Combining Seminorms in Adaptive Lifting Schemes and Applications to Image Analysis and Compression
Journal of Mathematical Imaging and Vision
An edge-sensing predictor in wavelet lifting structures for lossless image coding
Journal on Image and Video Processing
Representation and compression of multidimensional piecewise functions using surflets
IEEE Transactions on Information Theory
Critically sampled composite wavelets
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
The optimal free knot spline approximation of stochastic differential equations with additive noise
Journal of Computational and Applied Mathematics
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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