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

• Published in: IEEE transactions on information theory Vol. 48; no. 7; pp. 1895 - 1921
• Main Authors: Cohen, A., Daubechies, I., Guleryuz, O.G., Orchard, M.T.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2002; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Approximation; Approximation methods; Compressing; Data compression; Image coding; Mathematical analysis; Mathematical models; Nonlinearity; Rate distortion theory; Stochastic processes; Strategy; Transform coding; Transforms; Wavelet transforms
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2002.1013132

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.