Wim Sweldens

Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions

This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions. Several types of quadratures are discussed and compared for different classes of wavelets. Since their construction using monomials is ill-conditioned, also a modified, well-conditioned construction using Chebyshev polynomials is presented. The second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions. They are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm. This is illustrated with numerical examples.

Spherical wavelets: Texture processing

Wavelets are a powerful tool for planar image processing. The resulting algorithms are straightforward, fast, and efficient. With the recently developed spherical wavelets this framework can be transposed to spherical textures. We describe a class of processing operators which are diagonal in the wavelet basis and which can be used for smoothing and enhancement. Since the wavelets (filters) are local in space and frequency, complex localized constraints and spatially varying characteristics can be incorporated easily. Examples from environment mapping and the manipulation of topography/bathymetry data are given.

Wavelet probing for compression-based segmentation

In this paper we show how wavelets can be used for data segmentation. The basic idea is to split the data into smooth segments that can be compressed separately. A fast algorithm that uses wavelets on closed sets and wavelet probing is presented.

Signal compression with smooth local trigonometric bases

We discuss smooth local trigonometric bases and their applications to signal compression. In image compression, these bases can nreduce the blocking effect that occurs in the Joint Photographic Experts Group (JPEG). We present and compare two generalizations of the original construction of Malvar, Coifman, and Meyer: biorthogonal and equal parity bases. These have the advantage that constant and linear components, respectively, can be represented efficiently. We show how they reduce blocking effects and improve the SNR.

An Overview of the Theory and Applications of Wavelets

In this paper an attempt is made to give an overview of some existing wavelet techniques. The continuous wavelet transform and several wavelet-based multiresolution techniques leading to the fast wavelet transform algorithm are briefly discussed. Different families of wavelets and their construction are discussed and compared. The essentials of two major applications are outlined: data compression and compression of linear operators.

Multiplicative and zero-crossing representations of signals

The implicit sampling theorem of Bar-David gives a representation of band limited functions using their crossings with a cosine function. This cosine function is chosen such that its difference with the original function has sufficient zero crossings for a unique representation. We show how, on an interval, this leads to a multiplicative representation involving a Riesz product. This provides an alternative to the classic additive Fourier series. We discuss stability and implementation issues. Since we have an explicit reconstruction formula, there is no need for an iterative algorithm.

New folding operators for image compression

We discuss bases formed by smooth local trigonometric functions and their applications to image compression. It is known that these bases can reduce the blocking effect that occurs in JPEG. We present and compare two generalizations of the original construction of Coifman and Meyer: biorthogonal and equal parity folding. They have the advantage that constant and linear components can be represented efficiently. We show how they reduce the blocking effect and improve the mean square error.