Journal
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
Volume 31, Issue 1, Pages 59-73Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.acha.2010.10.002
Keywords
l(1)-minimization; Basis pursuit; Restricted isometry property; Redundant dictionaries; l(1) -analysis
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Funding
- ONR [N00014-10-1-0599, N00014-08-1-0749]
- NSF [DMS EMSW21-VIGRE]
- Division of Computing and Communication Foundations
- Direct For Computer & Info Scie & Enginr [0965028] Funding Source: National Science Foundation
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This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is via)le in this context, but also that accurate recovery is possible via an l(1)-analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary and our results may be the first of this kind. We discuss practical examples and the implications of our results on those applications, and complement our study by demonstrating the potential of l(1)-analysis for such problems. (C) 2010 Elsevier Inc. All rights reserved.
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