Wavelets and Their Applications

It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. Hence, if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow. Werner Heisenberg This volume contains papers presented at the August 1992 NATO Advanced Study Institute on Wavelets and Their Applications. The conference was held at the beautiful Il Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy. Once again we gathered at this idyllic spot to explore and extend the reciprocity between mathematics and engineering. The dynamic interaction between world-renowned scientists from the usu ally disparate communities of pure mathematicians and applied scientists, which occurred at our 1989 and 1991 ASI's, continued at this meeting. Wavelet theory and technology is in an important growth stage at which theoretical and practical results are being compared with existing methods. There have been spectacular wavelet successes and sobering comparisons with traditional ideas-but still there is a wide expanse of scientific problems to explore. Since these problems lie at the forefront of both pure mathematics and applied science, our NATO ASI was especially pertinent at this time.

Klappentext

Wavelets and wavelet packets provide a theory analogous to Fourier analysis and tools analogous to coherent state methods. Among their numerous applications, wavelets have been used to data compression in both image and sound processing. They are intimately related to splines, and wavelet applications in spline theory are significant. Wavelets have become a tool in analyzing fractals and iterative schemes associated with dynamical systems. Signal processing methods such as quadrature mirror filters go hand in hand with wavelet techniques in studying a host of communcations problems. The profound issues of classical turbulence are being studied using wavelet packets. Both wavelet packet software and wavelet transform microchips are now available. There are also applications of wavelet theory in theoretical physics, oil exploration, irregular sampling, and singular integral operators. br/ Many of the world's experts in the field of wavelets were principal speakers at the ASI, and their papers appear in this volume. Furthermore, these renowned scientists addressed their talks to an audience which consisted of a broad spectrum of pure and applied mathematicians, as well as a diverse group of engineers and scientists. Thus, the reader has the opportunity to both learn or reinforce the fundamental concepts from the individuals who have created and developed the blossoming field of wavelets, and to see them discuss in accessible terms their profound contributions and ideas for future research. br/

Inhalt
From a wavelet auditory model to definitions of the Fourier transform.- Some stability properties of wavelets and scaling functions.- Convergence: Fourier series vs. wavelet expansions.- Regularity analysis of functions and random processes using wavelets.- The generalized modulus of continuity and wavelets.- Approximate frames and the narrowband multitarget radar problem.- Time-scale analyses and self-similar stochastic processes.- Multiscale statistical modeling: isotropic processes on a homogeneous tree.- A wavelet-type approach to image analysis and vision.- Analog VLSI network models, cortical linking neural network models, and quantum holographic neural technology.- Clifford wavelets, Hardy spaces, and elliptic boundary value problems.- Wavelet transforms and downward continuation.- Bilinear time-frequency distributions.- Time-frequency localization operators of Cohen's class.- Some remarks about the scalograms of wavelet transform coefficients.- Problems in Gabor representation.- On the extension of the Heisenberg group to incorporate multiscale resolution.- Signal processing and compression with wavelet packets.- Texture analysis with Hermite basic elementary functions.