Orthonormal Compactly Supported Wavelets with Optimal Sobolev Regularity: Numerical Results

Numerical optimization is used to construct new orthonormal compactly supported wavelets with a Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (π/2,π). The results improve those obtained by I. Daubechies (1988, Comm. Pure Appl. Math.41, 909–996), H. Volkmer (1995, SIAM J. Math. Anal.26, 1075–1087), and P. G. Lemarié-Rieusset and E. Zahrouni (1998, Appl. Comput. Harmon. Anal.5, 92–105).     

Biorthogonal M -Channel Compactly Supported Wavelets

We study a class of M -channel subband coding schemes with perfect reconstruction. Along the lines of [8] and [10], we construct compactly supported biorthogonal wavelet bases of L ² (R) , with dilation factor M , associated to these schemes. In particular, we study the case of splines, and obtain explicitly simple expressions for all the relevant filters. The resulting wavelets have arbitrarily large regularity and we also obtain asymptotic estimates for the regularity exponent. September 17, 1998. Date revised: June 14, 1999. Date accepted: June 25, 1999.     

二元紧支撑正交小波的构造

§ 1.　Introduction　　SinceDAUBECHIES [1 ]gavethewellknownconstructionofunivariatecompactlysup portedorthonormalwavelets,considerableattertionhasbeenspentonconstructingmultivariatecompactlysupportedorthonormalwavelets [2— 5etc.] .Althoughmanyspecialbivariatenon separablewaveletshavebeenconstructed ,itisstillanopenproblemhowtoconstructbivariatecompactlyorthonormalwaveletsforanygivencompactlysupportedscalingfunction .Thepur poseofthispaperistoconstructcompactlysupportedorthogonalwaveletass…     

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.     

二元可分正交紧支集小波基的刻划

本文讨论了当特征函数具有某种特殊形式时,不可分二元紧支集正交小波基的存在性问题．结论为,当特征函数为所给的形式时,不可分二元紧文集正交小波基是不存在的．     

Compactly Supported Wavelets Through the Classical Umbral Calculus

We explore compactly supported scaling functions of wavelet theory by means of classical umbral calculus as reformulated by Rota and Taylor. We set a theory of orthonormal scaling umbra which leads to a very simple and elementary proof of Lawton's theorem for umbrae. When umbrae come from a wavelet setting, we recover the usual Lawton condition for the orthonormality of the integer translates of a scaling function.     

Construction of Two-Dimensional Compactly Supported Orthogonal Wavelets Filters with Linear Phase

In this paper, a large class of two-dimensional orthogonal wavelet filters, (lowpass and highpass), are presented in explicit expression. We also characterize the filters with linear phase in this case. Some examples are also given, including non-separable filters with linear phase. Received September 28, 1999, Accepted July 24, 2000     

Some Methods for Constructing Nonseparable,Orthonormal, Compactly Supported Wavelet Bases

We first show that by combining monodimensional filter banks one can obtain nonseparable filter banks. We then give necessary conditions for these filter banks to generate orthonormal and regular wavelets. Finally, we establish that some of these filter banks lead to arbitrarily smooth, nonseparable, orthonormal, compactly supported wavelet bases.     

一类紧支撑矩阵值正交小波的构造

In this paper,we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets.We obtain a necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets by means of paraunitary vector filter bank theory.A method for constructing a class of compactly supported orthogonal matrix-valued wavelets is proposed by using multiresolution analysis method and matrix theory.     

Platonic Orthonormal Wavelets

We classify all orthonormal wavelets which occur in theL²space of the faces of a platonic solid.     

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.     

Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities

We give a simple formula for the duals of the filters associated with bivariate box spline functions. We show how to construct bivariate non-separable compactly supported biorthogonal wavelets associated with box spline functions which have arbitrarily high regularities.     

Construction of Compactly SupportedM-Band Wavelets

In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples ofM-band scaling functions which are both orthonormal and cardinal forMϵ 3.     

Dimension Functions of Orthonormal Wavelets

The article is devoted to dimension functions of orthonormal wavelets on the real line with dyadic dilations. We describe properties of dimension functions and prove several characterization theorems. In addition, we provide a method of construction of dimension functions. Various new examples of dimension functions and orthonormal wavelets are included.     

多元正交小波基

本文构造了一组Haar型的多元正交小波基,它们不能用张量积的方法产生.     

Compactly Supported Correlation Functions

This article proposes compactly supported correlation functions, which parameterize the smoothness of the associated stationary and isotropic random field. The constructions are straightforward, and compact support is relevant for various ends: computationally efficient spatial prediction, fast and exact simulation, and appeal among practicioners.     

L~2[0,1]上的正交小波基

从尺度因子 M =4的正交小波基出发 ,利用折叠方法得到了 L 2 [0 ,1 ]空间的正交小波基 .这种小波不同于折叠前的小波基 ,它是完全限制在有限区间 [0 ,1 ]上 ,且保持小波基的正交性 ,并在使用过程中拥有更大的灵活性 .也可用类似方法对一般尺度小波进行折叠     

On Multivariate Compactly Supported Bi-Frames

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.