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.     

Subspace Design of Compactly Supported Orthonormal Wavelets

We derive a new matrix parameterization of compactly supported orthonormal wavelets where the coefficients of the wavelet filter are the solution of a linear system of equations that is parameterized by an arbitrary vector. The parameterization shows that the vector of the wavelet filter coefficients is the kernel of a subspace of the condition matrix row-space. This property is exploited to develop a new design procedure for orthonormal wavelets of compact support. The proposed parameterization also describes the class of two-channel orthogonal filter banks where in this case we have two extra degrees of freedom in the design. The effectiveness of the proposed procedure is illustrated by design examples of common orthonormal wavelets.     

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     

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).     

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.     

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.     

Frobenius Endomorphisms in the Umbral Calculus

Frobenius operators F_n are introduced on sequences of binomial type. The Laguerre polynomials are essentially characterized by the property that F_n coincides with n-fold binomial convolution.     

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.     

Computer algebra and Umbral Calculus

Rota's Umbral Calculus uses sequences of Sheffer polynomials to count certain combinatorial objects. We review this theory and some of its generalizations in light of our computer implementation (Maple V.3). A Mathematica version of this package is being developed in parallel.     

哑运算的概率解释

利用随机变量的矩以及期望运算,给出了哑运算一种简单、自然的概率解释,并且得到了Abel恒等式的一个广泛哑运算证明.     

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.     

Combinatorics for Wavelets: The Umbral Refinement Equation

The refinement equation is the most fundamental equation in wavelet theory.In this article, we study its combinatorial meanings and analogs. We show that the invariant properties of the Cauchy generating function are well adapted to the translation and dilation operators in the refinement equation. This leads to the discovery of analytic scaling functions and wavelets. The classical umbral calculus (or symbolic calculus) provides a powerful tool for moments analysis and defines the combinatorial analog of the ordinary refinement equation—the umbral refinement equation. By developing the existing theory of classical umbral calculus, we are able to solve the umbral refinement equation in a purely umbral manner. Many classical results in wavelet analysis are reestablished in the context of umbral calculus.     

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.     

Construction of Compactly Supported Shearlet Frames

Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital settings. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.     

Compactly Supported Frames for Decomposition Spaces

In this article we study a construction of compactly supported frame expansions for decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces. This is done by showing that finite linear combinations of shifts and dilates of a single function with sufficient decay in both direct and frequency space can constitute a frame for Triebel-Lizorkin type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to Triebel-Lizorkin type spaces and the associated modulation spaces. Next, we prove that two function systems which are sufficiently close have an almost diagonal “change of frame coefficient” matrix. Finally, we approximate to an arbitrary degree an already known frame for Triebel-Lizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both direct and frequency space.