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2015 | OriginalPaper | Chapter

Compressive Sensing

Authors : Massimo Fornasier, Holger Rauhut

Published in: Handbook of Mathematical Methods in Imaging

Publisher: Springer New York

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Abstract

Compressive sensing is a recent type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as 1-minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.

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Literature
1.
go back to reference Achlioptas, D.: Database-friendly random projections. In: Proceedings of the 20th Annual ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Santa Barbara, pp. 274–281 (2001) Achlioptas, D.: Database-friendly random projections. In: Proceedings of the 20th Annual ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Santa Barbara, pp. 274–281 (2001)
2.
go back to reference Affentranger, F., Schneider, R.: Random projections of regular simplices. Discret. Comput. Geom. 7(3), 219–226 (1992)MATHMathSciNet Affentranger, F., Schneider, R.: Random projections of regular simplices. Discret. Comput. Geom. 7(3), 219–226 (1992)MATHMathSciNet
3.
go back to reference Ailon, N., Liberty, E.: Almost optimal unrestricted fast Johnson-Lindenstrauss transform. In: Symposium on Discrete Algorithms (SODA), San Francisco, (2011) Ailon, N., Liberty, E.: Almost optimal unrestricted fast Johnson-Lindenstrauss transform. In: Symposium on Discrete Algorithms (SODA), San Francisco, (2011)
4.
go back to reference Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization (2012). arXiv:1210.7752 Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization (2012). arXiv:1210.7752
5.
go back to reference Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)MATHMathSciNet Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)MATHMathSciNet
6.
go back to reference Baraniuk, R.: Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007) Baraniuk, R.: Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)
7.
go back to reference Baraniuk, R.G., Davenport, M., DeVore, R.A., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)MATHMathSciNet Baraniuk, R.G., Davenport, M., DeVore, R.A., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)MATHMathSciNet
8.
go back to reference Bauschke, H.H., Combettes, P.-L., Luke, D.R.: Hybrid projection-reflection method for phase retrieval. J. Opt. Soc. Am. A 20(6), 1025–1034 (2003) Bauschke, H.H., Combettes, P.-L., Luke, D.R.: Hybrid projection-reflection method for phase retrieval. J. Opt. Soc. Am. A 20(6), 1025–1034 (2003)
9.
go back to reference Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MATHMathSciNet Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MATHMathSciNet
10.
go back to reference Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)MathSciNet Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)MathSciNet
11.
go back to reference Berinde, R., Gilbert, A.C., Indyk, P., Karloff, H., Strauss, M.: Combining geometry and combinatorics: a unified approach to sparse signal recovery. In: Proceedings of the 46th Annual Allerton Conference on Comunication, Control, and Computing 2008, Urbana, pp. 798–805. IEEE (2008) Berinde, R., Gilbert, A.C., Indyk, P., Karloff, H., Strauss, M.: Combining geometry and combinatorics: a unified approach to sparse signal recovery. In: Proceedings of the 46th Annual Allerton Conference on Comunication, Control, and Computing 2008, Urbana, pp. 798–805. IEEE (2008)
12.
go back to reference Blanchard, J.D., Cartis, C., Tanner, J., Thompson, A.: Phase transitions for greedy sparse approximation algorithms. Appl. Comput. Harmon. Anal. 30(2), 188–203 (2011)MATHMathSciNet Blanchard, J.D., Cartis, C., Tanner, J., Thompson, A.: Phase transitions for greedy sparse approximation algorithms. Appl. Comput. Harmon. Anal. 30(2), 188–203 (2011)MATHMathSciNet
13.
go back to reference Blumensath, T., Davies, M.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009)MATHMathSciNet Blumensath, T., Davies, M.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009)MATHMathSciNet
14.
go back to reference Bobin, J., Starck, J.-L., Ottensamer, R.: Compressed sensing in astronomy. IEEE J. Sel. Top. Signal Process. 2(5), 718–726 (2008) Bobin, J., Starck, J.-L., Ottensamer, R.: Compressed sensing in astronomy. IEEE J. Sel. Top. Signal Process. 2(5), 718–726 (2008)
15.
go back to reference Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Breaking the k 2-barrier for explicit RIP matrices. In: STOC’11, San Jose, pp. 637–644 (2011) Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Breaking the k 2-barrier for explicit RIP matrices. In: STOC’11, San Jose, pp. 637–644 (2011)
16.
go back to reference Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159(1), 145–185 (2011)MATHMathSciNet Bourgain, J., Dilworth, S., Ford, K., Konyagin, S., Kutzarova, D.: Explicit constructions of RIP matrices and related problems. Duke Math. J. 159(1), 145–185 (2011)MATHMathSciNet
17.
go back to reference Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge/New York (2004)MATH Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge/New York (2004)MATH
18.
19.
go back to reference Cai, T., Zhang, A.: Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory 60(1), 122–132 (2014)MathSciNet Cai, T., Zhang, A.: Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory 60(1), 122–132 (2014)MathSciNet
20.
go back to reference Cai, J.-F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MATHMathSciNet Cai, J.-F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MATHMathSciNet
21.
go back to reference Candès, E.J.: Compressive sampling. In: Proceedings of the International Congress of Mathematicians, Madrid (2006) Candès, E.J.: Compressive sampling. In: Proceedings of the International Congress of Mathematicians, Madrid (2006)
22.
go back to reference Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris Ser. I Math. 346, 589–592 (2008)MATH Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris Ser. I Math. 346, 589–592 (2008)MATH
23.
go back to reference Candès, E.J., Li, X.: Solving quadratic equations via PhaseLift when there are about as many equations as unknowns. Found. Comput. Math. 14(5), 1017–1026 (2014)MATHMathSciNet Candès, E.J., Li, X.: Solving quadratic equations via PhaseLift when there are about as many equations as unknowns. Found. Comput. Math. 14(5), 1017–1026 (2014)MATHMathSciNet
24.
go back to reference Candès, E.J., Plan, Y.: Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements. IEEE Trans. Inf. Theory 57(4), 2342–2359 (2011) Candès, E.J., Plan, Y.: Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements. IEEE Trans. Inf. Theory 57(4), 2342–2359 (2011)
25.
go back to reference Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)MATHMathSciNet Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)MATHMathSciNet
26.
go back to reference Candès, E.J., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MATH Candès, E.J., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MATH
27.
go back to reference Candès, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56(5), 2053–2080 (2010) Candès, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56(5), 2053–2080 (2010)
28.
go back to reference Candès, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008) Candès, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)
29.
go back to reference Candès, E.J., Tao, T., Romberg, J.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MATH Candès, E.J., Tao, T., Romberg, J.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MATH
30.
go back to reference Candès, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MATH Candès, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MATH
31.
go back to reference Candès, E., Li, X., Soltanolkotabi, M.: Phase retrieval from masked Fourier transforms (2013, preprint) Candès, E., Li, X., Soltanolkotabi, M.: Phase retrieval from masked Fourier transforms (2013, preprint)
32.
go back to reference Candès, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)MATH Candès, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)MATH
33.
go back to reference Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, Montréal, pp. 659–668 (electronic). ACM (2002) Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, Montréal, pp. 659–668 (electronic). ACM (2002)
34.
go back to reference Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)MATHMathSciNet Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)MATHMathSciNet
35.
go back to reference Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1999)MATHMathSciNet Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1999)MATHMathSciNet
36.
go back to reference Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003)MATH Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003)MATH
37.
38.
go back to reference Cohen, A., Dahmen, W., DeVore, R.A.: Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009)MATHMathSciNet Cohen, A., Dahmen, W., DeVore, R.A.: Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009)MATHMathSciNet
39.
go back to reference Combettes, P., Pesquet, J.-C.: A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1(4), 564–574 (2007) Combettes, P., Pesquet, J.-C.: A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1(4), 564–574 (2007)
40.
go back to reference Combettes, P., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011) Combettes, P., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)
41.
go back to reference Combettes, P., Wajs, V.: Signal recovery by proximal forward-backward splitting. Multisc. Model. Simul. 4(4), 1168–1200 (electronic) (2005) Combettes, P., Wajs, V.: Signal recovery by proximal forward-backward splitting. Multisc. Model. Simul. 4(4), 1168–1200 (electronic) (2005)
42.
go back to reference Cormode, G., Muthukrishnan, S.: Combinatorial algorithms for compressed sensing. In: CISS, Princeton (2006) Cormode, G., Muthukrishnan, S.: Combinatorial algorithms for compressed sensing. In: CISS, Princeton (2006)
43.
go back to reference Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MATH Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MATH
44.
go back to reference Daubechies, I., DeVore, R., Fornasier, M., Güntürk, C.: Iteratively re-weighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)MATH Daubechies, I., DeVore, R., Fornasier, M., Güntürk, C.: Iteratively re-weighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)MATH
45.
go back to reference Davies, M., Gribonval, R.: Restricted isometry constants where ℓ p sparse recovery can fail for 0 < p ≤ 1. IEEE Trans. Inf. Theory 55(5), 2203–2214 (2009)MathSciNet Davies, M., Gribonval, R.: Restricted isometry constants where p sparse recovery can fail for 0 < p ≤ 1. IEEE Trans. Inf. Theory 55(5), 2203–2214 (2009)MathSciNet
46.
go back to reference Do, B., Indyk, P., Price, E., Woodruff, D.: Lower bounds for sparse recovery. In: Proceedings of the SODA, Austin (2010) Do, B., Indyk, P., Price, E., Woodruff, D.: Lower bounds for sparse recovery. In: Proceedings of the SODA, Austin (2010)
48.
go back to reference Donoho, D.L.: For most large underdetermined systems of linear equations the minimal l 1 solution is also the sparsest solution. Commun. Pure Appl. Anal. 59(6), 797–829 (2006)MATHMathSciNet Donoho, D.L.: For most large underdetermined systems of linear equations the minimal l 1 solution is also the sparsest solution. Commun. Pure Appl. Anal. 59(6), 797–829 (2006)MATHMathSciNet
49.
go back to reference Donoho, D.L.: High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discret. Comput. Geom. 35(4), 617–652 (2006)MATHMathSciNet Donoho, D.L.: High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discret. Comput. Geom. 35(4), 617–652 (2006)MATHMathSciNet
50.
go back to reference Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proc. Natl. Acad. Sci. USA 100(5), 2197–2202 (2003)MATHMathSciNet Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proc. Natl. Acad. Sci. USA 100(5), 2197–2202 (2003)MATHMathSciNet
51.
go back to reference Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decompositions. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001)MATHMathSciNet Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decompositions. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001)MATHMathSciNet
52.
go back to reference Donoho, D., Logan, B.: Signal recovery and the large sieve. SIAM J. Appl. Math. 52(2), 577–591 (1992)MATHMathSciNet Donoho, D., Logan, B.: Signal recovery and the large sieve. SIAM J. Appl. Math. 52(2), 577–591 (1992)MATHMathSciNet
53.
go back to reference Donoho, D.L., Tanner, J.: Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)MATHMathSciNet Donoho, D.L., Tanner, J.: Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)MATHMathSciNet
54.
go back to reference Donoho, D.L., Tanner, J.: Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22(1), 1–53 (2009)MATHMathSciNet Donoho, D.L., Tanner, J.: Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22(1), 1–53 (2009)MATHMathSciNet
55.
go back to reference Donoho, D.L., Tsaig, Y.: Fast solution of l1-norm minimization problems when the solution may be sparse. IEEE Trans. Inf. Theory 54(11), 4789–4812 (2008)MATHMathSciNet Donoho, D.L., Tsaig, Y.: Fast solution of l1-norm minimization problems when the solution may be sparse. IEEE Trans. Inf. Theory 54(11), 4789–4812 (2008)MATHMathSciNet
56.
go back to reference Dorfman, R.: The detection of defective members of large populations. Ann. Stat. 14, 436–440 (1943) Dorfman, R.: The detection of defective members of large populations. Ann. Stat. 14, 436–440 (1943)
57.
go back to reference Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)MATHMathSciNet Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)MATHMathSciNet
58.
go back to reference Duarte, M., Davenport, M., Takhar, D., Laska, J., Ting, S., Kelly, K., Baraniuk, R.: Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 83–91 (2008) Duarte, M., Davenport, M., Takhar, D., Laska, J., Ting, S., Kelly, K., Baraniuk, R.: Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 83–91 (2008)
59.
go back to reference Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)MATHMathSciNet Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)MATHMathSciNet
60.
go back to reference Ehler, M., Fornasier, M., Sigl, J.: Quasi-linear compressed sensing. Multiscale Model. Simul. 12(2), 725–754 (2014)MathSciNet Ehler, M., Fornasier, M., Sigl, J.: Quasi-linear compressed sensing. Multiscale Model. Simul. 12(2), 725–754 (2014)MathSciNet
61.
go back to reference Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, New York (2010) Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, New York (2010)
62.
go back to reference Elad, M., Bruckstein, A.M.: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory 48(9), 2558–2567 (2002)MATHMathSciNet Elad, M., Bruckstein, A.M.: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory 48(9), 2558–2567 (2002)MATHMathSciNet
63.
go back to reference Eldar, Y., Kutyniok, G. (eds.): Compressed Sensing – Theory and Applications. Cambridge University Press, Cambridge/New York (2012) Eldar, Y., Kutyniok, G. (eds.): Compressed Sensing – Theory and Applications. Cambridge University Press, Cambridge/New York (2012)
64.
go back to reference Eldar, Y., Mendelson, S.: Phase retrieval: stability and recovery guarantees. Appl. Comput. Harmon. Anal. (to appear). doi:10.1016/j.acha.2013.08.003 Eldar, Y., Mendelson, S.: Phase retrieval: stability and recovery guarantees. Appl. Comput. Harmon. Anal. (to appear). doi:10.1016/j.acha.2013.08.003
65.
go back to reference Ender, J.: On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)MATH Ender, J.: On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)MATH
66.
go back to reference Fannjiang, A., Yan, P., Strohmer, T.: Compressed remote sensing of sparse objects. SIAM J. Imaging Sci. 3(3), 595–618 (2010)MATHMathSciNet Fannjiang, A., Yan, P., Strohmer, T.: Compressed remote sensing of sparse objects. SIAM J. Imaging Sci. 3(3), 595–618 (2010)MATHMathSciNet
67.
go back to reference Fazel, M.: Matrix rank minimization with applications. PhD thesis, Stanford University (2002) Fazel, M.: Matrix rank minimization with applications. PhD thesis, Stanford University (2002)
68.
go back to reference Fienup, J.R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21(15), 2758–2769 (1982) Fienup, J.R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21(15), 2758–2769 (1982)
69.
go back to reference Fornasier, M.: Numerical methods for sparse recovery. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Series on Computational and Applied Mathematics, vol. 9, pp. 93–200. deGruyter, Berlin (2010). Papers based on the presentations of the summer school “Theoretical Foundations and Numerical Methods for Sparse Recovery”, Vienna, Austria, 31 Aug-4 Sept 2009 Fornasier, M.: Numerical methods for sparse recovery. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Series on Computational and Applied Mathematics, vol. 9, pp. 93–200. deGruyter, Berlin (2010). Papers based on the presentations of the summer school “Theoretical Foundations and Numerical Methods for Sparse Recovery”, Vienna, Austria, 31 Aug-4 Sept 2009
70.
go back to reference Fornasier, M., March, R.: Restoration of color images by vector valued BV functions and variational calculus. SIAM J. Appl. Math. 68(2), 437–460 (2007)MATHMathSciNet Fornasier, M., March, R.: Restoration of color images by vector valued BV functions and variational calculus. SIAM J. Appl. Math. 68(2), 437–460 (2007)MATHMathSciNet
71.
go back to reference Fornasier, M., Ramlau, R., Teschke, G.: The application of joint sparsity and total variation minimization algorithms to a real-life art restoration problem. Adv. Comput. Math. 31(1–3), 157–184 (2009)MATHMathSciNet Fornasier, M., Ramlau, R., Teschke, G.: The application of joint sparsity and total variation minimization algorithms to a real-life art restoration problem. Adv. Comput. Math. 31(1–3), 157–184 (2009)MATHMathSciNet
72.
go back to reference Fornasier, M., Langer, A., Schönlieb, C.: A convergent overlapping domain decomposition method for total variation minimization. Numer. Math. 116(4), 645–685 (2010)MATHMathSciNet Fornasier, M., Langer, A., Schönlieb, C.: A convergent overlapping domain decomposition method for total variation minimization. Numer. Math. 116(4), 645–685 (2010)MATHMathSciNet
73.
go back to reference Fornasier, M., Rauhut, H., Ward, R.: Low-rank matrix recovery via iteratively reweighted least squares minimization. SIAM J. Optim. 21(4), 1614–1640 (2011)MATHMathSciNet Fornasier, M., Rauhut, H., Ward, R.: Low-rank matrix recovery via iteratively reweighted least squares minimization. SIAM J. Optim. 21(4), 1614–1640 (2011)MATHMathSciNet
74.
go back to reference Foucart, S.: A note on guaranteed sparse recovery via ℓ 1-minimization. Appl. Comput. Harmon. Anal. 29(1), 97–103 (2010)MATHMathSciNet Foucart, S.: A note on guaranteed sparse recovery via 1-minimization. Appl. Comput. Harmon. Anal. 29(1), 97–103 (2010)MATHMathSciNet
75.
go back to reference Foucart, S., Lai, M.: Sparsest solutions of underdetermined linear systems via ℓ q -minimization for 0 < q ≤ 1. Appl. Comput. Harmon. Anal. 26(3), 395–407 (2009)MATHMathSciNet Foucart, S., Lai, M.: Sparsest solutions of underdetermined linear systems via q -minimization for 0 < q ≤ 1. Appl. Comput. Harmon. Anal. 26(3), 395–407 (2009)MATHMathSciNet
76.
go back to reference Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2013)MATH Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2013)MATH
77.
go back to reference Foucart, S., Pajor, A., Rauhut, H., Ullrich, T.: The Gelfand widths of ℓ p -balls for 0 < p ≤ 1. J. Complex. 26(6), 629–640 (2010)MATHMathSciNet Foucart, S., Pajor, A., Rauhut, H., Ullrich, T.: The Gelfand widths of p -balls for 0 < p ≤ 1. J. Complex. 26(6), 629–640 (2010)MATHMathSciNet
78.
go back to reference Fuchs, J.J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory 50(6), 1341–1344 (2004)MATH Fuchs, J.J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory 50(6), 1341–1344 (2004)MATH
79.
go back to reference Garnaev, A., Gluskin, E.: On widths of the Euclidean ball. Sov. Math. Dokl. 30, 200–204 (1984)MATH Garnaev, A., Gluskin, E.: On widths of the Euclidean ball. Sov. Math. Dokl. 30, 200–204 (1984)MATH
80.
go back to reference Gilbert, A.C., Muthukrishnan, S., Guha, S., Indyk, P., Strauss, M.: Near-optimal sparse Fourier representations via sampling. In: Proceedings of the STOC’02, Montréal, pp. 152–161. Association for Computing Machinery (2002) Gilbert, A.C., Muthukrishnan, S., Guha, S., Indyk, P., Strauss, M.: Near-optimal sparse Fourier representations via sampling. In: Proceedings of the STOC’02, Montréal, pp. 152–161. Association for Computing Machinery (2002)
81.
go back to reference Gilbert, A.C., Muthukrishnan, S., Strauss, M.J.: Approximation of functions over redundant dictionaries using coherence. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, 12–14 Jan 2003, pp. 243–252. SIAM and Association for Computing Machinery (2003) Gilbert, A.C., Muthukrishnan, S., Strauss, M.J.: Approximation of functions over redundant dictionaries using coherence. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, 12–14 Jan 2003, pp. 243–252. SIAM and Association for Computing Machinery (2003)
82.
go back to reference Gilbert, A.C., Strauss, M., Tropp, J.A., Vershynin, R.: One sketch for all: fast algorithms for compressed sensing. In: Proceedings of the 39th ACM Symposium Theory of Computing (STOC), San Diego (2007) Gilbert, A.C., Strauss, M., Tropp, J.A., Vershynin, R.: One sketch for all: fast algorithms for compressed sensing. In: Proceedings of the 39th ACM Symposium Theory of Computing (STOC), San Diego (2007)
83.
go back to reference Glowinski, R., Le, T.: Augmented Lagrangian and Operator-Splitting Methods. SIAM, Philadelphia (1989)MATH Glowinski, R., Le, T.: Augmented Lagrangian and Operator-Splitting Methods. SIAM, Philadelphia (1989)MATH
84.
go back to reference Gluskin, E.: Norms of random matrices and widths of finite-dimensional sets. Math. USSR-Sb. 48, 173–182 (1984)MATH Gluskin, E.: Norms of random matrices and widths of finite-dimensional sets. Math. USSR-Sb. 48, 173–182 (1984)MATH
85.
go back to reference Goldfarb, D., Ma, S.: Convergence of fixed point continuation algorithms for matrix rank minimization. Found. Comput. Math. 11(2), 183–210 (2011)MATHMathSciNet Goldfarb, D., Ma, S.: Convergence of fixed point continuation algorithms for matrix rank minimization. Found. Comput. Math. 11(2), 183–210 (2011)MATHMathSciNet
86.
go back to reference Gorodnitsky, I., Rao, B.: Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997) Gorodnitsky, I., Rao, B.: Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997)
87.
go back to reference Gribonval, R., Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inf. Theory 49(12), 3320–3325 (2003)MATHMathSciNet Gribonval, R., Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inf. Theory 49(12), 3320–3325 (2003)MATHMathSciNet
88.
go back to reference Gross, D.: Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57(3), 1548–1566 (2011) Gross, D.: Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57(3), 1548–1566 (2011)
89.
go back to reference Gross, D., Liu, Y.-K., Flammia, S.T., Becker, S., Eisert, J.: Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010) Gross, D., Liu, Y.-K., Flammia, S.T., Becker, S., Eisert, J.: Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010)
90.
go back to reference Gross, D., Krahmer, F., Kueng, R.: Improved recovery guarantees for phase retrieval from coded diffraction patterns (2014, preprint) Gross, D., Krahmer, F., Kueng, R.: Improved recovery guarantees for phase retrieval from coded diffraction patterns (2014, preprint)
91.
go back to reference Gross, D., Krahmer, F., Kueng, R.: A partial derandomization of PhaseLift using spherical designs. J. Fourier Anal. Appl. (to appear) Gross, D., Krahmer, F., Kueng, R.: A partial derandomization of PhaseLift using spherical designs. J. Fourier Anal. Appl. (to appear)
92.
go back to reference He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5(1), 119–149 (2012)MATHMathSciNet He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5(1), 119–149 (2012)MATHMathSciNet
93.
go back to reference Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge/New York (1990)MATH Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge/New York (1990)MATH
94.
go back to reference Hügel, M., Rauhut, H., Strohmer, T.: Remote sensing via l1-minimization. Found. Comput. Math. 14, 115–150 (2014)MATHMathSciNet Hügel, M., Rauhut, H., Strohmer, T.: Remote sensing via l1-minimization. Found. Comput. Math. 14, 115–150 (2014)MATHMathSciNet
95.
go back to reference Johnson, W.B., Lindenstrauss, J. (eds.): Handbook of the Geometry of Banach Spaces, vol. I. North-Holland, Amsterdam (2001)MATH Johnson, W.B., Lindenstrauss, J. (eds.): Handbook of the Geometry of Banach Spaces, vol. I. North-Holland, Amsterdam (2001)MATH
96.
go back to reference Kashin, B.: Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR Izv. 11, 317–333 (1977)MATH Kashin, B.: Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR Izv. 11, 317–333 (1977)MATH
97.
go back to reference Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from a few entries. IEEE Trans. Inf. Theory 56, 2980–2998 (2010)MathSciNet Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from a few entries. IEEE Trans. Inf. Theory 56, 2980–2998 (2010)MathSciNet
98.
go back to reference Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from noisy entries. J. Mach. Learn. Res. 11, 2057–2078 (2010)MATHMathSciNet Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from noisy entries. J. Mach. Learn. Res. 11, 2057–2078 (2010)MATHMathSciNet
99.
go back to reference Krahmer, F., Rauhut, H.: Structured random measurements in signal processing. GAMM Mitteilungen. (to appear) Krahmer, F., Rauhut, H.: Structured random measurements in signal processing. GAMM Mitteilungen. (to appear)
100.
go back to reference Krahmer, F., Ward, R.: New and improved Johnson-Lindenstrauss embeddings via the restricted isometry property. SIAM J. Math. Anal. 43(3), 1269–1281 (2011)MATHMathSciNet Krahmer, F., Ward, R.: New and improved Johnson-Lindenstrauss embeddings via the restricted isometry property. SIAM J. Math. Anal. 43(3), 1269–1281 (2011)MATHMathSciNet
101.
go back to reference Krahmer, F., Mendelson, S., Rauhut, H.: Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. (to appear). doi:10.1002/cpa.21504 Krahmer, F., Mendelson, S., Rauhut, H.: Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. (to appear). doi:10.1002/cpa.21504
102.
go back to reference Lawson, C.: Contributions to the theory of linear least maximum approximation. PhD thesis, University of California, Los Angeles (1961) Lawson, C.: Contributions to the theory of linear least maximum approximation. PhD thesis, University of California, Los Angeles (1961)
103.
go back to reference Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin/New York (1991)MATH Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin/New York (1991)MATH
104.
go back to reference Lee, K., Bresler, Y.: ADMiRA: atomic decomposition for minimum rank approximation. IEEE Trans. Inf. Theory 56(9), 4402–4416 (2010)MathSciNet Lee, K., Bresler, Y.: ADMiRA: atomic decomposition for minimum rank approximation. IEEE Trans. Inf. Theory 56(9), 4402–4416 (2010)MathSciNet
105.
go back to reference Li, X., Voroninski, V.: Sparse signal recovery from quadratic measurements via convex programming (2013). arXiv:1209.4785 Li, X., Voroninski, V.: Sparse signal recovery from quadratic measurements via convex programming (2013). arXiv:1209.4785
106.
go back to reference Logan, B.: Properties of high-pass signals. PhD thesis, Columbia University (1965) Logan, B.: Properties of high-pass signals. PhD thesis, Columbia University (1965)
107.
go back to reference Lorentz, G.G., von Golitschek, M., Makovoz, Y.: Constructive Approximation: Advanced Problems. Springer, Berlin (1996)MATH Lorentz, G.G., von Golitschek, M., Makovoz, Y.: Constructive Approximation: Advanced Problems. Springer, Berlin (1996)MATH
108.
go back to reference Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)MATH Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)MATH
109.
go back to reference Marple, S.: Digital Spectral Analysis with Applications. Prentice-Hall, Englewood Cliffs (1987) Marple, S.: Digital Spectral Analysis with Applications. Prentice-Hall, Englewood Cliffs (1987)
110.
go back to reference Mendelson, S., Pajor, A., Tomczak Jaegermann, N.: Uniform uncertainty principle for Bernoulli and subgaussian ensembles. Constr. Approx. 28(3), 277–289 (2009)MathSciNet Mendelson, S., Pajor, A., Tomczak Jaegermann, N.: Uniform uncertainty principle for Bernoulli and subgaussian ensembles. Constr. Approx. 28(3), 277–289 (2009)MathSciNet
111.
go back to reference Millane, R.: Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7(3), 394–411 (1990) Millane, R.: Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7(3), 394–411 (1990)
112.
go back to reference Mixon, D.: Short, fat matrices. Blog (2013) Mixon, D.: Short, fat matrices. Blog (2013)
113.
go back to reference Mohan, K., Fazel, M.: Reweighted nuclear norm minimization with application to system identification. In: Proceedings of the American Control Conference, Baltimore, pp. 2953–2959 (2010) Mohan, K., Fazel, M.: Reweighted nuclear norm minimization with application to system identification. In: Proceedings of the American Control Conference, Baltimore, pp. 2953–2959 (2010)
114.
go back to reference Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)MATHMathSciNet Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)MATHMathSciNet
115.
go back to reference Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994) Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)
116.
go back to reference Netrapalli, P., Jain, P., Sanghavi, S.: Phase retrieval using alternating minimization (2013). arXiv:1306.0160 Netrapalli, P., Jain, P., Sanghavi, S.: Phase retrieval using alternating minimization (2013). arXiv:1306.0160
117.
go back to reference Novak, E.: Optimal recovery and n-widths for convex classes of functions. J. Approx. Theory 80(3), 390–408 (1995)MATHMathSciNet Novak, E.: Optimal recovery and n-widths for convex classes of functions. J. Approx. Theory 80(3), 390–408 (1995)MATHMathSciNet
118.
go back to reference Ohlsson, H., Yang, A.Y., Dong, R., Sastry, S.S.: Nonlinear basis pursuit. In: 47th Asilomar Conference on Signals, Systems and Computers, Pacific Grove (2013) Ohlsson, H., Yang, A.Y., Dong, R., Sastry, S.S.: Nonlinear basis pursuit. In: 47th Asilomar Conference on Signals, Systems and Computers, Pacific Grove (2013)
119.
go back to reference Osborne, M., Presnell, B., Turlach, B.: A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 20(3), 389–403 (2000)MATHMathSciNet Osborne, M., Presnell, B., Turlach, B.: A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 20(3), 389–403 (2000)MATHMathSciNet
120.
go back to reference Osborne, M., Presnell, B., Turlach, B.: On the LASSO and its dual. J. Comput. Graph. Stat. 9(2), 319–337 (2000)MathSciNet Osborne, M., Presnell, B., Turlach, B.: On the LASSO and its dual. J. Comput. Graph. Stat. 9(2), 319–337 (2000)MathSciNet
121.
go back to reference Oymak, S., Mohan, K., Fazel, M., Hassibi, B.: A simplified approach to recovery conditions for low-rank matrices. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), St. Petersburg (2011) Oymak, S., Mohan, K., Fazel, M., Hassibi, B.: A simplified approach to recovery conditions for low-rank matrices. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), St. Petersburg (2011)
122.
go back to reference Pfander, G.E., Rauhut, H.: Sparsity in time-frequency representations. J. Fourier Anal. Appl. 16(2), 233–260 (2010)MATHMathSciNet Pfander, G.E., Rauhut, H.: Sparsity in time-frequency representations. J. Fourier Anal. Appl. 16(2), 233–260 (2010)MATHMathSciNet
123.
go back to reference Pfander, G.E., Rauhut, H., Tanner, J.: Identification of matrices having a sparse representation. IEEE Trans. Signal Process. 56(11), 5376–5388 (2008)MathSciNet Pfander, G.E., Rauhut, H., Tanner, J.: Identification of matrices having a sparse representation. IEEE Trans. Signal Process. 56(11), 5376–5388 (2008)MathSciNet
124.
go back to reference Pfander, G.E., Rauhut, H., Tropp, J.A.: The restricted isometry property for time-frequency structured random matrices. Probab. Theory Relat. Fields 156, 707–737 (2013)MATHMathSciNet Pfander, G.E., Rauhut, H., Tropp, J.A.: The restricted isometry property for time-frequency structured random matrices. Probab. Theory Relat. Fields 156, 707–737 (2013)MATHMathSciNet
125.
go back to reference Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: IEEE International Conference Computer Vision (ICCV), Barcelona, pp. 1762–1769 (2011) Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: IEEE International Conference Computer Vision (ICCV), Barcelona, pp. 1762–1769 (2011)
126.
go back to reference Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford-Shah functional. In: ICCV Proceedings, Kyoto. Springer (2009) Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford-Shah functional. In: ICCV Proceedings, Kyoto. Springer (2009)
127.
go back to reference Prony, R.: Essai expérimental et analytique sur les lois de la Dilatabilité des uides élastique et sur celles de la Force expansive de la vapeur de loeau et de la vapeur de l’alkool, à différentes températures. J. École Polytechnique 1, 24–76 (1795) Prony, R.: Essai expérimental et analytique sur les lois de la Dilatabilité des uides élastique et sur celles de la Force expansive de la vapeur de loeau et de la vapeur de l’alkool, à différentes températures. J. École Polytechnique 1, 24–76 (1795)
128.
go back to reference Rauhut, H.: Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal. 22(1), 16–42 (2007)MATHMathSciNet Rauhut, H.: Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal. 22(1), 16–42 (2007)MATHMathSciNet
129.
go back to reference Rauhut, H.: Stability results for random sampling of sparse trigonometric polynomials. IEEE Trans. Inf Theory 54(12), 5661–5670 (2008)MATHMathSciNet Rauhut, H.: Stability results for random sampling of sparse trigonometric polynomials. IEEE Trans. Inf Theory 54(12), 5661–5670 (2008)MATHMathSciNet
130.
go back to reference Rauhut, H.: Circulant and Toeplitz matrices in compressed sensing. In: Proceedings of the SPARS’09 (2009) Rauhut, H.: Circulant and Toeplitz matrices in compressed sensing. In: Proceedings of the SPARS’09 (2009)
131.
go back to reference Rauhut, H.: Compressive sensing and structured random matrices. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Series on Computational and Applied Mathematics, vol. 9, pp. 1–92. deGruyter, Berlin (2010). Papers based on the presentations of the summer school “Theoretical Foundations and Numerical Methods for Sparse Recovery”, Vienna, Austria, 31 Aug-4 Sept 2009 Rauhut, H.: Compressive sensing and structured random matrices. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Series on Computational and Applied Mathematics, vol. 9, pp. 1–92. deGruyter, Berlin (2010). Papers based on the presentations of the summer school “Theoretical Foundations and Numerical Methods for Sparse Recovery”, Vienna, Austria, 31 Aug-4 Sept 2009
132.
go back to reference Rauhut, H., Ward, R.: Interpolation via weighted l1 minimization (2013). ArXiv:1308.0759 Rauhut, H., Ward, R.: Interpolation via weighted l1 minimization (2013). ArXiv:1308.0759
133.
go back to reference Rauhut, H., Romberg, J.K., Tropp, J.A.: Restricted isometries for partial random circulant matrices. Appl. Comput. Harmon. Anal. 32(2), 242–254 (2012)MATHMathSciNet Rauhut, H., Romberg, J.K., Tropp, J.A.: Restricted isometries for partial random circulant matrices. Appl. Comput. Harmon. Anal. 32(2), 242–254 (2012)MATHMathSciNet
134.
go back to reference Recht, B.: A simpler approach to matrix completion. J. Mach. Learn. Res. 12, 3413–3430 (2012)MathSciNet Recht, B.: A simpler approach to matrix completion. J. Mach. Learn. Res. 12, 3413–3430 (2012)MathSciNet
135.
go back to reference Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)MATHMathSciNet Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)MATHMathSciNet
136.
go back to reference Romberg, J.: Imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 14–20 (2008) Romberg, J.: Imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 14–20 (2008)
137.
go back to reference Romberg, J.K.: Compressive sensing by random convolution. SIAM J. Imaging Sci. 2(4), 1098–1128 (2009)MATHMathSciNet Romberg, J.K.: Compressive sensing by random convolution. SIAM J. Imaging Sci. 2(4), 1098–1128 (2009)MATHMathSciNet
138.
go back to reference Rudelson, M., Vershynin, R.: On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61, 1025–1045 (2008)MATHMathSciNet Rudelson, M., Vershynin, R.: On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61, 1025–1045 (2008)MATHMathSciNet
139.
go back to reference Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)MATH Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)MATH
140.
go back to reference Santosa, F., Symes, W.: Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput. 7(4), 1307–1330 (1986)MATHMathSciNet Santosa, F., Symes, W.: Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput. 7(4), 1307–1330 (1986)MATHMathSciNet
141.
go back to reference Schnass, K., Vandergheynst, P.: Dictionary preconditioning for greedy algorithms. IEEE Trans. Signal Process. 56(5), 1994–2002 (2008)MathSciNet Schnass, K., Vandergheynst, P.: Dictionary preconditioning for greedy algorithms. IEEE Trans. Signal Process. 56(5), 1994–2002 (2008)MathSciNet
142.
go back to reference Starck, J.-L., Murtagh, F., Fadili, J.: Sparse Image and Signal Processing Wavelets, Curvelets, Morphological Diversity, xvii, p. 316. Cambridge University Press, Cambridge (2010) Starck, J.-L., Murtagh, F., Fadili, J.: Sparse Image and Signal Processing Wavelets, Curvelets, Morphological Diversity, xvii, p. 316. Cambridge University Press, Cambridge (2010)
143.
go back to reference Strohmer, T., Heath, R.W., Jr.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)MATHMathSciNet Strohmer, T., Heath, R.W., Jr.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)MATHMathSciNet
144.
go back to reference Strohmer, T., Hermann, M.: Compressed sensing radar. In: IEEE Proceedings of the International Conference on Acoustic, Speech, and Signal Processing, Las Vegas, pp. 1509–1512 (2008) Strohmer, T., Hermann, M.: Compressed sensing radar. In: IEEE Proceedings of the International Conference on Acoustic, Speech, and Signal Processing, Las Vegas, pp. 1509–1512 (2008)
145.
go back to reference Tadmor, E.: Numerical methods for nonlinear partial differential equations. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science. Springer, New York/London (2009) Tadmor, E.: Numerical methods for nonlinear partial differential equations. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science. Springer, New York/London (2009)
146.
147.
go back to reference Tauböck, G., Hlawatsch, F., Eiwen, D., Rauhut, H.: Compressive estimation of doubly selective channels in multicarrier systems: leakage effects and sparsity-enhancing processing. IEEE J. Sel. Top. Signal Process. 4(2), 255–271 (2010) Tauböck, G., Hlawatsch, F., Eiwen, D., Rauhut, H.: Compressive estimation of doubly selective channels in multicarrier systems: leakage effects and sparsity-enhancing processing. IEEE J. Sel. Top. Signal Process. 4(2), 255–271 (2010)
148.
go back to reference Taylor, H., Banks, S., McCoy, J.: Deconvolution with the ℓ 1-norm. Geophys. J. Int. 44(1), 39–52 (1979) Taylor, H., Banks, S., McCoy, J.: Deconvolution with the 1-norm. Geophys. J. Int. 44(1), 39–52 (1979)
149.
go back to reference Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58(1), 267–288 (1996)MATHMathSciNet Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58(1), 267–288 (1996)MATHMathSciNet
150.
go back to reference Traub, J., Wasilkowski, G., Wo’zniakowski, H.: Information-Based Complexity. Computer Science and Scientific Computing. Academic, Boston (1988)MATH Traub, J., Wasilkowski, G., Wo’zniakowski, H.: Information-Based Complexity. Computer Science and Scientific Computing. Academic, Boston (1988)MATH
151.
go back to reference Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)MATHMathSciNet Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)MATHMathSciNet
152.
go back to reference Tropp, J.A.: Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inf. Theory 51(3), 1030–1051 (2006)MathSciNet Tropp, J.A.: Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inf. Theory 51(3), 1030–1051 (2006)MathSciNet
153.
go back to reference Tropp, J., Needell, D.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)MATHMathSciNet Tropp, J., Needell, D.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)MATHMathSciNet
154.
go back to reference Tropp, J.A., Laska, J.N., Duarte, M.F., Romberg, J.K., Baraniuk, R.G.: Beyond nyquist: efficient sampling of sparse bandlimited signals. IEEE Trans. Inf. Theory 56(1), 520–544 (2010)MathSciNet Tropp, J.A., Laska, J.N., Duarte, M.F., Romberg, J.K., Baraniuk, R.G.: Beyond nyquist: efficient sampling of sparse bandlimited signals. IEEE Trans. Inf. Theory 56(1), 520–544 (2010)MathSciNet
155.
go back to reference Unser, M.: Sampling—50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000) Unser, M.: Sampling—50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000)
156.
go back to reference van den Berg, E., Friedlander, M.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)MATHMathSciNet van den Berg, E., Friedlander, M.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)MATHMathSciNet
157.
158.
go back to reference Wagner, G., Schmieder, P., Stern, A., Hoch, J.: Application of non-linear sampling schemes to cosy-type spectra. J. Biomol. NMR 3(5), 569 (1993) Wagner, G., Schmieder, P., Stern, A., Hoch, J.: Application of non-linear sampling schemes to cosy-type spectra. J. Biomol. NMR 3(5), 569 (1993)
159.
go back to reference Willett, R., Marcia, R., Nichols, J.: Compressed sensing for practical optical imaging systems: a tutorial. Opt. Eng. 50(7), 072601–072601–13 (2011) Willett, R., Marcia, R., Nichols, J.: Compressed sensing for practical optical imaging systems: a tutorial. Opt. Eng. 50(7), 072601–072601–13 (2011)
160.
go back to reference Willett, R., Duarte, M., Davenport, M., Baraniuk, R.: Sparsity and structure in hyperspectral imaging: sensing, reconstruction, and target detection. IEEE Signal Proc. Mag. 31(1), 116–126 (2014) Willett, R., Duarte, M., Davenport, M., Baraniuk, R.: Sparsity and structure in hyperspectral imaging: sensing, reconstruction, and target detection. IEEE Signal Proc. Mag. 31(1), 116–126 (2014)
Metadata
Title
Compressive Sensing
Authors
Massimo Fornasier
Holger Rauhut
Copyright Year
2015
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-0790-8_6

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