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Complex Bases, Number Systems and Their Application to Fractal-Wavelet Image Coding

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dc.contributor.authorPiché, Daniel Guyen
dc.date.accessioned2006-08-22T14:21:54Z
dc.date.available2006-08-22T14:21:54Z
dc.date.issued2002en
dc.date.submitted2002en
dc.description.abstractThis thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.en
dc.formatapplication/pdfen
dc.format.extent10314636 bytes
dc.format.mimetypeapplication/pdf
dc.identifier.urihttp://hdl.handle.net/10012/1057
dc.language.isoenen
dc.pendingfalseen
dc.publisherUniversity of Waterlooen
dc.rightsCopyright: 2002, Piché, Daniel G.. All rights reserved.en
dc.subjectMathematicsen
dc.subjectcomplex basesen
dc.subjectnumber systemsen
dc.subjectfractal-waveletsen
dc.subjectwaveletsen
dc.subjectimage codingen
dc.subjectfractalsen
dc.titleComplex Bases, Number Systems and Their Application to Fractal-Wavelet Image Codingen
dc.typeDoctoral Thesisen
uws-etd.degreeDoctor of Philosophyen
uws-etd.degree.departmentApplied Mathematicsen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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