【作者】 全宏跃；

【导师】 王国秋；

【作者基本信息】 湖南师范大学 ， 基础数学， 2009， 博士

【摘要】 小波分析是在20世纪80年代末迅速发展起来的一个新兴数学分支,小波分析的出现在科学研究领域引起了广泛的关注.在最近20多年,小波理论及应用得到了迅速的发展并取得丰硕成果.小波分析与其它分析方法一样,都是利用特殊基函数来展开和研究任意信号函数,但小波基与其它函数基相比具有一定的优势,例如,小波基具有良好的局部化性质、多分辨率功能及可同时作时频分析等优点.小波基性质的多样性是小波分析具有广泛应用领域主要原因,人们可以根据应用于领域的不同来构造相适应的小波基.小波基的构造和性质分析是小波理论研究的一个重要课题,也是小波应用的前提和基础.本论文主要是关于多进小波和多小波基构造理论的研究及小波基相关的性质分析.具体内容安排如下:第1章介绍了小波分析发展历史、本论文研究目的、主要研究内容及得到相关主要结果.第2章总结和探讨具有特殊性质多进尺度函数构造方法,着重研究了紧支正交对称多进尺度函数构造一般方法.在逼近则阶一定条件下,探索支集最短正交对称多进尺度函数构造一般程序.第3章首先总结基于多分辨分析L~2(R)中的小波框架构造理论.利用UEP规则,研究正交对称多进小波基构造方法.利用多相矩阵酉矩阵分解结构,得到了一类正交对称小波滤波器簇角参数表示形式,根据角参数的选取可得到一系列性质不同正交对称多进小波系.第4章提出了最优多进Haar小波概念,并讨论其所具有基本性质及构造方法.在图像压缩中,用实验数据说明了最优Haar小波变换相比于传统的离散余弦变换具有一定优势.第5章利用小波矩阵之间一种运算,给出了具有特定结构的正交小波矩阵一种构造方法.着重研究成对对称和具有优美结构的多进小波函数的构造.第6章首先总结了正交多小波一此基本理论及一些构造的方法.为提高正交对称多小波消失矩和逼近阶,研究一类正交对称多小波维数扩充的一种算法.第7章是本文总结与展望,指出了与本文相关期待进一步研究几个问题.本论文的主要创新之处如下:1.给出了多进正交对称低通滤波器构造一般方法,得到了在正则阶一定条件下,支集最短正交对称尺度函数构造程序.2.给出了偶数进低通滤波器对称应的多相向量一种特殊酉分解,得到了正交对称多进小波的高通滤波器的一种构造方法.得到了一类正交对称多进小波滤波器簇角参数表示形式.以3进小波为例,给出了奇数进正交对称小波多相矩阵酉扩充的一种算法.3.提出了多进最优Haar小波概念和构造方法.用图像压缩效果的实验数据说明了最优Haar小波较传统离散余弦变换具有一定优势.4.利用小波矩阵之间一种运算,得到了一类成对对称和优美小波构造方法.为提升正交对称多小波函数消失矩,提出了多小波维数扩充的一种算法.更多还原

【Abstract】 Wavelet analysis is a new offset of mathematics which developing very fast in late 1980s.The appearance of wavelet has brought extensive effect in science research fields.The theory and application of wavelet analysis have obtained plentiful fruits in last two decates.Wavelet analysis,samilary as the traditional analysis,deploy and research an arbitrary function by base functions.But,compare to other bases,wavelet bases have some advantages.For example,wavelet bases have localize character,mathematical microtelescope feature and adaptive feature.Multiplicity of property is the main reason that wavelet analysis has been applied in many fields. people are possibility to tailor wavelet bases to a given application field.Therefore, the construction of wavelet bases is a very important task in the wavelet academic research.The main contents of this thesis are about the construction and application of multi-band wavelet and multiwavelet bases.The outline of this thesis is as follows:In chapter one,the history of wavelet development、the motivation and the primary results relating to the thesis are introduced.In chaper two,we summarize some methods to construct multi-band scaling functions firstly.A method to construct the compacted orthogonal symmtric multiband scaling function is given in details.Under the condition of given regular order, we consider the construction procedure for orthogonal symmetric scaling function with the shortest support.In chaper three,the theory on constructing multi-band wavelet frames is summarized at firstly.utilizing the UEP regulation,we consider in detail the method to construct the filter banks of symmetric and orthogonal multi-band wavelet.By analyzing the polyphase matrix of multi-band wavelet,we research a class of symmetric orthogonal filter banks of multi-band wavelet which are expressed by angle parameter.via choosing angle parameter,one can obtain multi-band wavelet bases with different property.In chaper four,we put forward the concept of optimal multi-band Haar wavelets and give a general method to construct the filter banks of optimal Haar wavelets. Optimal Haar wavelet transform is superior to the discrete cosine transform(DCT) which have been shown experimentally in image and video compression.In chaper five,we define a new operation between wavelet matrix.Applying the operation to orthonomal wavelet matrix,we obtain a method to construct orthononal wavelet matrix with especial structure such as pair-symmetric and beautiful structure.In chaper six,we firstly summarize some theory of multi-wavelet and method to construct orthogonal multi-wavelet.at last,in order to heighten p.p.o and vanishing moment,we research an algorithm for dimension extension of orthogonal symmetric multi-wavelets.In chaper seven,summarize and prospect,and we put forward some issue relating to the thesis which deserved to rescarch.The main innovations of this thsis are as follows:1.A general way to construct symmetric orthogonal multi-band scaling function is provided.A scheme for constructing scaling sequence with the shortest length and a given regularly order is obtained.2.The factorization of polyphase matrix of a class of orthogonal symmetric multi-band wavelet filter banks has been considered,method to construct orthogonal symmetric multi-band wavelet filter banks has been obtained.A class of orthogonal symmetric filter banks are provided by angle parameters.3.The concept of optimal multi-bank Haar wavelets was put forward,and a general method of constructing the optimal multi-bank Haar wavelets is provided.The optimal multi-bank Haar wavelets perform better than DCT in image compression has been shown experimentally.4.Appling wavelet matrix operation,we obtain a method to construct a class of orthononal wavelet matrix with especial structure such as pair-synunetric and beautiful structure wavelet matrix.In order to heighten vanishing moment,we put forward an algorithm for dimension extension of orthogonal symmetric multi-wavelets.更多还原