Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China.
IEEE Trans Image Process. 2010 Mar;19(3):612-24. doi: 10.1109/TIP.2009.2038762. Epub 2009 Dec 18.
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.
在本文中，我们提出了小波双框架的提升方案，并进行了理论分析、结构和算法的研究。我们展示了如何将任何小波双框架分解为一系列简单的滤波步骤。这种分解对应于小波双框架的多相矩阵的因式分解。基于这个概念，我们提出了一种构建小波双框架的新方法。对于对称双框架的构建，我们使用广义 Bernstein 基函数，这使我们能够设计对称预测和更新滤波器。这种构建允许更有效的实现，并提供了自定义设计小波双框架的工具。通过结合预测和更新步骤的不同设计滤波器，我们可以设计出实际无限形式的小波双框架。此外，我们还提出了一种通过所提出的提升方案将双框架的消失矩数量增加到任意阶的算法，该算法采用迭代算法并确保最短的提升方案。给出了几个构造实例来说明结果。
