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A multiresolutional approach to the construction of spline wavelets

Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: In this thesis we study a wavelet construction procedure based on a multiresolutional
method, before specializing to the case of spline wavelets.
First, we introduce and analyze the concepts of scaling functions and their
duals, after which we analyze the multiresolutional analysis (MM) which they
generate. The advantages of orthonormality in scaling functions are pointed out
and discussed. Following the methods which were introduced in two standard
texts of Chui, we next show how a minimally supported wavelet and its dual
can be explicitly constructed from a given MM, thereby yielding an orthogonal
decomposition of the space of square-(Lebesgue)integrable functions on the real
line. We show that our method applied to orthonormal scaling functions also
yields orthonormal wavelets, including as a special case the Daubechies wavelet.
General decomposition and reconstruction algorithms are explicitly formulated,
and the importance of the vanishing moments of a wavelet in practical applications
is shown.
We next introduce and analyze cardinal B-splines, in particular showing that
these functions are refinable, and that they satisfy the criteria of Riesz stability.
Thus the cardinal B-spline is an admissible choice for a scaling function, so that the
previously developed wavelet construction procedure based on a MM yields an
explicit formula for the minimally supported B-spline wavelet. The corresponding
vanishing moment order is calculated, and the resulting ability of the B-spline
wavelet to detect singularities in a given function is demonstrated by means of a
numerical example. Finally, we develop an explicit procedure for the construction
of minimally supported B-spline wavelets on a bounded interval. This method,
as developed in work by de Villiers and Chui, is then compared with a previous
boundary wavelet construction method introduced in work by Chui and Quak. / AFRIKAANSE OPSOMMING: In hierdie tesis bestudeer ons 'n golfie konstruksieprosedure wat gebaseer is op 'n
multiresolusiemetode, voordat ons spesialiseer na die geval van latfunksie-golfies.
Eerstens word die konsepte van skaalfunksies en hulle duale bekendgestel en geanaliseer,
waarna ons die multiresolusie analise (MM) wat sodoende gegenereer
word, analiseer. Die voordeel van ortonormaliteit by skaalfunksies word uitgewys
en bespreek. Deur die metodes te volg wat bekendgestel is in twee standaardtekste
van Chui, wys ons vervolgens hoe 'n minimaal-gesteunde golfie en die duaal
daarvan eksplisiet gekonstrueer kan word vanuit 'n gegewe MM, en daarmee 'n
ortogonale dekomposisie van die ruimte van kwadraties-(Lebesgue)integreerbare
funksies op die reële lyn lewer. Ons wys dat ons metode toegepas op ortonormale
skaalfunksies ook ortonormale golfies oplewer, insluitende as 'n spesiale geval die
Daubechies golfie. Algemene dekomposisie en rekonstruksie algoritmes word eksplisiet
geformuleer, en die belangrikheid in praktiese toepassings van 'n golfie met
die nulmomenteienskap word aangetoon.
Vervolgens word kardinale B-Iatfunksies bekendgestel, en word daar in die
besonder aangetoon dat hierdie funksies verfynbaar is, en dat hulle aan die Rieszstabiliteit
vereiste voldoen. Dus is die kardinale B-Iatfunksie 'n toelaatbare keuse
vir 'n skaalfunksie, sodat die golfie konstruksieprosedure gebaseer op 'n MM,
soos vantevore ontwikkel, 'n eksplisiete formule vir die minimaal-gesteunde Blatfunksiegolfie
oplewer. Die ooreenkomstige nulmomentorde word bereken, en die
gevolglike vermoë van 'n B-Iatfunksiegolfie om singulariteite in 'n gegewe funksie
raak te sien en uit te wys word gedemonstreer deur middel van 'n numeriese voorbeeld.
Laastens ontwikkelons 'n eksplis.iete prosedure vir die konstruksie van
minimaal-gesteunde B-Iatfunksiegolfies op 'n begrensde interval. Hierdie metode,
soos ontwikkel in werk deur de Villiers en Chui, word dan vergelyk met 'n vorige
randgolfie konstruksie wat bekendgestel is in werk deur Chui en Quak.

Identiferoai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/51580
Date04 1900
CreatorsRohwer, Birgit
ContributorsDe Villiers, J. M., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
PublisherStellenbosch : Stellenbosch University
Source SetsSouth African National ETD Portal
Languageen_ZA
Detected LanguageUnknown
TypeThesis
Format148 p.
RightsStellenbosch University

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