NATO Advanced Study Institute

Wavelets and their Applications

August 16-29, 1992,   Il Ciocco, Italy

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	Purpose
	The subject of our ASI was the exciting and expanding field of wavelets. 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 mostly there is a wide expanse of scientific problems to explore. Inasmuch as these problems lie at the forefront of both pure mathematics and applied science, a NATO ASI on "Wavelets and their Applications" was especially pertinent at this time. 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 for 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 communications 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. Each of the above-mentioned topics was the focus of at least one of the principal conference lectures. Thus, the participants learned about how the human auditory system does a wavelet transform of the incoming sound signal, with irregular sampling theorems leading to the Wavelet Auditory Model for speech data compression and reconstruction. The connection between time-frequency and time-scale analysis was examined at length in two lectures, one by a university-based research-oriented mathematician and the other by an applied mathematician working at a major industrial research laboratory. Another fascinating talk described coherent structures of vorticity in the atmosphere as a model of air turbulence. In short, an important scientific contribution of the Institute was the enhancement of the dynamic interaction between the usually disparate communities of pure mathematicians and applied scientists.
	Acknowledgements
	Support for the conference and this volume, from the following organizations and individuals, is gratefully acknowledged:
	NATO Scientific & Environmental Affairs Division, Dr. Louis V. daCunha
	NSF, Dr. Richard P. Metcalf
	ONR, Dr. Neil Gerr
	ONREUR, Dr. Robert Ryan
	Prometheus Inc.
	University of Massachusetts at Boston