Stability and Independence for Multivariate Refinable Distributions

Due to their so-called time-frequency localization properties, wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. In this paper, we derive necessary and sufficient conditions for the stability of the shifts of certain compactly supported refinable functions. These conditions are in terms of the zeros of the refinement mask. Our results are actually applicable to more general distributions which are not of function type, if we generalize the notion of stability appropriately. We also provide a similar characterization of the (global) linear independence of the shifts. We present several examples illustrating our results, as well as one example in which known results on box splines are derived using the theorems of this paper.     

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.     

Stability and independence of the shifts of finitely many refinable functions

Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g., symmetry and piecewise polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this article, we characterize stability and linear independence of the shifts of a finite refinable function set in terms of the refinement mask. Several illustrative examples are provided. The characterizations given in Section 2 actually require that the refinable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide sufficient conditions on the mask to ensure minimality. The conditions are shown to be necessary also under further assumptions on the refinement mask. An example is provided illustrating how the software package MAPLE can be used to investigate at least the case of two simultaneously refinable functions.     

Homogeneous wavelets and framelets with the refinable structure

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.     

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.     

The Sobolev Regularity of Refinable Functions

Refinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms on the other. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L₂-regularity of the refinable function φ in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function φ, the relevant invariant space is proved to be finite dimensional and is completely characterized in terms of the dependence relations among the shifts of φ together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported and that the shifts of φ are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may be abysmal. Our characterization is further extended to the FSI (i.e., vector) case, to the unisotropic dilation matrix case, and to even snore general setups. We also establish corresponding results for refinable distributions.     

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.     

Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

Summary. The cascade algorithm with mask a and dilation M generates a sequence by the iterative process from a starting function where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets. Received May 5, 1999 / Revised version received June 24, 1999 / Published online June 20, 2001     

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.     

A construction of interpolating wavelets on invariant sets

We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.

     

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.     

Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

Quadratic stable wavelet bases on general meshes

In this paper, we construct stable wavelet bases with piecewise quadratic functions for certain Sobolev spaces on non-uniform meshes. These wavelets have small support with 4 or 6 non-zero coefficients. Furthermore, a simple method is developed for verifying the global stability of wavelets derived from refinable functions with a complicated structure.     

Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].     

On divergence-free wavelets

This paper is concerned with the construction of compactly supported divergence-free vector wavelets. Our construction is based on a large class of refinable functions which generate multivariate multiresolution analyses which includes, in particular, the non tensor product case.For this purpose, we develop a certain relationship between partial derivatives of refinable functions and wavelets with modifications of the coefficients in their refinement equation. In addition, we demonstrate that the wavelets we construct form a Riesz-basis for the space of divergence-free vector fields.Work supported by the Deutsche Forschungsgemeinschaft in the Graduiertenkolleg Analyse und Konstruktion in der Mathematik at the RWTH Aachen.     

Differentiability of multivariate refinable functions and factorization

The paper develops a necessary condition for the regularity of a multivariate refinable function in terms of a factorization property of the associated subdivision mask. The extension to arbitrary isotropic dilation matrices necessitates the introduction of the concepts of restricted and renormalized convergence of a subdivision scheme as well as the notion of subconvergence, i.e., the convergence of only a subsequence of the iterations of the subdivision scheme. Since, in addition, factorization methods pass even from scalar to matrix valued refinable functions, those results have to be formulated in terms of matrix refinable functions or vector subdivision schemes, respectively, in order to be suitable for iterated application. Moreover, it is shown for a particular case that the the condition is not only a necessary but also a sufficient one. Dedicated to Charles A. Micchelli, a unique person, friend, mathematician and collaborator, on the occasion of his sixtieth birthday Mathematics subject classifications (2000) 65T60, 65D99.     

Algorithms for wavelet construction on Vilenkin groups

In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.     

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.     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995