An Invitation to Compressive Sensing

  • Simon Foucart
  • Holger Rauhut
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


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.


sparsity compressibility algorithms random matrices stability single-pixel camera magnetic resonance imaging radar sampling theory sparse approximation error correction statistics and machine learning low-rank matrix recovery and matrix completion 


  1. 5.
    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)CrossRefGoogle Scholar
  2. 18.
    M. Anthony, P. Bartlett, Neural Network Learning: Theoretical Foundations (Cambridge University Press, Cambridge, 1999)zbMATHCrossRefGoogle Scholar
  3. 23.
    F. Bach, R. Jenatton, J. Mairal, G. Obozinski, Optimization with sparsity-inducing penalties. Found. Trends Mach. Learn. 4(1), 1–106 (2012)CrossRefGoogle Scholar
  4. 27.
    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)CrossRefGoogle Scholar
  5. 29.
    R.G. Baraniuk, Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)CrossRefGoogle Scholar
  6. 30.
    R.G. Baraniuk, V. Cevher, M. Duarte, C. Hedge, Model-based compressive sensing. IEEE Trans. Inform. Theor. 56, 1982–2001 (April 2010)CrossRefGoogle Scholar
  7. 39.
    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. 41.
    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, 2008Google Scholar
  9. 48.
    P. Bickel, Y. Ritov, A. Tsybakov, Simultaneous analysis of lasso and Dantzig selector. Ann. Stat. 37(4), 1705–1732 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 73.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 76.
    P. Bühlmann, S. van de Geer, Statistics for High-dimensional Data. Springer Series in Statistics (Springer, Berlin, 2011)Google Scholar
  12. 84.
    E.J. Candès, Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006Google Scholar
  13. 90.
    E.J. Candès, B. Recht, Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 92.
    E.J. Candès, J. Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6(2), 227–254 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 94.
    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)zbMATHCrossRefGoogle Scholar
  16. 96.
    E.J. Candès, T. Tao, Decoding by linear programming. IEEE Trans. Inform. Theor. 51(12), 4203–4215 (2005)zbMATHCrossRefGoogle Scholar
  17. 97.
    E.J. Candès, T. Tao, Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theor. 52(12), 5406–5425 (2006)CrossRefGoogle Scholar
  18. 98.
    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. 99.
    E.J. Candès, T. Tao, The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theor. 56(5), 2053–2080 (2010)CrossRefGoogle Scholar
  20. 100.
    E.J. Candès, M. Wakin, An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)CrossRefGoogle Scholar
  21. 104.
    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. 263–340Google Scholar
  22. 105.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 106.
    A. Chambolle, P.-L. Lions, Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 107.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 114.
    S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by Basis Pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 122.
    A. Cohen, Numerical Analysis of Wavelet Methods (North-Holland, Amsterdam, 2003)zbMATHGoogle Scholar
  27. 124.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 125.
    A. Cohen, R. DeVore, S. Foucart, H. Rauhut, Recovery of functions of many variables via compressive sensing. In Proc. SampTA 2011, Singapore, 2011Google Scholar
  29. 131.
    G. Cormode, S. Muthukrishnan, Combinatorial algorithms for compressed sensing. In CISS, Princeton, 2006Google Scholar
  30. 133.
    F. Cucker, S. Smale, On the mathematical foundations of learning. Bull. Am. Math. Soc., New Ser. 39(1), 1–49 (2002)Google Scholar
  31. 134.
    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. 137.
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (SIAM, Philadelphia, 1992)Google Scholar
  33. 138.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 143.
    M. Davies, Y. Eldar, Rank awareness in joint sparse recovery. IEEE Trans. Inform. Theor. 58(2), 1135–1146 (2012)MathSciNetCrossRefGoogle Scholar
  35. 147.
    C. De Mol, E. De Vito, L. Rosasco, Elastic-net regularization in learning theory. J. Complex. 25(2), 201–230 (2009)zbMATHCrossRefGoogle Scholar
  36. 149.
    R.A. DeVore, G. Petrova, P. Wojtaszczyk, Approximation of functions of few variables in high dimensions. Constr. Approx. 33(1), 125–143 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 150.
    D.L. Donoho, De-noising by soft-thresholding. IEEE Trans. Inform. Theor. 41(3), 613–627 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 152.
    D.L. Donoho, Compressed sensing. IEEE Trans. Inform. Theor. 52(4), 1289–1306 (2006)MathSciNetCrossRefGoogle Scholar
  39. 154.
    D.L. Donoho, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35(4), 617–652 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 155.
    D.L. Donoho, M. Elad, Optimally sparse representations in general (non-orthogonal) dictionaries via 1 minimization. Proc. Nat. Acad. Sci. 100(5), 2197–2202 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 156.
    D.L. Donoho, M. Elad, On the stability of the basis pursuit in the presence of noise. Signal Process. 86(3), 511–532 (2006)zbMATHCrossRefGoogle Scholar
  42. 157.
    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)MathSciNetCrossRefGoogle Scholar
  43. 158.
    D.L. Donoho, X. Huo, Uncertainty principles and ideal atomic decompositions. IEEE Trans. Inform. Theor. 47(7), 2845–2862 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 159.
    D.L. Donoho, I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26(3), 879–921 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 160.
    D.L. Donoho, G. Kutyniok, Microlocal analysis of the geometric separation problem. Comm. Pure Appl. Math. 66(1), 1–47 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 161.
    D.L. Donoho, B. Logan, Signal recovery and the large sieve. SIAM J. Appl. Math. 52(2), 577–591 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 163.
    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 1989Google Scholar
  48. 164.
    D.L. Donoho, P. Stark, Uncertainty principles and signal recovery. SIAM J. Appl. Math. 48(3), 906–931 (1989)MathSciNetCrossRefGoogle Scholar
  49. 165.
    D.L. Donoho, J. Tanner, Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 166.
    D.L. Donoho, J. Tanner, Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. 102(27), 9446–9451 (2005)MathSciNetCrossRefGoogle Scholar
  51. 167.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 170.
    D.L. Donoho, M. Vetterli, R.A. DeVore, I. Daubechies, Data compression and harmonic analysis. IEEE Trans. Inform. Theor. 44(6), 2435–2476 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 171.
    R. Dorfman, The detection of defective members of large populations. Ann. Stat. 14, 436–440 (1943)CrossRefGoogle Scholar
  54. 173.
    D.-Z. Du, F. Hwang, Combinatorial Group Testing and Its Applications (World Scientific, Singapore, 1993)zbMATHGoogle Scholar
  55. 174.
    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)CrossRefGoogle Scholar
  56. 179.
    M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, New York, 2010)CrossRefGoogle Scholar
  57. 180.
    M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736 –3745 (2006)MathSciNetCrossRefGoogle Scholar
  58. 181.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 182.
    Y. Eldar, G. Kutyniok (eds.), Compressed Sensing: Theory and Applications (Cambridge University Press, New York, 2012)Google Scholar
  60. 183.
    Y. Eldar, M. Mishali, Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inform. Theor. 55(11), 5302–5316 (2009)MathSciNetCrossRefGoogle Scholar
  61. 184.
    Y. Eldar, H. Rauhut, Average case analysis of multichannel sparse recovery using convex relaxation. IEEE Trans. Inform. Theor. 56(1), 505–519 (2010)MathSciNetCrossRefGoogle Scholar
  62. 185.
    J. Ender, On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)zbMATHCrossRefGoogle Scholar
  63. 186.
    H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Springer, New York, 1996)zbMATHCrossRefGoogle Scholar
  64. 189.
    A. Fannjiang, P. Yan, T. Strohmer, Compressed remote sensing of sparse objects. SIAM J. Imag. Sci. 3(3), 596–618 (2010)MathSciNetCrossRefGoogle Scholar
  65. 190.
    M. Fazel, Matrix Rank Minimization with Applications. PhD thesis, 2002Google Scholar
  66. 195.
    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)MathSciNetzbMATHGoogle Scholar
  67. 203.
    M. Fornasier, H. Rauhut, Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46(2), 577–613 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 204.
    M. Fornasier, H. Rauhut, Compressive sensing. In Handbook of Mathematical Methods in Imaging, ed. by O. Scherzer (Springer, New York, 2011), pp. 187–228CrossRefGoogle Scholar
  69. 206.
    M. Fornasier, K. Schnass, J. Vybiral, Learning Functions of Few Arbitrary Linear Parameters in High Dimensions. Found. Comput. Math. 12, 229–262 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 215.
    J.J. Fuchs, On sparse representations in arbitrary redundant bases. IEEE Trans. Inform. Theor. 50(6), 1341–1344 (2004)CrossRefGoogle Scholar
  71. 219.
    A. Garnaev, E. Gluskin, On widths of the Euclidean ball. Sov. Math. Dokl. 30, 200–204 (1984)zbMATHGoogle Scholar
  72. 221.
    Q. Geng, J. Wright, On the local correctness of 1-minimization for dictionary learning. Preprint (2011)Google Scholar
  73. 222.
    A. Gilbert, M. Strauss, Analysis of data streams. Technometrics 49(3), 346–356 (2007)MathSciNetCrossRefGoogle Scholar
  74. 223.
    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, 2002Google Scholar
  75. 224.
    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, 2003Google Scholar
  76. 225.
    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, 2007Google Scholar
  77. 227.
    E. Gluskin, Norms of random matrices and widths of finite-dimensional sets. Math. USSR-Sb. 48, 173–182 (1984)zbMATHCrossRefGoogle Scholar
  78. 238.
    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, 2002Google Scholar
  79. 239.
    R. Gribonval, M. Nielsen, Sparse representations in unions of bases. IEEE Trans. Inform. Theor. 49(12), 3320–3325 (2003)MathSciNetCrossRefGoogle Scholar
  80. 241.
    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)MathSciNetzbMATHGoogle Scholar
  81. 242.
    R. Gribonval, K. Schnass, Dictionary identification—sparse matrix-factorisation via l 1-minimisation. IEEE Trans. Inform. Theor. 56(7), 3523–3539 (2010)MathSciNetCrossRefGoogle Scholar
  82. 244.
    K. Gröchenig, Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 2001)Google Scholar
  83. 245.
    D. Gross, Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theor. 57(3), 1548–1566 (2011)CrossRefGoogle Scholar
  84. 246.
    D. Gross, Y.-K. Liu, S.T. Flammia, S. Becker, J. Eisert, Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010)CrossRefGoogle Scholar
  85. 249.
    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. 252.
    M. Haacke, R. Brown, M. Thompson, R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design (Wiley-Liss, New York, 1999)Google Scholar
  87. 255.
    J. Haldar, D. Hernando, Z. Liang, Compressed-sensing MRI with random encoding. IEEE Trans. Med. Imag. 30(4), 893–903 (2011)CrossRefGoogle Scholar
  88. 261.
    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, 2012Google Scholar
  89. 262.
    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, 2012Google Scholar
  90. 266.
    T. Hemant, V. Cevher, Learning non-parametric basis independent models from point queries via low-rank methods. Preprint (2012)Google Scholar
  91. 267.
    W. Hendee, C. Morgan, Magnetic resonance imaging Part I—Physical principles. West J. Med. 141(4), 491–500 (1984)Google Scholar
  92. 268.
    M. Herman, T. Strohmer, High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57(6), 2275–2284 (2009)MathSciNetCrossRefGoogle Scholar
  93. 269.
    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)CrossRefGoogle Scholar
  94. 270.
    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. 271.
    J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, vol. 1 (Clarendon Press, Oxford, 1996)zbMATHGoogle Scholar
  96. 272.
    J.R. Higgins, R.L. Stens, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, vol. 2 (Oxford University Press, Oxford, 1999)zbMATHGoogle Scholar
  97. 278.
    D. Holland, M. Bostock, L. Gladden, D. Nietlispach, Fast multidimensional NMR spectroscopy using compressed sensing. Angew. Chem. Int. Ed. 50(29), 6548–6551 (2011)CrossRefGoogle Scholar
  98. 282.
    W. Huffman, V. Pless, Fundamentals of Error-correcting Codes (Cambridge University Press, Cambridge, 2003)zbMATHCrossRefGoogle Scholar
  99. 283.
    M. Hügel, H. Rauhut, T. Strohmer, Remote sensing via 1-minimization. Found. Comput. Math., to appear. (2012)Google Scholar
  100. 285.
    P. Indyk, A. Gilbert, Sparse recovery using sparse matrices. Proc. IEEE 98(6), 937–947 (2010)CrossRefGoogle Scholar
  101. 287.
    M. Iwen, Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 288.
    M. Iwen, Improved approximation guarantees for sublinear-time Fourier algorithms. Appl. Comput. Harmon. Anal. 34(1), 57–82 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 289.
    M. Iwen, A. Gilbert, M. Strauss, Empirical evaluation of a sub-linear time sparse DFT algorithm. Commun. Math. Sci. 5(4), 981–998 (2007)MathSciNetzbMATHGoogle Scholar
  104. 290.
    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)MathSciNetCrossRefGoogle Scholar
  105. 294.
    A.J. Jerri, The Shannon sampling theorem—its various extensions and applications: A tutorial review. Proc. IEEE. 65(11), 1565–1596 (1977)zbMATHCrossRefGoogle Scholar
  106. 299.
    B. Kashin, Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR, Izv. 11, 317–333 (1977)Google Scholar
  107. 316.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 330.
    Y. Liu, Universal low-rank matrix recovery from Pauli measurements. In NIPS, pp. 1638–1646, 2011Google Scholar
  109. 331.
    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)CrossRefGoogle Scholar
  110. 332.
    B. Logan, Properties of High-Pass Signals. PhD thesis, Columbia University, New York, 1965Google Scholar
  111. 338.
    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)CrossRefGoogle Scholar
  112. 342.
    S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)zbMATHCrossRefGoogle Scholar
  113. 344.
    S. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, 1987)Google Scholar
  114. 353.
    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)CrossRefGoogle Scholar
  115. 358.
    M. Murphy, M. Alley, J. Demmel, K. Keutzer, S. Vasanawala, M. Lustig, Fast 1-SPIRiT Compressed Sensing Parallel Imaging MRI: Scalable Parallel Implementation and Clinically Feasible Runtime. IEEE Trans. Med. Imag. 31(6), 1250–1262 (2012)CrossRefGoogle Scholar
  116. 359.
    B.K. Natarajan, Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 364.
    D. Needell, R. Ward, Stable image reconstruction using total variation minimization. Preprint (2012)Google Scholar
  118. 370.
    E. Novak, Optimal recovery and n-widths for convex classes of functions. J. Approx. Theor. 80(3), 390–408 (1995)zbMATHCrossRefGoogle Scholar
  119. 371.
    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. 384.
    G. Pfander, H. Rauhut, J. Tanner, Identification of matrices having a sparse representation. IEEE Trans. Signal Process. 56(11), 5376–5388 (2008)MathSciNetCrossRefGoogle Scholar
  121. 385.
    G. Pfander, H. Rauhut, J. Tropp, The restricted isometry property for time-frequency structured random matrices. Prob. Theor. Relat. Field. to appearGoogle Scholar
  122. 393.
    Y. Plan, R. Vershynin, One-bit compressed sensing by linear programming. Comm. Pure Appl. Math. 66(8), 1275–1297 (2013)zbMATHCrossRefGoogle Scholar
  123. 394.
    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)MathSciNetCrossRefGoogle Scholar
  124. 397.
    L. Potter, E. Ertin, J. Parker, M. Cetin, Sparsity and compressed sensing in radar imaging. Proc. IEEE 98(6), 1006–1020 (2010)CrossRefGoogle Scholar
  125. 401.
    D. Potts, M. Tasche, Parameter estimation for exponential sums by approximate Prony method. Signal Process. 90(5), 1631–1642 (2010)zbMATHCrossRefGoogle Scholar
  126. 402.
    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. 406.
    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. 201–262Google Scholar
  128. 407.
    M. Raphan, E. Simoncelli, Optimal denoising in redundant representation. IEEE Trans. Image Process. 17(8), 1342–1352 (2008)MathSciNetCrossRefGoogle Scholar
  129. 408.
    H. Rauhut, Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal. 22(1), 16–42 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 409.
    H. Rauhut, On the impossibility of uniform sparse reconstruction using greedy methods. Sampl. Theor. Signal Image Process. 7(2), 197–215 (2008)MathSciNetzbMATHGoogle Scholar
  131. 411.
    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. 1–92Google Scholar
  132. 412.
    H. Rauhut, G.E. Pfander, Sparsity in time-frequency representations. J. Fourier Anal. Appl. 16(2), 233–260 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 416.
    H. Rauhut, R. Ward, Sparse Legendre expansions via 1-minimization. J. Approx. Theor. 164(5), 517–533 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 417.
    B. Recht, A simpler approach to matrix completion. J. Mach. Learn. Res. 12, 3413–3430 (2011)MathSciNetGoogle Scholar
  135. 418.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 427.
    J.K. Romberg, Imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 14–20 (March, 2008)CrossRefGoogle Scholar
  137. 429.
    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. 431.
    M. Rudelson, R. Vershynin, Geometric approach to error-correcting codes and reconstruction of signals. Int. Math. Res. Not. 64, 4019–4041 (2005)MathSciNetCrossRefGoogle Scholar
  139. 436.
    L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)zbMATHCrossRefGoogle Scholar
  140. 441.
    F. Santosa, W. Symes, Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput. 7(4), 1307–1330 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 444.
    B. Schölkopf, A. Smola, Learning with Kernels (MIT Press, Cambridge, 2002)Google Scholar
  142. 447.
    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)CrossRefGoogle Scholar
  143. 450.
    J.-L. Starck, E.J. Candès, D.L. Donoho, The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)MathSciNetCrossRefGoogle Scholar
  144. 451.
    J.-L. Starck, F. Murtagh, J. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  145. 455.
    T. Strohmer, B. Friedlander, Analysis of sparse MIMO radar. Preprint (2012)Google Scholar
  146. 468.
    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)CrossRefGoogle Scholar
  147. 469.
    H. Taylor, S. Banks, J. McCoy, Deconvolution with the 1-norm. Geophysics 44(1), 39–52 (1979)CrossRefGoogle Scholar
  148. 472.
    V. Temlyakov, Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 20 (Cambridge University Press, Cambridge, 2011)Google Scholar
  149. 473.
    R. Tibshirani, Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58(1), 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  150. 474.
    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. 476.
    J.A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theor. 50(10), 2231–2242 (2004)MathSciNetCrossRefGoogle Scholar
  152. 478.
    J.A. Tropp, Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Process. 86(3), 589–602 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  153. 479.
    J.A. Tropp, Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theor. 51(3), 1030–1051 (2006)MathSciNetCrossRefGoogle Scholar
  154. 482.
    J.A. Tropp, On the linear independence of spikes and sines. J. Fourier Anal. Appl. 14(5–6), 838–858 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  155. 487.
    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)zbMATHCrossRefGoogle Scholar
  156. 488.
    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)MathSciNetCrossRefGoogle Scholar
  157. 497.
    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)CrossRefGoogle Scholar
  158. 507.
    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)CrossRefGoogle Scholar
  159. 508.
    P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge University Press, Cambridge, 1997)zbMATHCrossRefGoogle Scholar
  160. 512.
    G. Wright, Magnetic resonance imaging. IEEE Signal Process. Mag. 14(1), 56–66 (1997)CrossRefGoogle Scholar
  161. 513.
    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)CrossRefGoogle Scholar
  162. 519.
    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)MathSciNetCrossRefGoogle Scholar
  163. 520.
    A. Zymnis, S. Boyd, E.J. Candès, Compressed sensing with quantized measurements. IEEE Signal Process. Lett. 17(2), 149–152 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simon Foucart
    • 1
  • Holger Rauhut
    • 2
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany

Personalised recommendations