Original article: The lifting factorization of wavelet bi-frames with arbitrary generators

Authors:
Yan Shi;Xiaoyuan Yang
Affiliations:
State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China;Key Laboratory of Mathematics, Informatics, and Behavioral Semantics of Ministry of Education, Department of Mathematics and Systems Science, Beihang University, Beijing 100191, China
Venue:
Mathematics and Computers in Simulation
Year:
2011

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Abstract

In this paper, we present the lifting scheme of wavelet bi-frames with arbitrary generators. The Euclidean algorithm for arbitrary n Laurent polynomials and the factorization theorem of polyphase matrices of wavelet bi-frames are proposed. We prove that any wavelet bi-frame with arbitrary generators can be factorized into a finite number of alternating lifting and dual lifting steps. Based on this concept, we present a new idea for constructing bi-frames by lifting. For the construction, by using generalized Bernstein basis functions, we realize a lifting scheme of wavelet bi-frames with arbitrary prediction and update filters and establish explicit formulas for wavelet bi-frame transforms. By combining the different designed filters for the prediction and update steps, we can devise practically unlimited forms of wavelet bi-frames. Furthermore, we present an algorithm for increasing the number of vanishing moments of wavelet bi-frames to arbitrary order by the presented lifting scheme, which adopts an iterative algorithm. Several examples are constructed to illustrate the conclusion.