Construction and decomposition of biorthogonal vector-valued wavelets with compact support

In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented.     

向量值双正交小波的存在性及滤波器的构造

引进了向量值多分辨分析与向量值双正交小波的概念.讨论了向量值双正交小波的存在性.运用多分辨分析和矩阵理论,给出一类紧支撑向量值双正交小波滤波器的构造算法.最后,给出4-系数向量值双正交小波滤波器的的构造算例.     

Construction of trivariate biorthogonal compactly supported wavelets

In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonseparable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scaling functions.     

三元双正交小波滤波器的刻画

研究由三元双正交插值尺度函数构造对应的双正交小波滤波器的矩阵扩充问题.当给定的一对三元双正交尺度函数中有一个为插值函数时,利用提升思想与矩阵多相分解方法,给出一类三元双正交小波滤波器的显示构造公式和一个计算实例.讨论了三元双正交小波包的的性质.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.     

一类向量值正交小波的设计与性质

引入整数因子伸缩的向量值正交小波与向量值小波包的概念.运用仿酉向量滤波器理论和矩阵理论,给出具有整数因子伸缩的向量值正交小波存在的充要条件.提供了紧支撑向量值正交的构建算法,给出了相应的构建算例.利用时频分析方法与算子理论,刻画了一类向量值正交小波包的性质,得到了整数伸缩的向量值小波包的正交公式.     

矩阵伸缩的多元多重向量值小波包的双正交性

1引言 小波分析是二十世纪八十年代中期发展起来的一个数学分枝,其应用涉及自然科学与工程技术的许多领域[1-3].向量值小波从属多小波理论范畴.文献[4]引入向量值小波的概念,讨论了多重向量值双正交小波的存在性及其构造.Bacchelli等[5]证明了多重向量值双正交小波的存在性.文献[6]运用多重向量值双正交小波变换研究海洋涡流现象.     

Craya decomposition using compactly supported biorthogonal wavelets

We present a new local Craya–Herring decomposition of three-dimensional vector fields using compactly supported biorthogonal wavelets. Therewith vector-valued function spaces are split into two orthogonal components, i.e., curl-free and divergence-free spaces. The latter is further decomposed into toroidal and poloidal parts to decorrelate horizontal from vertical contributions which are of particular interest in geophysical turbulence. Applications are shown for isotropic, rotating and stratified turbulent flows. A comparison between isotropic and anisotropic orthogonal Craya–Herring wavelets, built in Fourier space and thus not compactly supported, is also given.     

向量值正交小波的构造与向量值小波包的特征

The notion of vector-valued multiresolution analysis is introduced and the concept of orthogonal vector-valued wavelets with 3-scale is proposed.A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is given by means of paraunitary vector filter bank theory.An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented.Their characteristics is discussed by virtue of operator theory,time-frequency method.Moreover,it is shown how to design various orthonormal bases of space L2(R,Cn) from these wavelet packets.     

The Wavelet Element Method Part II. Realization and Additional Features in 2D and 3D

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D, and 3D cases. We address additional features such as symmetry, vanishing moments, and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.     

Construction of trivariate compactly supported biorthogonal box spline wavelets

We give a formula for the duals of the masks associated with trivariate box spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions. The biorthogonal wavelets may have arbitrarily high regularities.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.     

双正交多重小波的一种构造方法

多重小波是近年来新兴的小波研究方向,它具有许多一维小波所不具备的优越性质．完全正交的多重小波在构造上有很大的难度,所以在许多应用中人们都可以用双正交多重小波作为分析的工具     

向量值正交小波包

引进对应于2尺度向量值尺度函数的多分辨分析和向量值小波的概念.给出向量值小波包的定义及其构造算法,研究了向量值正交小波包的正交性,并讨论了空间L2(R,CN)的正交分解.     

多尺度紧支撑向量值正交小波的构造

引入了多尺度向量值多分辨分析．利用矩阵理论,给出多尺度紧支撑向量值正交小波存在的必要条件及其构造方法．     

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing-as a model problem-image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.     

A study on orthogonal two-direction vector-valued wavelets and two-direction wavelet packets

In this paper, the notion of two-direction vector-valued multiresolution analysis and the two-direction orthogonal vector-valued wavelets are introduced. The definition for two-direction orthogonal vector-valued wavelet packets is proposed. An algorithm for constructing a class of two-direction orthogonal vector-valued compactly supported wavelets corresponding to the two-direction orthogonal vector-valued compactly supported scaling functions is proposed by virtue of matrix theory and time-frequency analysis method. The properties of the two-direction vector-valued wavelet packets are investigated. At last, the direct decomposition relation for space L₂(R)^r is presented.