【作者】 刘占卫；

【导师】 胡国恩；

【作者基本信息】 解放军信息工程大学 ， 信号与信息处理， 2008， 博士

【摘要】 自1986年以来,小波分析已成为科学研究中的热点领域,其应用涉及自然科学与工程技术的许多领域。目前,小波分析已成为研究和解决自然科学与工程计算中许多复杂问题的强有力工具。在小波分析受到广泛应用的信号处理和通信领域中,抽样定理起着举足轻重的作用,它建立了离散信号和模拟信号的等价关系。小波子空间中抽样定理的研究是当今国际上抽样定理发展的一个重要方向。作为正交小波基的推广,Parseval框架小波保持了正交小波基除正交性以外的几乎所有的良好性质。如今,框架理论已经被成功地应用在信号处理、图像处理、数据压缩、抽样理论等领域,因此研究Parseval框架小波具有非常重要的意义。本文分为两个部分。第一部分研究了Parseval框架小波与Parseval框架尺度函数的构造和性质;第二部分研究了L2 ( R )子空间中非整点抽样定理和基于框架的多小波子空间的抽样定理。主要结果概括为以下几个方面:1.综述了小波分析产生的背景及小波理论和小波子空间中抽样定理若干重要方面的研究现状,并介绍了本文的主要工作。2.考虑到通过广义多分辨分析(GMRA)构造的小波都具有半正交性,因此,我们首先研究矩阵伸缩半正交框架小波的性质,给出其成立的必要条件,证明具有附加条件的半正交框架小波成立的充要条件。其次,发现所有GMRA Parseval框架小波与{T kψ: k∈Zd}为一闭子空间W0的Parseval框架的等价性,从而进一步完善了广义多分辨分析的理论基础。最后,通过小波的最小向量滤波器刻画了GMRA Parseval框架小波的一个重要性质。3.研究了Parseval框架尺度函数集与MSF Parseval框架小波,得出许多有意义的结论。首先给出可测集S为Parseval框架尺度函数集的充要条件。其次,通过研究可测集S的性质,得到一种构造Parseval框架尺度函数集的方法,并证明所有的Parseval框架尺度函数集都由这种方法构成。最后,依靠揭示MSF Parseval框架小波与Parseval框架尺度函数集之间的关系,我们得到两个结果:一是每个Parseval框架尺度函数集生成一个满足一定条件的MSF Parseval框架小波ψ;二是对于维数函数几乎处处满足Dψ(ξ) =χ? (ξ)≤1的MSF Parseval框架小波ψ,一定存在一个Parseval框架尺度函数集S使得ψ? =χBS \S成立。由于MSF Parseval框架小波ψ可以由它的Fourier变换ψ? =χW来定义,研究小波的性质就可以简化为研究可测集W的性质。因此,我们的结果不但提供了一种构造MSF Parseval框架小波的新方法,而且也加深了人们对小波的理解。4.把文献[101]的结果推广到Parseval框架尺度函数的情况,得到可测集G为Parseval框架尺度函数的频域支集的充要条件。特别是通过讨论集合G、τ( G)、12G和G \12G之间的关系,揭示了基于多分辨分析的尺度函数与基于Parseval框架多分辨分析的尺度函数之间的不同。同时,通过研究有限带宽Parseval框架小波的频域支集,得到它的一个重要性质。5.讨论了L2 ( R )子空间中基于框架的非整点抽样定理。我们依靠平移算子和伸缩算子性质来证明( )L2 R子空间中a > 0步长非整点抽样定理成立的充要条件,同时也给出了其抽样函数在频域中的表示形式。6.研究了多小波子空间中非整点规则抽样与非规则抽样。首先,通过Zak变换提供一种构造满足一定条件的非整点规则抽样函数的方法,同时建立了基于框架的非整点规则抽样定理。其次,注意到非规则抽样具有很强的实用价值,因此,我们也研究了与框架相关的非规则抽样定理,得到其成立的必要条件。然后,基于所得结果,讨论非整点规则抽样的扰动问题,给出计算扰动的方法。最后,对整个论文的工作和研究成果进行了总结,讨论了一些有待于进一步研究的问题以及下一步的研究设想和目标。更多还原

【Abstract】 Since 1980’s, wavelet analysis has been a popular field in scientific research. Its application almost involves all the branches in natural science and engineering technology. Nowadays, it has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. Wavelet analysis is applied widely in signal processing and digital communications, and the sampling theorem plays a crucial role in signal processing and digital communications too: it tells us how to convert an analog sinal into a sequence of numbers. The sampling theorem in the wavelet subspaces is a popular field in modern stage. As the generalization of orthogonal bases, Parseval frame wavelet almost preserve the nice properties of orthogonal bases except the orthogonality. So far, frame theory is successfully used by many fields such as signal processing, mapping processing, data compress and sampling theorem. Thus, it is important to study Parseval frame wavelet.The dissertation is composed of two parts. In part one, we discuss the construction and property of Parseval frame wavelets and Parseval frame scaling functions. In part two, we investigate the lattic sampling theorem in subspace of L2 ( R ) and sampling theorem associated to frame in multiwavelet subspace. The important and significant results obtained in this dissertation can be sumarized as follows:1.We outline the development histories of wavelet analysis and some current research situations of wavelet theory and the sampling theorem in the wavelet subspaces. Main results of the thesis will be presented.2.Notice that all the waveltes constructed by general multiresolution analysis(GMRA) have the property of sime-orthogonal, so we firstly study the property of semi-orthogonal frame wavelet, and get the necessary conditions for the semi-orthogonal frame wavelet to hold, and prove the necessary and sufficent conditions for the semi-orthogonal frame wavelet to hold. Then, we discover that all the GMRA Parseval wavelets are equivalent to the affine system {T kψ: k∈Zd} being the Parseval frame of a closed subspace W0 such that improving the theoretical basis of GMRA. Finally, by the minimal vector-filter, we depict an important property of frame wavelets.3.We consider the Parseval frame scaling function set and MSF Parseval frame wavelets, and get several significant results. At first, we show the necessary and sufficent conditions for a measurable set S to be the Parseval frame scaling function set. Then, by discussing the property of the measurable set S, we provide a method of construction of Parseval frame scaling function set, and prove that all the Parseval frame scaling function sets are obtained by the construction method. At last, by discovering the relation between the MSF Parseval frame wavelet and the Parseval frame scaling function set, we get two important results: Firstly, each MSF Parseval frame wavelet with the additional property arise from the Parseval frame scaling function set. Secondly, for the dimention function of MSF Parseval frame waveletψsatisfies Dψ(ξ) =χ? (ξ)≤1, there must exist a Parseval frame scaling function set s such thatψ? =χBS \S. By the MSF Parseval frame wavelet defined by its Fourier transformψ? =χW, the study of the property of the wavelet is reduced to the study of the property of the set W. So, we not only provide a new construction method of the MSF Parseval frame wavelet, but also have a better understanding of wavelets in general.4.We generalize the results in [101] to the Parseval frame scaling function, and derive the necessary and sufficent conditions for the measurable set G to be the support G of the Fourier transform of the Parseval frame scaling functions. In the special case, by studying the relations among the sets G,τ( G)、12G and G \12G , we discover the difference between the MRA scaling function and the FMRA scaling function. And, by considering the support of the Fourier transform of the band-limited scaling function, we get an important property of the band-limited scaling function.5.We discuss the lattice sampling theorem based on frame in subspace of L2 ( R ). Through investigating the property of the translation operator and the dilation operator, we prove a necessary and sufficient condition for a > 0 lattice sampling theorem to hold in subspace of L2 ( R ), and obtain the formula of lattice sampling function in frequency domain.6 . We study the uniform noninteger sampling theorem and irregular sampling in multiwavelet subspace. Firstly, by Zak transform, we provide a construction method for the uniform noninteger sampling function with the additional property, and establish the uniform noninteger sampling theorem based on frame in multiwavelet subspace. Then, by irregular sampling be also useful in practice, we consider the irregular sampling theorem based on frame, and obtain the necessary condition for the irregular sampling theorem to hold in multiwavelet subspace. At last, from our obtained results, we discuss the perturbations of uniform noninteger sampling, and establish the algorithm for perturbations of uniform noninteger sampling in multiwavelet subspace.Finally , sums up the work and research results, brings forward future research considerations and objects.更多还原