This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M; c) refinable function OE in terms of the eigenvalues and eigenvectors of the transition operator W cau defined by the autocorrelation c au of its refinement mask c; where M is an arbitrary dilation matrix. Another consequence is that if the shifts of OE form a Riesz basis, then W cau has a unique eigenvector of eigenvalue 1; and all its other eigenvalues lie inside the unit circle. The general theory is applied to two-dimensional nonseparable (M, c) refinable functions whose masks are constructed from Daubechies' conjugate quadrature filters.

. In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 p 1) smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1 \Gamma 2r; 2r \Gamma 1] s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br . Here the sequence br on Z is the unique univariate interpolatory refinement mask which is supported on [1 \Gamma 2r; 2r \Gamma 1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal...

A convenient setting for studying multiscale techniques in various areas of applications is usually based on a sequence of nested closed subspaces of some function space F which is often referred to as multiresolution analysis. The concept of wavelets is a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of the orthogonal difference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of IR s , say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding to biorthogonal wavelets. The main objective of this paper is therefore to study possibly non-orthogonal but in a certain sense stable and even local decompositions of nested spaces and to develop tools which are not necessarily confined to ...

Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increases the size of the matrix and therefore make the computation very expensive. In this paper, we shall present a simple algorithm to efficiently numerically compute the smoothness exponent of a symmetric refinable function with a general dilation matrix. By taking into account of symmetry of a refinable function, our algorithm greatly reduces the size of the matrix and enables us to numerically compute the Sobolev smoothness exponents of a large class of symmetric refinable functions. Step by step numerically stable algorithms and details of the numerical implementation are given. To illustrate our results by performing some numerical experiments, we construct a family of dyadic interpolatory masks in any dimension and we compute the smoothness exponents of their refinable functions in dimension three. Several examples will also be presented for computing smoothness exponents of symmetric refinable functions on the quincunx lattice and on the hexagonal lattice.

Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power ( 1 55), which, while slower than the optimal rate ( 2), is much better than ( 1) achieved with wavelets, but at similar complexity. Index Terms—Directional vanishing moments, directionlets, filter banks, geometry, multidirection, multiresolution, separable filtering, sparse image representation, wavelets. I.

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levels of detail are widely used for large-scale two- and three-dimensional data sets. We present a four-dimensional multiresolution approach for time-varying volume data. This approach supports a hierarchy with spatial and temporal scalability.

We present here the mathematical foundations of the wavelet transform, multiresolution analysis and discrete-time transforms and algorithms. This article serves as background material for the rest of the special issue. 1 Introduction When we deal with a given physical object, we encounter many of its different faces, or, representations. For example, we can represent numbers in various systems depending on the application; in everyday life, we use the decimal system, while for use in computers we employ the binary representation. Consequently, in many fields, such as numerical analysis or signal processing, a preliminary task is to find an adapted representation of the signal that may be particularly suitable for a problem at hand. For example, in images, one of the common tasks is to attempt a representation that will facilitate extraction of features. A way to obtain a specific representation is to decompose a signal x into elementary building blocks x i , of some importance, as fo...

Abstract. The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L2(Rd) from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness. Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are

Refinable functions underlie the theory and constructions of wavelet systems on the one hand, and the theory and convergence analysis of uniform subdivision algorithms. 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 L 2 -regularity of the refinable function OE 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 OE, the relevant invariant space is proved to be finite dimensional, and is completely characterized in terms of the dependence relat...