Analysis and Construction of Multivariate Interpolating Refinable Function Vectors

In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this paper, we characterize a multivariate interpolating refinable function vector in terms of its mask and analyze the underlying sum rule structure of its generalized interpolatory matrix mask. We also discuss the symmetry property of multivariate interpolating refinable function vectors. Based on these results, we construct a family of univariate generalized interpolatory matrix masks with increasing orders of sum rules and with symmetry for interpolating refinable function vectors. Such a family includes several known important families of univariate refinable function vectors as special cases. Several examples of bivariate interpolating refinable function vectors with symmetry will also be presented.     

L2(Rs)中插值小波包的分裂技巧和精细分解

对具有任意伸缩矩阵A的插值加细函数，给出对应于L^2(R^s)中的小波包的一个构造方法．采样空间被直接分解来取代对加细函数的符号分解．按照这个方法构造的插值小波包能对基插值空间提供较为精细的分解，因而对自适应的插值给出较好的局部化．     

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.     

The construction ofr-regular wavelets for arbitrary dilations

Given a dilation matrix A and a natural number r we construct an associated r-regular multiresolution analysis with r-regular wavelet basis. Here a dilation is an n×n expansive matrix A (all eigenvalues λ of A satisfy |λ|>1) with integer entries. This extends a theorem of Strichartz which assumes the existence of a self-affine tiling associated with the dilation A. We also prove that regular wavelets have vanishing moments.     

Construction of multivariate compactly supported tight wavelet frames

Two simple constructive methods are presented to compute compactly supported tight wavelet frames for any given refinable function whose mask satisfies the QMF or sub-QMF conditions in the multivariate setting. We use one of our constructive methods in order to find tight wavelet frames associated with multivariate box splines, e.g., bivariate box splines on a three or four directional mesh. Moreover, a construction of tight wavelet frames with maximum vanishing moments is given, based on rational masks for the generators. For compactly supported bi-frame pairs, another simple constructive method is presented.     

Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Full rank positive matrix symbols: interpolation and orthogonality

We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case. In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive method to obtain these scaling functions. AMS subject classification (2000) 42C40, 65T60, 65D05     

The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks

In this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric Hurwitz non-interpolatory one. This brings back to a polynomial equation involving the symbol of the non-interpolatory scheme we start with. The solution of the polynomial equation here proposed, tailored for symmetric Hurwitz subdivision symbols, leads to an efficient procedure for the computation of the coefficients of the corresponding family of interpolatory masks. Several examples of interpolatory masks associated with classical approximating masks are given.     

Convergence and error estimate of cascade algorithms with infinitely supported masks in L p (? s )

The cascade algorithm plays an important role in computer graphics and wavelet analysis.In this paper,we first investigate the convergence of cascade algorithms associated with a polynomially decaying mask and a general dilation matrix in L p (R s) (1 p ∞) spaces,and then we give an error estimate of the cascade algorithms associated with truncated masks.It is proved that under some appropriate conditions if the cascade algorithm associated with a polynomially decaying mask converges in the L p-norm,then the cascade algorithms associated with the truncated masks also converge in the L p-norm.Moreover,the error between the two resulting limit functions is estimated in terms of the masks.     

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.     

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying p[`(p)]^T=1\mathbf {p}\overline{\mathbf{p}}^{T}=1 . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.     

Smoothness of multivariate refinable functions with infinitely supported masks

In this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported.     

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution

In this extension of earlier work, we point out several ways how a multiresolution analysis can be derived from a finitely supported interpolatory matrix mask which has a positive definite symbol on the unit circle except at −1. A major tool in this investigation will be subdivision schemes that are obtained by using convolution or correlation operations based on replacing the usual matrix multiplications by Kronecker products.     

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.     

Compactly supported orthonormal complex wavelets with dilation 4 and symmetry

In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that their generating wavelet functions have symmetry and the shortest possible supports with respect to their increasing orders of vanishing moments.     

Interpolatory Subdivision Schemes Induced by Box Splines

This paper is devoted to a study of interpolatory refinable functions. If a refinable function φ on ^sis continuous and fundamental, i.e., φ(0)=1 and φ(α)=0 for α ^s\{0}, then its corresponding mask bsatisfies b(0)=1 and b(2α)=0 for all α ^s\{0}. Such a refinement mask is called an interpolatory mask. We establish the existence and uniqueness of interpolatory masks which are induced by masks of box splines whose shifts are linearly independent.     

基于逼近型细分的诱导细分格式

利用逼近型细分构造插值型细分是细分领域中的一个重要问题,目前可以给出插值型细分生成函数的研究还非常少.本文给出一个生成函数的统一公式,该公式由逼近型细分的生成函数与一个子生成函数构成.该公式对应一个插值型细分或者逼近型细分,这个取决于子生成函数的选取.该公式在理论和实际中都很重要.首先,这个公式适用于任意伸缩矩阵的多元基本型细分;其次,不论是一元细分还是多元细分,推导这个统一公式都不需要求解线性方程组;再次,这个公式具有显著的几何意义,应用方便;最后,从理论上分析诱导细分的零条件和多项式再生性,本文发现这些性质不仅与逼近型细分的零条件有关,而且与逼近型细分的多项式再生性有关,从而对细分格式的构造有指导意义.本文给出3个例子来说明这个统一公式.     

Anisotropic Meshless Frames on ℝ n

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝ ⁿ . The wavelets we construct are C ^∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C ^∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.