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Directional Tensor Product Complex Tight Framelets

  • Author / Creator
    Zhao, Zhenpeng
  • 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.

  • Subjects / Keywords
  • Graduation date
    2015-11
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R3DF6KB58
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Applied Mathematics
  • Supervisor / co-supervisor and their department(s)
    • Han, Bin (Mathematical and Statistical Sciences)
    • Wong, Yau Shu (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Han, Bin (Mathematical and Statistical Sciences)
    • Selesnick, Ivan (Electrical and Computer Engineering, NYU Polytechnic School of Engineering)
    • Li, Michael Yi (Mathematical and Statistical Sciences)
    • Wong, Yau Shu (Mathematical and Statistical Sciences)
    • Jia, Rong-Qing (Mathematical and Statistical Sciences)
    • Zhao, Vicky (Electrical and Computer Engineering)