In the past several years, two disjoint clusters of researchers have been exploring the application of time-frequency (``space-frequency" when applied to imagery) analysis tools to image coding. On one hand, image coding researchers have been developing increasingly powerful ``wavelet-based" image coding algorithms that currently far outperform non-wavelet methods. On the other hand, a growing community of mathematicians and statisticians have been studying the approximation of spaces of functions using ``nonlinear approximation" methods with wavelet basis. Results in this direction have established some fundamental advantages of the wavelet basis compared with the Fourier basis for approximating certain classes of functions.

This talk will try to bridge the gap between these two communities. We begin with an overview of the state-of-the-art in image coding, focusing on the role of wavelets in addressing issues critical to efficient compression. We then sketch some recent mathematical results in nonlinear approximation, and show how the analysis reaches similar conclusions about the important issues in image compression. While the mathematical results characterize asymptotic coding performance, we describe how realistic coding regimes can be identified for which the mathematical results provide good qualitative characterization of performance. We briefly review several state-of-the-art wavelet coding algorithms, and show how their superior performance is well predicted by the mathematical analysis. Finally, looking beyond today's top-performing wavelet algorithms, we offer a simple exercise that points to the potential for future advances.

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1996-1997 Mathematics in High Performance Computing