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Directional Tensor Product Complex Tight Framelets Open Access


Other title
image processing
tensor product
Type of item
Degree grantor
University of Alberta
Author or creator
Zhao, Zhenpeng
Supervisor and department
Wong, Yau Shu (Mathematical and Statistical Sciences)
Han, Bin (Mathematical and Statistical Sciences)
Examining committee member and department
Han, Bin (Mathematical and Statistical Sciences)
Jia, Rong-Qing (Mathematical and Statistical Sciences)
Selesnick, Ivan (Electrical and Computer Engineering, NYU Polytechnic School of Engineering)
Wong, Yau Shu (Mathematical and Statistical Sciences)
Li, Michael Yi (Mathematical and Statistical Sciences)
Zhao, Vicky (Electrical and Computer Engineering)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
This thesis concentrates on the construction of directional tensor product complex tight framelets. It uses a complex tight framelet filter bank in one dimension and the tensor product of the one-dimensional filter bank to obtain high-dimensional filter bank. It has a number of advantages over the traditional tensor product real wavelet transform. Motivated by two-dimensional dual tree complex wavelet transform, the complex tight framelet filter banks with frequency separation are constructed in the frequency domain. Then the high-dimensional framelet filter banks via tensor product and corresponding frames will have directional selectivity. The computational cost increases exponentially as dimension and redundancy rate grow, which restricts the application of framelet filter banks in high-dimensional data processing. In the frequency domain, we propose complex tight framelet filter banks with mixed sampling factor to reduce the redundancy rate. The tensor product complex tight framelet filter banks constructed in the frequency domain are bandlimited. They are not finitely supported in the time domain. Compactly supported wavelets or framelets are essential to many applications due to their good space-frequency localization and fast computational algorithm. We have proved a theoretical result on directional selectivity and provided step-by-step algorithms to construct compactly supported complex tight framelet filter banks. Then the directional compactly supported tensor product complex tight framelet filter banks in high dimensions can be obtained via tensor product. The directional tensor product complex tight framelet is used to the application of image denoising and video denoising. Experimental results show that our constructed complex tight framelets succeeds in providing improved image denoising results combined with advanced statistical models comparing with many other state-of-the-art transform based image denoising methods.
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. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before 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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