The lifting scheme for wavelet bi-frames: theory, structure, and algorithm

Authors:
Xiaoyuan Yang;Yan Shi;Liuhe Chen;Zongfeng Quan
Affiliations:
Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China and Department of Mathematics and Systems Science, Beihang University, China;Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China and Department of Mathematics and Systems Science, Beihang University, China;Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China and Department of Mathematics and Systems Science, Beihang University, China;Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China and Department of Mathematics and Systems Science, Beihang University, China
Venue:
IEEE Transactions on Image Processing
Year:
2010

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Abstract

In this paper, we present the lifting scheme of wavelet bi-frames along with theory analysis, structure, and algorithm. We show how any wavelet bi-frame can be decomposed into a finite sequence of simple filtering steps. This decomposition corresponds to a factorization of a polyphase matrix of a wavelet bi-frame. Based on this concept, we present a new idea for constructing wavelet bi-frames. For the construction of symmetric bi-frames, we use generalized Bernstein basis functions, which enable us to design symmetric prediction and update filters. The construction allows more efficient implementation and provides tools for custom design of wavelet bi-frames. By combining the different designed filters for the prediction and update steps, we can devise practically unlimited forms of wavelet bi-frames. Moreover, we present an algorithm of increasing the number of vanishing moments of bi-framelets to arbitrary order via the presented lifting scheme, which adopts an iterative algorithm and ensures the shortest lifting scheme. Several construction examples are given to illustrate the results.