Celebratio Mathematica — Daubechies

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A. Cohen, I. Daubechies, O. G. Guleryuz, and M. T. Orchard: “On the importance of combining wavelet-based nonlinear approximation with coding strategies,” IEEE Trans. Inform. Theory 48 : 7 (2002), pp. 1895–1921. MR 1929999 Zbl 1061.94004 article

This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based nonlinear 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 nonlinear approximation results do not always translate to efficient compress on strategies in a rate-distortion 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/nonlinear 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.

Albert Cohen	Related
Onur G. Guleryuz	Related
Michael T. Orchard	Related