Secretary
Department of Statistics
Stanford University
Stanford, CA 94305

or via anonymous ftp

at "galton.uchicago.edu" in ~pub/kolaczyk/Thesis .

There is a README file in the above directory, detailing the
files which should be transfered.


Abstract:

In this dissertation we explore the use of wavelets in certain
linear inverse problems with discrete, noisy data. We observe discrete
samples of a process $y(u) = (Kf)(u) + z(u)$, where $K$ is a homogeneous
linear operator, $z$ is a noise process, and $f$ is a function we wish
to recover from the data. In the problems which we consider, the inverse
of $K$, $K^{-1}$, either does not exist or is poorly behaved.
We work within the theoretical framework of the wavelet-vaguelette
decomposition, as a means for decomposing $K$. When $Kf$ is
observed in the presence of noise, we can recover a denoised version
of $f$ by shrinking the wavelet coefficients towards zero, as Donoho
& Johnstone have done in a simpler class of problems (i.e. when
$K$ is the identity).
Our intent ultimately is to attack the problem of reconstructing
images from tomographic data using wavelets. In order to do so, we
begin by developing tools for the calculation and use of Meyer
wavelets. These wavelets have, in the frequency domain,
compact support and a closed-form expression, making them
particularly amenable to calculating the biorthogonal decompositions
of operators which are well-understood in that domain.
We develop in detail algorithms for fast one- and two-dimensional
discrete, periodic Meyer wavelet transforms, and use these to calculate
biorthogonal decompositions of certain self-adjoint, homogeneous
operators. Examples include differentiation, fractional differentiation,
and convolution.
With this background structure in place, we develop expressions
for a biorthogonal decomposition of the Radon transform operator. A
multiresolution analogue of the traditional filtering of backprojected
projections methods is implemented to calculate this decomposition, for
data arising in the contexts of parallel beam X-ray tomography and
positron emission tomography.
The biorthogonal decompositions of the Radon transform and other
operators are finally used with wavelet shrinkage methods for recovery of
signals and images in the context of inverse problems with noisy data. The
shrinkage methods are adjusted so that the simple threshold used in
non-inverse problems is replaced by a set of thresholds of increasing
size. We describe moderate-deviation probability calculations
that give theoretical justification to these thresholds.
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