Doctoral Degrees (Mathematical Sciences)

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Browsing Doctoral Degrees (Mathematical Sciences) by browse.metadata.advisor "Chui, C. K."

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    Bivariate wavelet construction based on solutions of algebraic polynomial identities
    (Stellenbosch : Stellenbosch University, 2012-03) Van der Bijl, Rinske; De Villiers, J. M.; Chui, C. K.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Multi-resolution analysis (MRA) has become a very popular eld of mathematical study in the past two decades, being not only an area rich in applications but one that remains lled with open problems. Building on the foundation of re nability of functions, MRA seeks to lter through levels of ever-increasing detail components in data sets { a concept enticing to an age where development of digital equipment (to name but one example) needs to capture more and more information and then store this information in di erent levels of detail. Except for designing digital objects such as animation movies, one of the most recent popular research areas in which MRA is applied, is inpainting, where \lost" data (in example, a photograph) is repaired by using boundary values of the data set and \smudging" these values into the empty entries. Two main branches of application in MRA are subdivision and wavelet analysis. The former uses re nable functions to develop algorithms with which digital curves are created from a nite set of initial points as input, the resulting curves (or drawings) of which possess certain levels of smoothness (or, mathematically speaking, continuous derivatives). Wavelets on the other hand, yield lters with which certain levels of detail components (or noise) can be edited out of a data set. One of the greatest advantages when using wavelets, is that the detail data is never lost, and the user can re-insert it to the original data set by merely applying the wavelet algorithm in reverse. This opens up a wonderful application for wavelets, namely that an existent data set can be edited by inserting detail components into it that were never there, by also using such a wavelet algorithm. In the recent book by Chui and De Villiers (see [2]), algorithms for both subdivision and wavelet applications were developed without using Fourier analysis as foundation, as have been done by researchers in earlier years and which have left such algorithms unaccessible to end users such as computer programmers. The fundamental result of Chapter 9 on wavelets of [2] was that feasibility of wavelet decomposition is equivalent to the solvability of a certain set of identities consisting of Laurent polynomials, referred to as Bezout identities, and it was shown how such a system of identities can be solved in a systematic way. The work in [2] was done in the univariate case only, and it will be the purpose of this thesis to develop similar results in the bivariate case, where such a generalization is entirely non-trivial. After introducing MRA in Chapter 1, as well as discussing the re nability of functions and introducing box splines as prototype examples of functions that are re nable in the bivariate setting, our fundamental result will also be that wavelet decomposition is equivalent to solving a set of Bezout identities; this will be shown rigorously in Chapter 2. In Chapter 3, we give a set of Laurent polynomials of shortest possible length satisfying the system of Bezout identities in Chapter 2, for the particular case of the Courant hat function, which will have been introduced as a linear box spline in Chapter 1. In Chapter 4, we investigate an application of our result in Chapter 3 to bivariate interpolatory subdivision. With the view to establish a general class of wavelets corresponding to the Courant hat function, we proceed in the subsequent Chapters 5 { 8 to develop a general theory for solving the Bezout identities of Chapter 2 separately, before suggesting strategies for reconciling these solution classes in order to be a simultaneous solution of the system.