On the importance of combining wavelet-based nonlinear approximation with coding strategies

This paper provides a mathematical analysis of transform compression in its relationship to linear and non-linear approximation theory. Contrasting linear and non-linear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based non-linear approximation. These classes include locally smooth signals that have singularities, and provide a model for many signals encountered in practice, in particular for images. However, we also show that non-linear approximation results do not always translate to efficient compression strategies in a ratedistortion sense. Based on this observation, we construct compression techniques and formulate the family of functions/stochastic processes for which they provide efficient descriptions in a rate-distortion sense. We show that this family invariably leads to Besov spaces, yielding a natural relationship among Besov smoothness, linear/non-linear approximation order, and compression performance in a rate-distortion sense. The designed compression techniques show similarities to modern high-performance transform codecs, allowing us to establish relevant rate-distortion estimates and identify performance limits.