We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to re-interpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function.

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...

. We describe wavelet methods for recovery of objects from noisy and incomplete data. The common themes: (a) the new methods utilize nonlinear operations in the wavelet domain; (b) they accomplish tasks which are not possible by traditional linear/Fourier approaches to such problems. We attempt to indicate the heuristic principles, theoretical foundations, and possible application areas for these methods. Areas covered: (1) Wavelet De-Noising. (2) Wavelet Approaches to Linear Inverse Problems. (4) Wavelet Packet De-Noising. (5) Segmented MultiResolutions. (6) Nonlinear Multi-resolutions. 1. Introduction. With the rapid development of computerized scientific instruments comes a wide variety of interesting problems for data analysis and signal processing. In fields ranging from Extragalactic Astronomy to Molecular Spectroscopy to Medical Imaging to Computer Vision, one must recover a signal, curve, image, spectrum, or density from incomplete, indirect, and noisy data. What can wavelets ...

This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit d-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions. Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods fo...

is a library of routines for wavelet analysis, wavelet-packet analysis, cosine-packet analysis and matching pursuit. The library is available free of charge over the Internet. Versions are provided for Macintosh, UNIX and Windows machines.

. We construct orthogonal and biorthogonal wavelets on a given closed subset of the real line. We also study wavelets satisfying certain types of boundary conditions. We introduce the concept of "wavelet probing ", which is closely related to our construction of wavelets. This technique allows us to very quickly perform a number of different numerical tasks associated with wavelets. x1. Introduction Wavelets and multiscale analysis have emerged in a number of different fields, from harmonic analysis and partial differential equations in pure mathematics to signal and image processing in computer science and electrical engineering. Typically a general function, signal, or image is broken up into linear combinations of translated and scaled versions of some simple, basic building blocks. Multiscale analysis comes with a natural hierarchical structure obtained by only considering the linear combinations of building blocks up to a certain scale. This hierarchical structure is particularly...

Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold cross-validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of two. The cross-validation algorithm is then extended to data sets of any length and to multi-dimensional data sets. The algorithms are compared to established threshold choosers using simulation. An application to a real data set arising from anaesthesia is presented.

We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of average-interpolation -- synthesizing a smooth function on the line having prescribed boxcar averages -- and the link between average-interpolation and Dubuc-Deslauriers interpolation. We also emphasize characterizations of smooth functions via their coefficients. We describe boundary-corrected expansions for the interval, which have a simple and revealing form. We use these results to re-interpret the empirical wavelet transform -- i.e. finite, discrete wavelet transforms of data arising from boxcar integrators (e.g. CCD devices).

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We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in R d . Our analysis is based on the theory of additive subspace correction methods and applies to finite-element- and prewavelet-schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.