Chinese Journal of Computers   Full Text
Title	Constructing Biorthogonal Wavelets Using Lifting Scheme
Authors	CHEN Zhi-Neng1),2),3) GAO Xie-Ping1)
Address	1)(Information Engineering College, Xiangtan University, Xiangtan, Hunan 411105) 2)(Key Laboratory of Intelligent Information Processing,Institute of Computing Technology,Chinese Academy of Sciences, Beijing 100190) 3)(Graduate University of Chinese Academy of Sciences, Beijing 100049)
Year	2009
Issue	No.2(288—298)
Abstract &   Background	Abstract This paper investigates central properties such as support length, linear phase, vanishing moments of wavelet filters within the lifting scheme framework. A general algorithm that constructs biorthogonal wavelet with short support length, symmetry, and arbitrary vanishing moments is proposed. The definition of Best Lifting Scheme (BLS) is given. The construction of several famous biorthogonal wavelet systems, and a new system of biorthogonal wavelets with desirable properties, named odd-ordered BLS Wavelet (BLSW) system, is explicitly presented. Moreover, properties of BLSW system are listed, and experimental results of applying BLSW to image nonlinear approximation have shown that the performance of several BLSWs is promising because of their attractive properties. Keywords lifting scheme; biorthogonal wavelet; support length; symmetry; vanishing moments; nonlinear approximation Background The construction of biorthogonal wavelets with desirable properties is a challenging task in the past two decades. Generally, the construction methods fall mostly into two categories: spectral factorization and lifting scheme. The former one has been extensively studied. As a result almost all popular biorthogonal wavelets are firstly introduced using this technique. However, for the latter one, the general relationship between lifting scheme and properties of wavelet filters is still unclear. This paper studies the relationship between lifting scheme and properties of wavelet filters. The general theory for constructing of desirable biorthogonal wavelets within lifting scheme framework is established, and a general algorithm which can construct lots of biorthogonal wavelets with desirable properties in a straightforward way is proposed. This paper exhibits a new angle of view in biorthogonal wavelets construction and offers researchers a new series of ready-to-use wavelet filters. Furthermore, the capability of several newly constructed wavelets in image nonlinear approximation is discussed, which shows that the performance of these wavelets outperform both the popular CDF9-7 wavelet and the corresponding D-DW. The research of this paper is supported by the National Natural Science Foundation of China “Research on Parameterization of Wavelets and Self-adaptive Wavelet Neural Networks” (No.60375021). This project mainly aims at exploring theories and techniques for constructing wavelets and wavelet neural networks with desirable properties, which is a fundamental issue in many practical problems. The research group has got a lot of research results in this area. Some of them are published in area premium journals such as Science in China Series E: Information Science, Journal of Computer Science and Technology, International Journal of Computer Mathematics and so on. This paper deals with the construction of biorthogonal wavelets using lifting scheme. As an essential part of the project, both the theory and experimental results will deepen our research and contribute to our projects.