title: The lifting scheme for wavelet bi-frames: theory, structure, and algorithm.
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.
authors.affiliation: Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China. xiaoyuanyang@vip.163.com
authors.forename: Xiaoyuan
authors.initials: X
authors.lastname: Yang
authors.forename: Yan
authors.initials: Y
authors.lastname: Shi
authors.forename: Liuhe
authors.initials: L
authors.lastname: Chen
authors.forename: Zongfeng
authors.initials: Z
authors.lastname: Quan