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   -> Volume 5, Issue 1

Preprint: Two papers on wavelets and denoising
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Eric Kolaczyk (

PostPosted: Tue Dec 03, 2002 4:15 pm    
Subject: Preprint: Two papers on wavelets and denoising
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Preprint: Two papers on wavelets and denoising

The following two manuscripts, relating to de-noising with wavelets,
are available via anonymous ftp. The file format is compressed
postscript (*.ps.Z)

1.) Title: ``A Wavelet Shrinkage Approach to Tomographic
Image Reconstruction"
Author: Eric D. Kolaczyk

A method is proposed for reconstructing images from tomographic data
with respect to a two-dimensional wavelet basis. The
Wavelet-Vaguelette Decomposition is used as a framework within which
expressions for the necessary wavelet coefficients may be derived.
These coefficients are calculated using a version of the filtered
backprojection algorithm, as a computational tool, in a
multiresolution fashion. The necessary filters are defined in terms
of the underlying wavelets. Denoising is accomplished through an
adaptation of the Wavelet Shrinkage approach of Donoho et al., and
amounts to a form of regularization. Combining the above two steps
yields the proposed WVD/WS reconstruction algorithm, which is compared
to the traditional filtered backprojection method in a small study.

Authors: Hugh A. Chipman, Eric D. Kolaczyk, Robert E. McCulloch

When fitting wavelet based models, shrinkage of the wavelet
coefficients is an effective tool for de-noising the data. This paper
outlines a Bayesian approach to shrinkage, obtained by placing priors
on the wavelet coefficients. The prior for each coefficient consists
of a mixture of two normal distributions with different standard
deviations, a form simple enough to allow closed form calculation of
the posterior distribution. This means that a closed form shrinkage
function is available, and reconstructions for large data sets can be
computed quickly. The simple and intuitive form of prior allows us to
propose automatic choices of prior parameters. These parameters are
chosen adaptively according to the resolution level of the
coefficients, typically shrinking high resolution (frequency)
coefficients more heavily. The closed form of the posterior
distribution means that posterior variances of the coefficients may be
used to assess uncertainty in the reconstruction. Several examples
are used to illustrate the method, and comparisons are made with other
shrinkage methods.

Please direct communication regarding these manuscripts to
Eric Kolaczyk at
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