Making Do with Less: An Introduction to Compressed Sensing

This article offers an accessible but rigorous and essentially self-contained account of some of the central ideas in compressed sensing, aimed at nonspecialists and undergraduates who have had linear algebra and some probability. The basic premise is first illustrated by considering the problem of detecting a few defective items in a large set. We then build up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.

[1]  Dimitris Achlioptas,et al.  Database-friendly random projections: Johnson-Lindenstrauss with binary coins , 2003, J. Comput. Syst. Sci..

[2]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[3]  Brian Hayes The best bits , 2009 .

[4]  Ting Sun,et al.  Single-pixel imaging via compressive sampling , 2008, IEEE Signal Process. Mag..

[5]  V. Chandar A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property , 2008 .

[6]  Piotr Indyk,et al.  Sparse Recovery Using Sparse Matrices , 2010, Proceedings of the IEEE.

[7]  F. Downton,et al.  Introduction to Mathematical Statistics , 1959 .

[8]  Farrokh Marvasti,et al.  Deterministic Construction of Binary, Bipolar, and Ternary Compressed Sensing Matrices , 2009, IEEE Transactions on Information Theory.

[9]  D. Mackenzie,et al.  Compressed sensing makes every pixel count , 2009 .

[10]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[11]  J. Romberg,et al.  Imaging via Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[12]  Emmanuel J. Candès,et al.  Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions , 2004, Found. Comput. Math..

[13]  P. Shearer,et al.  Compressive sensing of the Tohoku‐Oki Mw 9.0 earthquake: Frequency‐dependent rupture modes , 2011 .

[14]  D. Donoho,et al.  Atomic Decomposition by Basis Pursuit , 2001 .

[15]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[16]  E. Candès,et al.  Sparsity and incoherence in compressive sampling , 2006, math/0611957.

[17]  Aswin C. Sankaranarayanan,et al.  Compressive Sensing , 2008, Computer Vision, A Reference Guide.

[18]  Richard G. Baraniuk,et al.  Compressive Sensing DNA Microarrays , 2008, EURASIP J. Bioinform. Syst. Biol..

[19]  Richard G. Baraniuk,et al.  A simple proof that random matrices are democratic , 2009, ArXiv.

[20]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[21]  Michael C. Ferris,et al.  Linear programming with MATLAB , 2007, MPS-SIAM series on optimization.

[22]  Kai Lai Chung,et al.  Elementary Probability Theory , 1974 .

[23]  Otmar Scherzer,et al.  Handbook of Mathematical Methods in Imaging , 2015, Handbook of Mathematical Methods in Imaging.

[24]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[25]  Frédéric Lesage,et al.  The Application of Compressed Sensing for , 2009 .

[26]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[27]  Stephen Becker,et al.  Practical Compressed Sensing: Modern data acquisition and signal processing , 2011 .

[28]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[29]  Necdet Batir,et al.  Inequalities for the gamma function , 2008 .

[30]  Dirk Werner,et al.  What's Happening in the Mathematical Sciences? , 2001 .

[31]  E. Candès The restricted isometry property and its implications for compressed sensing , 2008 .

[32]  Ronald A. DeVore,et al.  Deterministic constructions of compressed sensing matrices , 2007, J. Complex..

[33]  Emmanuel J. Cand The Restricted Isometry Property and Its Implications for Compressed Sensing , 2008 .

[34]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[35]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[36]  Massimo Fornasier,et al.  Compressive Sensing , 2015, Handbook of Mathematical Methods in Imaging.

[37]  D. Donoho For most large underdetermined systems of equations, the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution , 2006 .

[38]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[39]  R. DeVore,et al.  Compressed sensing and best k-term approximation , 2008 .