A multiresolutional approach to the construction of spline wavelets

Rohwer, Birgit (2000-04)

Thesis (MSc) -- University of Stellenbosch, 2000.

Thesis

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.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/51580
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