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 http://hdl.handle.net/10048/1224
 Interpolating refinable function vectors and matrix extension with symmetry
 Zhuang, Xiaosheng
 en

refinable function vector
matrix extension
interpolation
symmetry
wavelets
orthonormal multiwavelets
biorthogonal multiwavelets
filter banks
perfect reconstruction
tight framelets
directionality
symmetry groups  Jul 15, 2010 5:38 PM
 Thesis
 en
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 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 stepbystep 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 frequencybased nonstationary tight wavelet frames with directionality. We demonstrate that one can construct a frequencybased 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.
 Doctoral
 Doctor of Philosophy
 Department of Mathematical and Statistical Sciences
 Fall 2010
 Bin Han (Mathematical and Statistical Sciences)

Bin Han (Mathematical and Statistical Sciences)
John C. Bowman (Mathematical and Statistical Sciences)
RongQing Jia (Mathematical and Statistical Sciences)
Yau Shu Wong (Mathematical and Statistical Sciences)
Mrinal Mandal (Electrical and Computer Engineering)
DingXuan Zhou (Mathematics, City University of Hongkong)
Faculty of Graduate Studies and Research
Theses and Dissertations Spring 2009 to present
Department of Mathematical and Statistical Sciences
Theses and Dissertations Spring 2009 to present
Department of Mathematical and Statistical Sciences
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