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> Volume 7, Issue 3
Preprint: Preprint from Tim Downie

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Tim Downie (tim.downie@cmis.csiro.au) Guest

Posted: Thu Mar 05, 1998 4:37 am Subject: Preprint: Preprint from Tim Downie




#3 Preprint: Preprint from Tim Downie
The following are available in postscript format from my
web page http://www.dms.csiro.au/~timd/research.html
Thanks for your interest
Tim
1) "Wavelets in statistics" PhD Thesis Downie T.R.
University of Bristol, 1997.
Abstract
Two different aspects of wavelet methods in statistics are studied,
preceded by a review of existing methods. The first aspect is
thresholding using multiple wavelets, and the second is fitting a
wavelet basis to the deformable template model.
A multiple wavelet basis is a wavelet basis generated by $L geq 2$
mother wavelets. The discrete multiple wavelet transform DMWT
transforms a set of scalar data into a set of $L$vector
coefficients. In order to obtain an efficient multiwavelet
decomposition the data must be preprocessed first. Four criteria
are developed to assess different preprocessing methods. A
multivariate thresholding method is proposed which adapts each
coefficient vector as one entity. This accommodates the
timefrequency properties of the discrete multiple wavelet
transform, and the correlation within vector coefficients. Hard
and Soft thresholding rules, and a universal threshold are
developed. Results from a simulation study demonstrate that
multivariate thresholding gives a lower mean square error than
using univariate thresholding of multiple wavelets and thresholding
using Daubechies wavelets with the universal threshold. Using
different preprocessing methods gave markedly different results.
The method of deformable templates is used to deform one
image(template) into the shape of another similar image. The
template is deformed by a deformation function. This deformation
function is often modelled using a Fourier basis so that a
deformation can be obtained numerically. A wavelet basis should be
better at representing deformations that have localised features.
A data set where the deformation is likely to be localised is a
collection of femoral condyle images. A wavelet model using a
mixture distribution is proposed and a penalised least squares
method is developed in order to obtain a deformation between a
template and image. Three algorithms are developed and compared by
fitting a deformation between two femoral condyle images.
2) "Signal Preprocessing for Multiwavelets" Downie T.R.
Abstract:
Orthogonal wavelet bases have been developed using multiple mother
wavelet functions. Applying the discrete multiple wavelet
transform requires the input data to be preprocessed to obtain a
more economical decomposition. Four general properties of
preprocessing methods are reviewed: length, degree, orthogonality
and frequency response. A minimal prefilter is defined to have
length one and highest attainable degree. For the GHM Multiwavelet
specific prefilters are discussed, including a minimal matrix
prefilter and a repeated signal prefilter. The criteria for each
prefilter are obtained and how this compares to the performance of
prefilters.
3) "The Discrete Multiple Wavelet Transform and Thresholding
Methods." Downie T.R. & Silverman B.W. (revised version)
abstract:
We propose thresholding for multiwavelets considering the
coefficient vectors as a whole rather than thresholding individual
elements. A multivariate universal threshold is obtained using the
$chi ^2$ distribution. Simulations indicate that, using the GHM
multiwavelet with appropriate preprocessing, our method outperforms
univariate thresholding of both GHM and Daubechies wavelet
decompositions.
4) "Economical Representation of Image Deformation Functions
Using a Wavelet Mixture Model"
Downie T.R. Shepstone L.& Silverman B.W.
Abstract: One method of analysing two similar images is to obtain a
deformation function that maps one image onto the other. The
modelling of this deformation function in terms of a functional
basis leads to numerical methods of obtaining a good deformation.
It is advantageous to be able to represent a wide class commonly
observed deformations economically, ie where most coefficients are
zero. We proposed a wavelet model for the deformation, where each
wavelet coefficient has the mixture distribution $pi delta _0 +
(1pi) N(0,sigma ^2)$. This distribution reflects our prior
belief in the wavelet coefficients and results in an economical
representation for the deformation. To implement this method, a
penalised least squares methodology is adopted and three algorithms
are devised. A numerical assessment of the method is made by
applying the algorithms to some femoral condyle images, which
require the deformations to have localised features.

Tim Downie
Analysis of Large and Complex Datasets Project
CSIRO Mathematical and Information Sciences
Locked Bag 17, North Ryde, NSW 2113, Australia
Phone: +61 2 9325 3256 Fax: +61 2 9325 3200
email:tim.downie@cmis.csiro.au
url:http://www.dms.csiro.au/~timd/research.html 





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