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ATOMIC DECOMPOSITION BY BASIS PURSUIT (1995)
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@MISC{Chen95atomicdecomposition,
author = {Scott Shaobing Chen and David L. Donoho and Michael A. Saunders},
title = {ATOMIC DECOMPOSITION BY BASIS PURSUIT },
year = {1995}
}
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Abstract
The Time-Frequency and Time-Scale communities have recently developed a large number of overcomplete waveform dictionaries -- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the Method of Frames (MOF), Matching Pursuit (MP), and, for special dictionaries, the Best Orthogonal Basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP and BOB, including better sparsity, and super-resolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation de-noising, and multi-scale edge denoising. Basis Pursuit in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
Keyphrases
atomic decomposition basis pursuit basis pursuit linear programming large-scale optimization problem multi-scale edge denoising large number abstract harmonic analysis primal-dual logarithmic barrier method ill-posed problem wavelet packet dictionary dictionary element special dictionary several advantage best orthogonal basis overcomplete dictionary interesting relation conjugate-gradient solver overcomplete system reasonable success overcomplete waveform dictionary stationary wavelet recent advance time-scale community interior-point method cosine packet wavelet packet total variation de-noising equivalent linear program several method optimal superposition
