关于MRA超小波的研究

【作者】 王凤兰；

【导师】 李万社；

【作者基本信息】 陕西师范大学 ， 应用数学， 2008， 硕士

【摘要】 小波分析应用到信号处理和图象处理已是众所周知,其成功之处在于空间L²（R）的多分辨分析（MRA）的引入,正因为有MRA结构,我们可以给出实现小波分析的快速算法.经典的MRA小波是一种重要的正交小波,在应用中许多著名例子及小波都属于正交小波类。超小波是指希尔伯特空间L²（R）^（m）=L²（R）（?）…（?）L²（R）中的小波,是基于小波分析基础上的信号分析方法。超小波的独特之处在于,它对解决信号的多重性问题有小波无法比拟的优势。可构造L²（R）^（m）空间的紧支撑小波的正交基,理论上,这些基要比已知的小波基较好。在数据压缩方面,当有同样数据时,超小波可以同时压缩多个信号。类似地,研究空间L²（R）^（m）中的MRA超小波有十分重要作用。在2000年,Han D.及Larson D.已证明了完全可以构造L²（R）^（m）中的MRA超小波。于2005年,Bildea S.,Dutkay D E.及Galriel P.给出了构造空间L²（R）^（m）中的MRA超小波的一种方法,并且构造了紧支撑小波的正交基。建立在对规范紧框架小波分析的基础之上,本文给出了关于MRA超小波的判定定理,即一个超小波是MRA超小波的充分和必要条件。给出并证明了一个超小波是MRA结构当且仅当每个特殊线性子空间的维数是0或者1。特别地,当E_i是有界规范紧框架小波集时,一个超小波是MRA结构当且仅当E_i^s=∪_k≥12^-kE_i是2π平移同余到[-π,π)的一个子集。除此之外还给出一个例子。本文共分四部分:第一章,概述了小波分析及超小波的产生、发展发展过程。第二章,主要讨论了超小波的基本性质。规范紧框架小波扩展为超小波是超小波的一个重要的方面。因此,我们有必要在此介绍可扩展的一些性质。第三章,着重研究了超小波的构造方法。如何根据所处理的信号特征构造或者选择最佳的超小波函数应该成为我们要研究的问题之一。第四章,是本文的核心内容,讨论了直和希尔伯特空间L²（R）^（m）中的MRA超小波的判定条件。更多还原

【Abstract】 The applications of wavelet theory to signal processing and image processing are now well-know.Probably the main reason for the success of the wavelet theory was the introduction of the concept of multiresolution analysis（MRA）,which provided the right framework to construct orthogonal wavelet bases with good localization properties.The classical MRA wavelets are probably the most important class of orthogonal wavelets.Many of the better known examples as well as those often used in applications belong to this class.The super-wavelets is a wavelet in the space L²（R）^（m） = L²（R）（?）···（?）L²（R）.It is a signal analysis method that bases on wavelet analysis.An advantage of superwavelet is solving multiple signals.We constucted orthonarmal bases of compactly supported wavelets for the space L²（R）^（m）.Theoretically,these bases appear to be better than the know wavelets because,with the same amount of data,one cancompress not just one signal,but several.Similarly,It is important to study MRA super-wavelets in the space L²（R）^（m）.Han D.and Larson D.have shown that the technique of multiresolution analysis breaks down for the super-wavelets in 2000.Bildea S.,Dutkay D E.and Picioraga G.gave a method of constructing MRA super-wavelet and constructed orthogonal bases of compactly supported wavelets for the space L²（R）^（m） in 2005.Building up from a careful analysis of normalized tight frame wavelet,we developed a theory of the MRA super-wavelets,which is necessary and sufficient conditions of super-wavelets arises from MRA construction.It is shown that a superwavelet arises from our MRA construction if and only if the dimension of each particular linear space is either zero or one.In particular,when E_i is a bounded normalized tight frame wavelet set,it is shown that a super-wavelet arise from our MRA construction if and only if E_i^s =∪_κ≥12^-κE_i is 2πtranslation congruent to a subset of[—π,π).In addition,an example is given.This paper is composed of four parts:The chapter 1 is an introduction which summarizes the emergence,develepment of wavelet analysis and super-wavelet theory. The chapter 2 presents the basic properties of super-wavelet.It is an important that the normalized tight frame wavelet extending to the super-wavelet.The chapter 3 studies the construction method of super-wavelet.It is one of our studing problem that how to choose better function according to the proceeding signals.The chapter 4 is the key of the paper,we dicuss the condition of a super-wavelets arises arises from MRA construction.更多还原