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Structured random measurements in signal processing
Author(s) -
Krahmer Felix,
Rauhut Holger
Publication year - 2014
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201410010
Subject(s) - compressed sensing , circulant matrix , algorithm , random matrix , computer science , matrix (chemical analysis) , restricted isometry property , signal (programming language) , rank (graph theory) , gaussian , signal processing , fourier transform , mathematics , digital signal processing , mathematical analysis , physics , combinatorics , eigenvalues and eigenvectors , materials science , quantum mechanics , computer hardware , composite material , programming language
Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)