DATA-COMPRESSION AND HARMONIC-ANALYSIS

In this paper we review some recent interactions between harmonic anal ysis and data compression, The story goes hack of course to Shannon's R(D) theory in the case of Gaussian stationary processes, which says t hat transforming into a Fourier basis followed by block coding gives a n optimal lossy compression technique; practical developments like tra nsform-based image compression have been inspired by this result. In t his paper we also discuss connections perhaps less familiar to the Inf ormation Theory community, growing out of the field of harmonic analys is. Recent harmonic analysis constructions, such as wavelet transforms and Gabor transforms, are essentially optimal transforms for transfor m coding in certain settings. Some of these transforms are under consi deration for Future compression standards. We discuss some of the less ons of harmonic analysis in this century. Typically the problems and a chievements of this field have involved goals that were not obviously related to practical data compression, and have used a language not im mediately accessible to outsiders. Nevertheless, through an extensive generalization of what Shannon called the ''sampling theorem,'' harmon ic analysis has succeeded in developing new forms of functional repres entation which turn out to have significant data compression interpret ations. We explain why harmonic analysis has interacted with data comp ression, and we describe some interesting recent ideas in the field th at may affect data compression in the future.