Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity Properties

This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFW's that are MSF tight frame wavelets.     

L~2(R~n)中MRA Parseval框架小波和它们的乘子

刻画了L~2(R~n)中具有扩展矩阵伸缩的广义低通滤波器和多尺度分析Parseval框架小波(缩写为MRA PFW).首先,研究了伪逆的尺度函数、广义的低通滤波器和MRA PFW,给出它们的一些刻画.接着,我们给出与MRA PFW相联系的几类乘子的一些刻画.最后,给出了一个例子来证明的结论.     

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

复合伸缩Parseval框架小波

推广了AB-多尺度分析的概念,在一定条件下,复合伸缩Parseval框架小波能被AB-多尺度分析得到.接着给出了通过古典小波构造复合伸缩Paxseval框架小波的方法.     

The Closure of the Set of Tight Frame Wavelets

We study properties of the closure of the set of tight frame wavelets. We give a necessary condition and a sufficient condition for a function to be in this closure. In particular, we show that the collection of tight frame wavelets is not dense in L ²(? ⁿ ), which answers a question posed by D. Han and D. Larson (Preprint, 2008).     

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L ²(ℝ ⁿ ) under the actions of translations from a full rank lattice and dilations by products of elements of groups A and B. A minimally supported frequency composite dilation Parseval frame wavelet has generating functions whose Fourier transforms are characteristic functions of sets contained in a lattice tiling set. Constructive proofs are used to establish the existence of minimally supported frequency composite dilation Parseval frame wavelets in arbitrary dimension using any finite group B, any full rank lattice, and an expanding matrix generating the group A and normalizing the group B. Moreover, every such system is derived from a Parseval frame multiresolution analysis. Multiple examples are provided including examples that capture directional information.      

A Characterization of Generalized Frame MRAs Deriving Orthonormal Wavelets

In 2000, Papadakis announced that any orthonormal wavelet must be derived by a generalized frame MRA （GFMRA）. In this paper, we give a characterization of GFMRAs which can derive orthonormal wavelets, and show a general approach to the constructions of non-MRA wavelets. Finally we present two examples to illustrate the theory.     

一类L2(Rd)上含有膨胀矩阵的非频谱有限的标架波

通过Rd上一种集合的刻画 ,给出了一类L2 (Rd)上含有膨胀矩阵的非频谱有限的标架波 .举例和已有结果进行了比较     

Wavelets with differential relation

Divergence-free wavelets are successfully applied to numerical solutions of Navier-Stokes equation and to analysis of incompressible flows. They closely depend on a pair of one-dimensional wavelets with some differential relations. In this paper, we point out some restrictions of those wavelets and study scaling functions with the differential relation; Wavelets and their duals are discussed; In addition to the differential relation, we are particularly interested in a class of examples with the interpolatory property; It turns out there is a connection between our examples and Micchelli’s work.     

构造三角样条小波的一种新方法

首先给出了三角样条函数及其性质,然后在此基础上给出了一种构造三角样条小波的新方法.该方法简单易行,而且构造出的小波具有许多良好的性质,如对称性(具有线性相位或广义线性相位)、良好的时频局部特性、短支集及半正交性等,这些对信号处理是非常重要的.     

Wavelets with Crystal Symmetry Shifts

A generalization of multi-dimensional wavelet theory is introduced in which the usual lattice of translational shifts is replaced by a discrete subgroup of the group of affine, area preserving, transformations of Euclidean space. The dilation matrix must now be compatible with the group of shifts. An existence theorem for a multiwavelet in the presence of a multiresolution analysis is established and examples are given to illustrate the theory with two dimensional crystal symmetry groups as shifts.     

具有特殊伸缩矩阵的三元不可分小波的构造

多元小波分析是分析和处理高维数字信号的有力工具.不可分多元小波被广泛地应用在模式识别、纹理分析和边缘检测等领域.本文给出了构造一类特殊伸缩矩阵的紧支撑三元不可分小波的算法,利用该算法得到的小波函数继承了来源于尺度函数和符号函数的对称性和消失矩性质,由于符号函数中的参数选取具有很大的自由度,因此可以根据不同的实际情况来动态地确定符号函数,从而为这类小波在信号处理方面的应用提供了便利.最后给出了相应的数值算例.     

具有特殊伸缩矩阵的三元不可分正交小波的构造

多元小波分析是分析和处理多维数字信号的有力工具.不可分多元小波被广泛地应用在模式识别、纹理分析和边缘检测等领域.给出了构造具有伸缩矩阵(101-1-110-10)的紧支撑三元不可分正交小波的算法,利用该算法得到的小波函数继承了来源于尺度函数和符号函数的对称性和消失矩性质,从而为这类小波在信号处理方面的应用提供了便利.最后给出了数值算例.     

小波分析及信源熵在股市预测分析中的应用

提出了一种预测股市行情的新指标——信源熵 ,结合小波分析和 MATLAB工具来分析预测股票的未来走势 .并用实例说明了信源熵和小波分析在预测股票的走势上有很大优越性和准确性 .     

Wavelets of Wilson Type with Arbitrary Shapes

Motivated by the Gaussian bases of Coifman and Meyer and the need of bases with arbitrary shapes which may have to be different at different locations, we derive complete characterizations of window functions and their duals for localization of all appropriate sines and cosines that give rise to biorthogonal Schauder bases, Riesz bases, and frames. In addition, when the window functions are simply integer translates of a single window function, we give an explicit formulation of its dual that generates the biorthogonal basis, regardless of the shape and support of the window function. Besides the Coifman–Meyer Gaussian bases, several other examples of wavelets of Wilson type are given.     

Characterization of Multiplier Spaces with Daubechies Wavelets

We characterize the multiplier spaces from Sobolev spaces to L² with Daubechies wavelets without using capacity.     

二元正交多项式小波的构造

本文中,给出了一个构造二元张量积正交多项式小波的构造准则,还给出了一个二元张量积正交多项式小波的例子.