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Interpolating refinable function vectors and matrix extension with symmetry Open Access


Other title
refinable function vector
tight framelets
perfect reconstruction
symmetry groups
biorthogonal multiwavelets
filter banks
matrix extension
orthonormal multiwavelets
Type of item
Degree grantor
University of Alberta
Author or creator
Zhuang, Xiaosheng
Supervisor and department
Bin Han (Mathematical and Statistical Sciences)
Examining committee member and department
Yau Shu Wong (Mathematical and Statistical Sciences)
Rong-Qing Jia (Mathematical and Statistical Sciences)
John C. Bowman (Mathematical and Statistical Sciences)
Ding-Xuan Zhou (Mathematics, City University of Hongkong)
Bin Han (Mathematical and Statistical Sciences)
Mrinal Mandal (Electrical and Computer Engineering)
Department of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctor of Philosophy
Degree level
In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results. In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1. In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed.
License granted by Xiaosheng Zhuang ( on 2010-07-14T17:45:17Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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