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An Invitation to Compressive Sensing

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A Mathematical Introduction to Compressive Sensing

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This first chapter formulates the objectives of compressive sensing. It introduces the standard compressive problem studied throughout the book and reveals its ubiquity in many concrete situations by providing a selection of motivations, applications, and extensions of the theory. It concludes with an overview of the book that summarizes the content of each of the following chapters.

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References

  1. M. Aharon, M. Elad, A. Bruckstein, The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)

    Article  Google Scholar 

  2. M. Anthony, P. Bartlett, Neural Network Learning: Theoretical Foundations (Cambridge University Press, Cambridge, 1999)

    Book  MATH  Google Scholar 

  3. F. Bach, R. Jenatton, J. Mairal, G. Obozinski, Optimization with sparsity-inducing penalties. Found. Trends Mach. Learn. 4(1), 1–106 (2012)

    Article  Google Scholar 

  4. W. Bajwa, J. Haupt, A.M. Sayeed, R. Nowak, Compressed channel sensing: a new approach to estimating sparse multipath channels. Proc. IEEE 98(6), 1058–1076 (June 2010)

    Article  Google Scholar 

  5. R.G. Baraniuk, Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)

    Article  Google Scholar 

  6. R.G. Baraniuk, V. Cevher, M. Duarte, C. Hedge, Model-based compressive sensing. IEEE Trans. Inform. Theor. 56, 1982–2001 (April 2010)

    Article  Google Scholar 

  7. J.J. Benedetto, P.J.S.G. Ferreira (eds.), Modern Sampling Theory: Mathematics and Applications. Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 2001)

    Google Scholar 

  8. R. Berinde, A. Gilbert, P. Indyk, H. Karloff, M. Strauss, Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Proc. of 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 798–805, 2008

    Google Scholar 

  9. P. Bickel, Y. Ritov, A. Tsybakov, Simultaneous analysis of lasso and Dantzig selector. Ann. Stat. 37(4), 1705–1732 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Bruckstein, D.L. Donoho, M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Bühlmann, S. van de Geer, Statistics for High-dimensional Data. Springer Series in Statistics (Springer, Berlin, 2011)

    Google Scholar 

  12. E.J. Candès, Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006

    Google Scholar 

  13. E.J. Candès, B. Recht, Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. E.J. Candès, J. Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6(2), 227–254 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theor. 52(2), 489–509 (2006)

    Article  MATH  Google Scholar 

  16. E.J. Candès, T. Tao, Decoding by linear programming. IEEE Trans. Inform. Theor. 51(12), 4203–4215 (2005)

    Article  MATH  Google Scholar 

  17. E.J. Candès, T. Tao, Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theor. 52(12), 5406–5425 (2006)

    Article  Google Scholar 

  18. E.J. Candès, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351, (2007)

    Google Scholar 

  19. E.J. Candès, T. Tao, The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theor. 56(5), 2053–2080 (2010)

    Article  Google Scholar 

  20. E.J. Candès, M. Wakin, An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)

    Article  Google Scholar 

  21. A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, An introduction to total variation for image analysis. In Theoretical Foundations and Numerical Methods for Sparse Recovery, ed. by M. Fornasier. Radon Series on Computational and Applied Mathematics, vol. 9 (de Gruyter, Berlin, 2010), pp. 263–340

    Google Scholar 

  22. A. Chambolle, R.A. DeVore, N.-Y. Lee, B.J. Lucier, Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Chambolle, P.-L. Lions, Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vis. 40, 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by Basis Pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Cohen, Numerical Analysis of Wavelet Methods (North-Holland, Amsterdam, 2003)

    MATH  Google Scholar 

  27. A. Cohen, I. Daubechies, R. DeVore, G. Kerkyacharian, D. Picard, Capturing Ridge Functions in High Dimensions from Point Queries. Constr. Approx. 35, 225–243 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Cohen, R. DeVore, S. Foucart, H. Rauhut, Recovery of functions of many variables via compressive sensing. In Proc. SampTA 2011, Singapore, 2011

    Google Scholar 

  29. G. Cormode, S. Muthukrishnan, Combinatorial algorithms for compressed sensing. In CISS, Princeton, 2006

    Google Scholar 

  30. F. Cucker, S. Smale, On the mathematical foundations of learning. Bull. Am. Math. Soc., New Ser. 39(1), 1–49 (2002)

    Google Scholar 

  31. F. Cucker, D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint. Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  32. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (SIAM, Philadelphia, 1992)

    Google Scholar 

  33. I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Davies, Y. Eldar, Rank awareness in joint sparse recovery. IEEE Trans. Inform. Theor. 58(2), 1135–1146 (2012)

    Article  MathSciNet  Google Scholar 

  35. C. De Mol, E. De Vito, L. Rosasco, Elastic-net regularization in learning theory. J. Complex. 25(2), 201–230 (2009)

    Article  MATH  Google Scholar 

  36. R.A. DeVore, G. Petrova, P. Wojtaszczyk, Approximation of functions of few variables in high dimensions. Constr. Approx. 33(1), 125–143 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. D.L. Donoho, De-noising by soft-thresholding. IEEE Trans. Inform. Theor. 41(3), 613–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  38. D.L. Donoho, Compressed sensing. IEEE Trans. Inform. Theor. 52(4), 1289–1306 (2006)

    Article  MathSciNet  Google Scholar 

  39. D.L. Donoho, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35(4), 617–652 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. D.L. Donoho, M. Elad, Optimally sparse representations in general (non-orthogonal) dictionaries via ℓ 1 minimization. Proc. Nat. Acad. Sci. 100(5), 2197–2202 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. D.L. Donoho, M. Elad, On the stability of the basis pursuit in the presence of noise. Signal Process. 86(3), 511–532 (2006)

    Article  MATH  Google Scholar 

  42. D.L. Donoho, M. Elad, V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theor. 52(1), 6–18 (2006)

    Article  MathSciNet  Google Scholar 

  43. D.L. Donoho, X. Huo, Uncertainty principles and ideal atomic decompositions. IEEE Trans. Inform. Theor. 47(7), 2845–2862 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. D.L. Donoho, I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26(3), 879–921 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. D.L. Donoho, G. Kutyniok, Microlocal analysis of the geometric separation problem. Comm. Pure Appl. Math. 66(1), 1–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. D.L. Donoho, B. Logan, Signal recovery and the large sieve. SIAM J. Appl. Math. 52(2), 577–591 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  47. D.L. Donoho, P. Stark, Recovery of a sparse signal when the low frequency information is missing. Technical report, Department of Statistics, University of California, Berkeley, June 1989

    Google Scholar 

  48. D.L. Donoho, P. Stark, Uncertainty principles and signal recovery. SIAM J. Appl. Math. 48(3), 906–931 (1989)

    Article  MathSciNet  Google Scholar 

  49. D.L. Donoho, J. Tanner, Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  50. D.L. Donoho, J. Tanner, Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. 102(27), 9446–9451 (2005)

    Article  MathSciNet  Google Scholar 

  51. D.L. Donoho, J. Tanner, Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22(1), 1–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. D.L. Donoho, M. Vetterli, R.A. DeVore, I. Daubechies, Data compression and harmonic analysis. IEEE Trans. Inform. Theor. 44(6), 2435–2476 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. R. Dorfman, The detection of defective members of large populations. Ann. Stat. 14, 436–440 (1943)

    Article  Google Scholar 

  54. D.-Z. Du, F. Hwang, Combinatorial Group Testing and Its Applications (World Scientific, Singapore, 1993)

    MATH  Google Scholar 

  55. M. Duarte, M. Davenport, D. Takhar, J. Laska, S. Ting, K. Kelly, R.G. Baraniuk, Single-Pixel Imaging via Compressive Sampling. IEEE Signal Process. Mag. 25(2), 83–91 (2008)

    Article  Google Scholar 

  56. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, New York, 2010)

    Book  Google Scholar 

  57. M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736 –3745 (2006)

    Article  MathSciNet  Google Scholar 

  58. M. Elad, A.M. Bruckstein, A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inform. Theor. 48(9), 2558–2567 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  59. Y. Eldar, G. Kutyniok (eds.), Compressed Sensing: Theory and Applications (Cambridge University Press, New York, 2012)

    Google Scholar 

  60. Y. Eldar, M. Mishali, Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inform. Theor. 55(11), 5302–5316 (2009)

    Article  MathSciNet  Google Scholar 

  61. Y. Eldar, H. Rauhut, Average case analysis of multichannel sparse recovery using convex relaxation. IEEE Trans. Inform. Theor. 56(1), 505–519 (2010)

    Article  MathSciNet  Google Scholar 

  62. J. Ender, On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)

    Article  MATH  Google Scholar 

  63. H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Springer, New York, 1996)

    Book  MATH  Google Scholar 

  64. A. Fannjiang, P. Yan, T. Strohmer, Compressed remote sensing of sparse objects. SIAM J. Imag. Sci. 3(3), 596–618 (2010)

    Article  MathSciNet  Google Scholar 

  65. M. Fazel, Matrix Rank Minimization with Applications. PhD thesis, 2002

    Google Scholar 

  66. P.J.S.G. Ferreira, J.R. Higgins, The establishment of sampling as a scientific principle—a striking case of multiple discovery. Not. AMS 58(10), 1446–1450 (2011)

    MathSciNet  MATH  Google Scholar 

  67. M. Fornasier, H. Rauhut, Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46(2), 577–613 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  68. M. Fornasier, H. Rauhut, Compressive sensing. In Handbook of Mathematical Methods in Imaging, ed. by O. Scherzer (Springer, New York, 2011), pp. 187–228

    Chapter  Google Scholar 

  69. M. Fornasier, K. Schnass, J. Vybiral, Learning Functions of Few Arbitrary Linear Parameters in High Dimensions. Found. Comput. Math. 12, 229–262 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  70. J.J. Fuchs, On sparse representations in arbitrary redundant bases. IEEE Trans. Inform. Theor. 50(6), 1341–1344 (2004)

    Article  Google Scholar 

  71. A. Garnaev, E. Gluskin, On widths of the Euclidean ball. Sov. Math. Dokl. 30, 200–204 (1984)

    MATH  Google Scholar 

  72. Q. Geng, J. Wright, On the local correctness of ℓ 1-minimization for dictionary learning. Preprint (2011)

    Google Scholar 

  73. A. Gilbert, M. Strauss, Analysis of data streams. Technometrics 49(3), 346–356 (2007)

    Article  MathSciNet  Google Scholar 

  74. A.C. Gilbert, S. Muthukrishnan, S. Guha, P. Indyk, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing, STOC ’02, pp. 152–161, ACM, New York, NY, USA, 2002

    Google Scholar 

  75. A.C. Gilbert, S. Muthukrishnan, M.J. Strauss, Approximation of functions over redundant dictionaries using coherence. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’03, pp. 243–252. SIAM, Philadelphia, PA, 2003

    Google Scholar 

  76. A.C. Gilbert, M. Strauss, J.A. Tropp, R. Vershynin, One sketch for all: fast algorithms for compressed sensing. In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC ’07, pp. 237–246, ACM, New York, NY, USA, 2007

    Google Scholar 

  77. E. Gluskin, Norms of random matrices and widths of finite-dimensional sets. Math. USSR-Sb. 48, 173–182 (1984)

    Article  MATH  Google Scholar 

  78. R. Gribonval, Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture. In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP’02), vol. 3, pp. 3057–3060, 2002

    Google Scholar 

  79. R. Gribonval, M. Nielsen, Sparse representations in unions of bases. IEEE Trans. Inform. Theor. 49(12), 3320–3325 (2003)

    Article  MathSciNet  Google Scholar 

  80. R. Gribonval, H. Rauhut, K. Schnass, P. Vandergheynst, Atoms of all channels, unite! Average case analysis of multi-channel sparse recovery using greedy algorithms. J. Fourier Anal. Appl. 14(5), 655–687 (2008)

    MathSciNet  MATH  Google Scholar 

  81. R. Gribonval, K. Schnass, Dictionary identification—sparse matrix-factorisation via l 1-minimisation. IEEE Trans. Inform. Theor. 56(7), 3523–3539 (2010)

    Article  MathSciNet  Google Scholar 

  82. K. Gröchenig, Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 2001)

    Google Scholar 

  83. D. Gross, Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theor. 57(3), 1548–1566 (2011)

    Article  Google Scholar 

  84. D. Gross, Y.-K. Liu, S.T. Flammia, S. Becker, J. Eisert, Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010)

    Article  Google Scholar 

  85. C. Güntürk, M. Lammers, A. Powell, R. Saab, Ö. Yilmaz, Sobolev duals for random frames and ΣΔ quantization of compressed sensing measurements. Found. Comput. Math. 13(1), 1–36, Springer-Verlag (2013)

    Google Scholar 

  86. M. Haacke, R. Brown, M. Thompson, R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design (Wiley-Liss, New York, 1999)

    Google Scholar 

  87. J. Haldar, D. Hernando, Z. Liang, Compressed-sensing MRI with random encoding. IEEE Trans. Med. Imag. 30(4), 893–903 (2011)

    Article  Google Scholar 

  88. H. Hassanieh, P. Indyk, D. Katabi, E. Price, Nearly optimal sparse Fourier transform. In Proceedings of the 44th Symposium on Theory of Computing, STOC ’12, pp. 563–578, ACM, New York, NY, USA, 2012

    Google Scholar 

  89. H. Hassanieh, P. Indyk, D. Katabi, E. Price, Simple and practical algorithm for sparse Fourier transform. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’12, pp. 1183–1194. SIAM, 2012

    Google Scholar 

  90. T. Hemant, V. Cevher, Learning non-parametric basis independent models from point queries via low-rank methods. Preprint (2012)

    Google Scholar 

  91. W. Hendee, C. Morgan, Magnetic resonance imaging Part I—Physical principles. West J. Med. 141(4), 491–500 (1984)

    Google Scholar 

  92. M. Herman, T. Strohmer, High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57(6), 2275–2284 (2009)

    Article  MathSciNet  Google Scholar 

  93. F. Herrmann, M. Friedlander, O. Yilmaz, Fighting the curse of dimensionality: compressive sensing in exploration seismology. Signal Process. Mag. IEEE 29(3), 88–100 (2012)

    Article  Google Scholar 

  94. F. Herrmann, H. Wason, T. Lin, Compressive sensing in seismic exploration: an outlook on a new paradigm. CSEG Recorder 36(4), 19–33 (2011)

    Google Scholar 

  95. J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, vol. 1 (Clarendon Press, Oxford, 1996)

    MATH  Google Scholar 

  96. J.R. Higgins, R.L. Stens, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, vol. 2 (Oxford University Press, Oxford, 1999)

    MATH  Google Scholar 

  97. D. Holland, M. Bostock, L. Gladden, D. Nietlispach, Fast multidimensional NMR spectroscopy using compressed sensing. Angew. Chem. Int. Ed. 50(29), 6548–6551 (2011)

    Article  Google Scholar 

  98. W. Huffman, V. Pless, Fundamentals of Error-correcting Codes (Cambridge University Press, Cambridge, 2003)

    Book  MATH  Google Scholar 

  99. M. Hügel, H. Rauhut, T. Strohmer, Remote sensing via ℓ 1-minimization. Found. Comput. Math., to appear. (2012)

    Google Scholar 

  100. P. Indyk, A. Gilbert, Sparse recovery using sparse matrices. Proc. IEEE 98(6), 937–947 (2010)

    Article  Google Scholar 

  101. M. Iwen, Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  102. M. Iwen, Improved approximation guarantees for sublinear-time Fourier algorithms. Appl. Comput. Harmon. Anal. 34(1), 57–82 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  103. M. Iwen, A. Gilbert, M. Strauss, Empirical evaluation of a sub-linear time sparse DFT algorithm. Commun. Math. Sci. 5(4), 981–998 (2007)

    MathSciNet  MATH  Google Scholar 

  104. L. Jacques, J. Laska, P. Boufounos, R. Baraniuk, Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors. IEEE Trans. Inform. Theor. 59(4), 2082–2102 (2013)

    Article  MathSciNet  Google Scholar 

  105. A.J. Jerri, The Shannon sampling theorem—its various extensions and applications: A tutorial review. Proc. IEEE. 65(11), 1565–1596 (1977)

    Article  MATH  Google Scholar 

  106. B. Kashin, Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR, Izv. 11, 317–333 (1977)

    Google Scholar 

  107. J. Laska, P. Boufounos, M. Davenport, R. Baraniuk, Democracy in action: quantization, saturation, and compressive sensing. Appl. Comput. Harmon. Anal. 31(3), 429–443 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  108. Y. Liu, Universal low-rank matrix recovery from Pauli measurements. In NIPS, pp. 1638–1646, 2011

    Google Scholar 

  109. A. Llagostera Casanovas, G. Monaci, P. Vandergheynst, R. Gribonval, Blind audiovisual source separation based on sparse redundant representations. IEEE Trans. Multimed. 12(5), 358–371 (August 2010)

    Article  Google Scholar 

  110. B. Logan, Properties of High-Pass Signals. PhD thesis, Columbia University, New York, 1965

    Google Scholar 

  111. M. Lustig, D.L. Donoho, J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)

    Article  Google Scholar 

  112. S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  113. S. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, 1987)

    Google Scholar 

  114. M. Mishali, Y.C. Eldar, From theory to practice: Sub-nyquist sampling of sparse wideband analog signals. IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (April 2010)

    Article  Google Scholar 

  115. M. Murphy, M. Alley, J. Demmel, K. Keutzer, S. Vasanawala, M. Lustig, Fast ℓ 1-SPIRiT Compressed Sensing Parallel Imaging MRI: Scalable Parallel Implementation and Clinically Feasible Runtime. IEEE Trans. Med. Imag. 31(6), 1250–1262 (2012)

    Article  Google Scholar 

  116. B.K. Natarajan, Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  117. D. Needell, R. Ward, Stable image reconstruction using total variation minimization. Preprint (2012)

    Google Scholar 

  118. E. Novak, Optimal recovery and n-widths for convex classes of functions. J. Approx. Theor. 80(3), 390–408 (1995)

    Article  MATH  Google Scholar 

  119. E. Novak, H. Woźniakowski, Tractability of Multivariate Problems. Vol. 1: Linear Information. EMS Tracts in Mathematics, vol. 6 (European Mathematical Society (EMS), Zürich, 2008)

    Google Scholar 

  120. G. Pfander, H. Rauhut, J. Tanner, Identification of matrices having a sparse representation. IEEE Trans. Signal Process. 56(11), 5376–5388 (2008)

    Article  MathSciNet  Google Scholar 

  121. G. Pfander, H. Rauhut, J. Tropp, The restricted isometry property for time-frequency structured random matrices. Prob. Theor. Relat. Field. to appear

    Google Scholar 

  122. Y. Plan, R. Vershynin, One-bit compressed sensing by linear programming. Comm. Pure Appl. Math. 66(8), 1275–1297 (2013)

    Article  MATH  Google Scholar 

  123. Y. Plan, R. Vershynin, Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Trans. Inform. Theor. 59(1), 482–494 (2013)

    Article  MathSciNet  Google Scholar 

  124. L. Potter, E. Ertin, J. Parker, M. Cetin, Sparsity and compressed sensing in radar imaging. Proc. IEEE 98(6), 1006–1020 (2010)

    Article  Google Scholar 

  125. D. Potts, M. Tasche, Parameter estimation for exponential sums by approximate Prony method. Signal Process. 90(5), 1631–1642 (2010)

    Article  MATH  Google Scholar 

  126. R. Prony, Essai expérimental et analytique sur les lois de la Dilatabilité des fluides élastiques et sur celles de la Force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. École Polytechnique 1, 24–76 (1795)

    Google Scholar 

  127. R. Ramlau, G. Teschke, Sparse recovery in inverse problems. In Theoretical Foundations and Numerical Methods for Sparse Recovery, ed. by M. Fornasier. Radon Series on Computational and Applied Mathematics, vol. 9 (de Gruyter, Berlin, 2010), pp. 201–262

    Google Scholar 

  128. M. Raphan, E. Simoncelli, Optimal denoising in redundant representation. IEEE Trans. Image Process. 17(8), 1342–1352 (2008)

    Article  MathSciNet  Google Scholar 

  129. H. Rauhut, Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal. 22(1), 16–42 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  130. H. Rauhut, On the impossibility of uniform sparse reconstruction using greedy methods. Sampl. Theor. Signal Image Process. 7(2), 197–215 (2008)

    MathSciNet  MATH  Google Scholar 

  131. H. Rauhut, Compressive sensing and structured random matrices. In Theoretical Foundations and Numerical Methods for Sparse Recovery, ed. by M. Fornasier. Radon Series on Computational and Applied Mathematics, vol. 9 (de Gruyter, Berlin, 2010), pp. 1–92

    Google Scholar 

  132. H. Rauhut, G.E. Pfander, Sparsity in time-frequency representations. J. Fourier Anal. Appl. 16(2), 233–260 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  133. H. Rauhut, R. Ward, Sparse Legendre expansions via ℓ 1-minimization. J. Approx. Theor. 164(5), 517–533 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  134. B. Recht, A simpler approach to matrix completion. J. Mach. Learn. Res. 12, 3413–3430 (2011)

    MathSciNet  Google Scholar 

  135. B. Recht, M. Fazel, P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  136. J.K. Romberg, Imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 14–20 (March, 2008)

    Article  Google Scholar 

  137. R. Rubinstein, M. Zibulevsky, M. Elad, Double sparsity: learning sparse dictionaries for sparse signal approximation. IEEE Trans. Signal Process. 58(3, part 2), 1553–1564 (2010)

    Google Scholar 

  138. M. Rudelson, R. Vershynin, Geometric approach to error-correcting codes and reconstruction of signals. Int. Math. Res. Not. 64, 4019–4041 (2005)

    Article  MathSciNet  Google Scholar 

  139. L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MATH  Google Scholar 

  140. F. Santosa, W. Symes, Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput. 7(4), 1307–1330 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  141. B. Schölkopf, A. Smola, Learning with Kernels (MIT Press, Cambridge, 2002)

    Google Scholar 

  142. Y. Shrot, L. Frydman, Compressed sensing and the reconstruction of ultrafast 2D NMR data: Principles and biomolecular applications. J. Magn. Reson. 209(2), 352–358 (2011)

    Article  Google Scholar 

  143. J.-L. Starck, E.J. Candès, D.L. Donoho, The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)

    Article  MathSciNet  Google Scholar 

  144. J.-L. Starck, F. Murtagh, J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University Press, Cambridge, 2010)

    Book  Google Scholar 

  145. T. Strohmer, B. Friedlander, Analysis of sparse MIMO radar. Preprint (2012)

    Google Scholar 

  146. G. Tauböck, F. Hlawatsch, D. Eiwen, H. Rauhut, Compressive estimation of doubly selective channels in multicarrier systems: leakage effects and sparsity-enhancing processing. IEEE J. Sel. Top. Sig. Process. 4(2), 255–271 (2010)

    Article  Google Scholar 

  147. H. Taylor, S. Banks, J. McCoy, Deconvolution with the ℓ 1-norm. Geophysics 44(1), 39–52 (1979)

    Article  Google Scholar 

  148. V. Temlyakov, Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 20 (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  149. R. Tibshirani, Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  150. J. Traub, G. Wasilkowski, H. Woźniakowski, Information-based Complexity. Computer Science and Scientific Computing (Academic Press Inc., Boston, MA, 1988) With contributions by A.G.Werschulz, T. Boult.

    Google Scholar 

  151. J.A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theor. 50(10), 2231–2242 (2004)

    Article  MathSciNet  Google Scholar 

  152. J.A. Tropp, Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Process. 86(3), 589–602 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  153. J.A. Tropp, Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theor. 51(3), 1030–1051 (2006)

    Article  MathSciNet  Google Scholar 

  154. J.A. Tropp, On the linear independence of spikes and sines. J. Fourier Anal. Appl. 14(5–6), 838–858 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  155. J.A. Tropp, A.C. Gilbert, M.J. Strauss, Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit. Signal Process. 86(3), 572–588 (2006)

    Article  MATH  Google Scholar 

  156. J.A. Tropp, J.N. Laska, M.F. Duarte, J.K. Romberg, R.G. Baraniuk, Beyond Nyquist: Efficient sampling of sparse bandlimited signals. IEEE Trans. Inform. Theor. 56(1), 520–544 (2010)

    Article  MathSciNet  Google Scholar 

  157. S. Vasanawala, M. Alley, B. Hargreaves, R. Barth, J. Pauly, M. Lustig, Improved pediatric MR imaging with compressed sensing. Radiology 256(2), 607–616 (2010)

    Article  Google Scholar 

  158. Y. Wiaux, L. Jacques, G. Puy, A. Scaife, P. Vandergheynst, Compressed sensing imaging techniques for radio interferometry. Mon. Not. Roy. Astron. Soc. 395(3), 1733–1742 (2009)

    Article  Google Scholar 

  159. P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge University Press, Cambridge, 1997)

    Book  MATH  Google Scholar 

  160. G. Wright, Magnetic resonance imaging. IEEE Signal Process. Mag. 14(1), 56–66 (1997)

    Article  Google Scholar 

  161. J. Wright, A. Yang, A. Ganesh, S. Sastry, Y. Ma, Robust Face Recognition via Sparse Representation. IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 210–227 (2009)

    Article  Google Scholar 

  162. J. Zou, A.C. Gilbert, M. Strauss, I. Daubechies, Theoretical and experimental analysis of a randomized algorithm for sparse Fourier transform analysis. J. Comput. Phys. 211, 572–595 (2005)

    Article  MathSciNet  Google Scholar 

  163. A. Zymnis, S. Boyd, E.J. Candès, Compressed sensing with quantized measurements. IEEE Signal Process. Lett. 17(2), 149–152 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    Simon Foucart

  2. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Aachen, Germany

    Holger Rauhut

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Foucart, S., Rauhut, H. (2013). An Invitation to Compressive Sensing. In: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4948-7_1

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