Wavelets: Theory and Applications

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Daubechies ABSTRACT Daubechies will concentrate on several aspects of wavelet algorithms and their applications: a) Riesz bases of wavelets on the sphere: Recent constructions of wavelet algorithms on the sphere lead to good numerical performance; so far no proof exists that they correspond to Riesz bases for the square integrable functions on the sphere. Traditional methods for planar constructions exploit translational invariance, absent for triangulations on the sphere. b) Irregular subdivision schemes construct curves or surfaces from irregular grids, placing new grid points in positions that are not equally spaced (even locally) as the grids are refined. Similar to the equally spaced case, preservation of polynomial behavior plays an important role in determining the smoothness of the limit curve or surface. Such irregular subdivision schemes are being explored. c) Approaches to decompose wavelet filters into elementary matrices in an efficient manner, and possibly using only integers. d) Frames of wavelets can be used for multiplexing of signals, similar to CDMA, as well as for sending information efficiently over multiple channels. e) The role of smart coding strategies in realizing the full potential of wavelet-based nonlinear approximation theorems. Wavelets are a mathematical technique that allows us to view a complex structure as consisting of several, increasingly detailed layers. This "multi-resolution analysis" is analogous to viewing a painting from a distance, where only large features can be distinguished, and then, as we approach closer, perceiving more detailed, smaller features, until, when we are very close, we can detect even individual brush strokes. As a mathematical tool that has this potential to "fill in" detail at increasingly small scales, and only where needed, wavelets have turned out to be useful in computer aided graphic design, in image and data compression, and in mathematical analysis and numerical computation, in particular for composite materials. Several of the mathematical problems in this proposal are inspired by such applications of wavelets, and their solutions will widen the scope of settings in which we can use multi-resolution analysis.