NDLI: Atomic Decomposition by Basis Pursuit

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Saunders, Michael A. Donoho, David L. Chen, Scott Shaobing
Copyright Year	1998
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 l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising.BP 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.
Starting Page	33
Ending Page	61
Page Count	29
File Format	PDF
ISSN	10648275
DOI	10.1137/S1064827596304010
e-ISSN	10957197
Issue Number	1
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	Approximation by arbitrary linear expressions $\ell^1$ norm optimization Signal theory wavelet packets interior-point methods for linear programming multiscale edges matching pursuit cosine packets total variation denoising denoising Algorithms for functional approximation time-frequency analysis wavelets Mathematical programming methods time-scale analysis overcomplete signal representation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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