对称反对称紧支撑正交多小波的构造

对于给定的对称反对称紧支撑正交r重尺度函数,给出一种构造对称反对称紧支撑正交多小波的方法.通过此方法构造的多小波与尺度函数有相同的对称性与反对称性,并且给出算例.     

高维紧支撑正交对称的小波

基于仿酉矩阵的对称扩充方法,该文提出了一种尺度因子为3的紧支撑高维正交对称小波构造算法.即设φ(x)∈L~2(R~d)是尺度因子为3的紧支撑d维正交对称尺度函数,P(ξ)是它的两尺度符号,p_(0,v)(ξ)为P(ξ)的相位符号.首先提出一种向量的对称正交变换,应用对称正交变换对3~d维向量(p_(0,v)(ξ))_v,v∈E_d的分量进行对称化.通过仿酉矩阵的对称扩充,给出了3~d-1个紧支撑高维正交对称小波构造.这种方法构造的小波支撑不超过尺度函数的支撑.最后给出一个构造算例.     

矩阵的正交扩充与多正交小波

给出一种由a尺度紧支撑正交多尺度函数构造短支撑正交多小波的方法,其过程仅仅应用矩阵的正交扩充和求解方程组。如果r重尺度函数的支撑区间较大,可以将其转化为ar重短支撑情形,从而使得本文的方法适用于任意紧支撑正交多小波的构造,文后给出多小波的构造算例。     

区间[-1,1]上的a尺度双正交多小波的构造

本文研究了一元a尺度紧支撑、双正交多小波的构造.在区间[-1,1],给出了利用a尺度双正交尺度向量构造a尺度双正交多小波的推导过程得到了一种有效的小波构造算法,并给出了数值算例.     

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.     

三维插值对称正交小波的构造

高维小波分析是分析和处理多维数字信号的有力工具.基于任意的三维正交尺度函数及相应的正交小波,提出一种构造三维插值对称尺度函数和对称小波的方法,并建立了多维信号采样定理,这一点在信号处理中具有很好的应用价值.最后给出了数值算例.     

具有线性相位的4带正交滤波器的参数化

该文得到了具有线性相位的4带正交尺度滤波器的参数化形式,同时给出了构造相应的小波滤波器的一个简单的构造方法.应用所给出的参数化形式,得到了具有紧支撑的对称正交的尺度函数,进而也获得了相应的小波.     

不同尺度下多项式滤波器的优化算法

１　　引　言 在小波分析的应用中,紧支撑正交对称的小波是非常可贵的．尤其是对称性,它在实际应用中具有非常重要的意义．但Daubechies的具有紧支撑正交小波无任何对称性和反对称性（除Haar小波外）．为了克服这一不足,崔锦泰和王建忠［１］提出了样条小波,样条小波用失去正交性换来了小波的对称性．A.Cohen［２］等引入了双正交小波似乎解决了这一问题,但它需要两个对偶的小波．匡正［３］等采用了小波的分式滤波器构造出了既正交又对称的小波,但却没有有限的支撑区间．本文欲采用优化的方法给出了一种构造具有任意正则性的多项式…     

区间[-1，1]上的α尺度双正交多小波的构造

本文研究了一元α尺度紧支撑、双正交多小波的构造．在区间[-1，1]，给出了利用α尺度双正交尺度向量构造α尺度双正交多小波的推导过程得到了一种有效的小波构造算法，并给出了数值算例．     

紧支撑正交对称和反对称小波的构造

１．引言 近年来,人们分别从数学和信号的观点对正交小波进行了广泛的研究．尤其是２尺度小波,它克服了短时 Ｆｏｕｒｉｅｒ变换的一些缺陷．目前最常用的 ２尺度小波是 Ｄａｕｂｅｃｈｉｅｓ 小波,但 ２尺度小波也存在一些问题：如 Ｄａｕｂｅｃｈｉｅｓ［２］已证明了除 Ｈａａｒ小波外不存在既正交又对称的紧支撑 ２尺度小波．因此人们提出了 ａ尺度小波理论［３］－［６］,文献［４］－［６］对 ４尺度小波迸行研究．本文的目的是研究４尺度因子时紧支撑正交对称和反对称小波的构造方法．并指出对同一紧支撑正交对称尺度函数而言,…     

Biorthogonal radial multiresolution in dimension three

In this paper, we present the definition and the relative theorems of the biorthogonal radial multiresolution in dimension three. Unlike the orthogonal case, there exist real-valued dual radial scaling functions with compact support in the biorthogonal case. The associated Mallat algorithm can be simply performed in terms of classical biorthogonal filters.     

Parameterization and algebraic structure of 3-band orthogonal wavelet systems

In this paper, a complete parameterization for the 3-band compact wavelet systems is presented. Using the parametric result, a program of the filterbank design is completed, which can give not only the filterbanks but also the graphs of all possible scaling functions and their corresponding wavelets. Especially some symmetric wavelets with small supports are given. Finally an algebraic structure for this kind of wavelet systems is characterized.     

The M-band cardinal orthogonal scaling function

The M-band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory, signal processing, computer graphics, and numerical algorithms. In this paper, we provide a complete mathematical analysis for the M-band symmetric cardinal orthogonal scaling function. Firstly, we generalize some results of the cardinal orthogonal scaling function from the special case M=2 to the most general case M?2. Also, we find some new results. Secondly, we obtain the characterizations of the M-band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory.     

New Stable Biorthogonal Spline-Wavelets on the Interval

In this paper we present the construction of new stable biorthogonal spline-wavelet bases on the interval [0, 1] for arbitrary choice of spline-degree. As starting point, we choose the well-known family of compactly supported biorthogonal spline-wavelets presented by Cohen, Daubechies and Feauveau. Firstly, we construct biorthogonal MRAs (multiresolution analysis) on [0, 1]. The primal MRA consists of spline-spaces concerning equidistant, dyadic partitions of [0, 1], the so called Schoenberg-spline bases. Thus, the full degree of polynomial reproduction is preserved on the primal side. The construction, that we present for the boundary scaling functions on the dual side, guarantees the same for the dual side. In particular, the new boundary scaling functions on both, the primal and the dual side have staggered supports. Further, the MRA spaces satisfy certain Jackson- and Bernstein-inequalities, which lead by general principles to the result, that the associated wavelets are in fact L ₂([0, 1])-stable. The wavelets however are computed with aid of the method of stable completion. Due to the compact support of all occurring functions, the decomposition and reconstruction transforms can be implemented efficiently with sparse matrices. We also illustrate how bases with complementary or homogeneous boundary conditions can be easily derived from our construction.     

Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

Compactly Supported Tight Frames Associated with Refinable Functions

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L²(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ¹,…,ψ^N} for L²(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

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.