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   -> Volume 7, Issue 8


Thesis: Interpolation Schemes and Wavelets
 
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Daniel Lemire (daniel.lemire@tintin.net)
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PostPosted: Thu Aug 13, 1998 11:11 pm    
Subject: Thesis: Interpolation Schemes and Wavelets
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#8 Thesis: Interpolation Schemes and Wavelets

The following wavelet-related thesis is now available as a postscript
file.

TITLE : SCH=C9MAS D'INTERPOLATION ET ONDELETTES

(INTERPOLATION SCHEMES AND WAVELETS)

=C9cole Polytechnique de Montr=E9al and Universit=E9 de Montr=E9al

Montr=E9al, Qu=E9bec

AUTHOR : D. Lemire

JURY : A. Fortin, G. Deslauriers, S.Dubuc, R. Vaillancourt and J.-M.
Lina

DATE : March 1998

LANGUAGE : French

ENGLISH ABSTRACT :

In the early 80's, the convergence of research in engineering (signal
processing), mathematics (harmonic analysis), and physics (quantum
mechanics) gave birth to wavelet theory. At the same time, Gilles
Deslauriers and Serge Dubuc (and others) came up with iterative
interpolation. It is only later, in the early 90's, when wavelet
theory became a mature subject, that the link between iterative
interpolation and wavelets became obvious. Now, many researchers in
Quebec, Alberta, France, U.S.A., Israel, Scandinavia and, Singapore
are working on iterative interpolation for its applications in
Computer Science (graphics) and in engineering
(modelisation). However, few researchers have worked on building
wavelets from iterative interpolation schemes. This is what we wanted
to do.

On the one hand, we have discovered and shown that some well-known
wavelets, the Cohen--Daubechies--Feauveau biorthogonal wavelets, are
in fact, the derivatives of a certain family of functions (called
fundamental) obtained by the iterative interpolation scheme. This same
family of functions was adapted to the interval by Mongeau in the
early 90's. Using these facts, we adapted these wavelets to the
interval. We can then handle bounded regions of the real line.

Iterative interpolation can be easily casted into a multidimensional
context. It was therefore natural to try to build multidimensional
wavelets from some iterative interpolation schemes and, in order to
apply them, it was necessary to be able to work, for example, on
rectangular regions in the plane. We have done this in the last part
of this thesis. We have included an example of an image analysis by
means of these new wavelets.

Available for download with other documents as a zip file containing a
Postscript file at :

http://www.technopole.le-village.com/lemire/publications.html

http://cafe.rapidus.net/danilemi/publications.html

Some of the boundary adapted filters derived in this thesis are available
as a text file (with explanations in English) at

http://www.technopole.le-village.com/lemire/filtres.txt

http://cafe.rapidus.net/danilemi/filtres.txt

Finally, Java implementations of most of the ideas of this thesis and
of boundary some other boundary-handling wavelets

(including Daubechies-Meyer wavelets) with applications

to signal processing (ECG, medical data,...) and numerical analysis
(O.D.E., matrix representation of operators,...) are available at

http://www.technopole.le-village.com/lemire/java_contributions.html

http://cafe.rapidus.net/danilemi/java_contributions.html

(Page is in French, but English documentation is being written.)

Thanks!

Daniel Lemire

Daniel.Lemire@Tintin.net

Daniel Lemire

http://cafe.rapidus.net/danilemi/
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