. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...

We propose a new progressive compression scheme for arbitrary topology, highly detailed and densely sampled meshes arising from geometry scanning. We observe that meshes consist of three distinct components: geometry, parameter, and connectivity information. The latter two do not contribute to the reduction of error in a compression setting. Using semi-regular meshes, parameter and connectivity information can be virtually eliminated. Coupled with semi-regular wavelet transforms, zerotree coding, and subdivision based reconstruction we see improvements in error by a factor four (12dB) compared to other progressive coding schemes. CR Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - hierarchy and geometric transformations; G.1.2 [Numerical Analysis]: Approximation - approximation of surfaces and contours, wavelets and fractals; I.4.2 [Image Processing and Computer Vision]: Compression (Coding) - Approximate methods Additional K...

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This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shiftinvariant spaces, in particular, covering those induced by various types of wavelets. Key words: G...

After briefly reviewing the interrelation between Riesz-bases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, K-method, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...

: A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors from L 2 (IR d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution. AMS (MOS) Subject Classifications: primary: 41A63, 46C99; secondary: 41A30, 41A15, 42B99, 46E20. Key Words and phrases: wavelets, multiresolution, shift-invariant spaces, box splines. Authors' affiliation and address: 1 Center for Mathematical Sciences University of Wisconsin-Madison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work was carried out while t...

Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain, and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(IR d), let Sk be the 2 k-dilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolu-tion of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

Abstract. This paper gives a practical method of extending an n × r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, toa square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z)hasrankrfor each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter. 1.

. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integerto -integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...