Compressed sensing

IEEE Transactions on Information Theory

Volumn 52, Issue 4, 2006, Pages 1289-1306

  Donoho, David L ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Adaptive sampling; Almost spherical sections of Banach spaces; Basis Pursuit; Eigenvalues of random matrices; Gel'fand n widths; Information based complexity; Integrated sensing and processing; Minimum 1 norm decomposition; Optimal recovery; Quotient of a Subspace theorem; Sparse solution of linear equations

Indexed keywords

APPROXIMATION THEORY; CODING ERRORS; COMPUTATIONAL COMPLEXITY; EIGENVALUES AND EIGENFUNCTIONS; IMAGE RECONSTRUCTION; LINEAR EQUATIONS; MATRIX ALGEBRA; OPTIMIZATION; SIGNAL ENCODING; VECTORS;

ADAPTIVE SAMPLING; BANANA SPACES; BASIS PURSUIT; COMPRESSED SENSING; GEL'FAND N-WIDTHS; INFORMATION-BASED COMPLEXITY; MINIMUM DECOMPOSITION; OPTIMAL RECOVERY; QUOTIENT-OF-A-SUBSPACE THEOREM; SPARSE SOLUTION;

IMAGE COMPRESSION;

EID: 33645712892     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2006.871582     Document Type: Article

References (47)

1. D. L. Donoho, M. Vetterli, R. A. DeVore, and I. C. Daubechies, "Data compression and harmonic analysis," IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2435-2476, Oct. 1998.
2. Y. Meyer, Wavelets and Operators. Cambridge, U.K.: Cambridge Univ. Press, 1993.
3. D. L. Donoho, "Unconditional bases are optimal bases for data compression and for statistical estimation," Appl. Comput. Harmonic Anal., vol. 1, pp. 100-115, 1993.
4. D. L. Donoho, "Sparse components of images and optimal atomic decomposition," Constructive Approx., vol. 17, pp. 353-382, 2001.
5. A. Pinkus, "n-widths and optimal recovery in approximation theory," in Proc. Symp. Applied Mathematics, vol. 36, C. de Boor, Ed., Providence, RI, 1986, pp. 51-66.
6. J. F. Traub and H. Woziakowski, A General Theory of Optimal Algorithms. New York: Academic, 1980.
7. D. L. Donoho, "Unconditional bases and bit-level compression," Appl. Comput. Harmonic Anal., vol. 3, pp. 388-92, 1996.
8. A. Pinkus, n-Widths in Approximation Theory. New York: Springer-Verlag, 1985.
9. A. Y. Garnaev and E. D. Gluskin, "On widths of the Euclidean ball" (in English), Sov. Math.-Dokl., vol. 30, pp. 200-203, 1984.
10. B. S. Kashin, "Diameters of certain finite-dimensional sets in classes of smooth functions," Izv. Akad. Nauk SSSR, Ser. Mat., vol. 41, no. 2, pp. 334-351, 1977.
11. 1-norm solution is also the sparsest solution," Commun. Pure Appl. Math., to be published.
12. A. C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, and M. Strauss, "Near-optimal sparse fourier representations via sampling," in Proc 34th ACM Symp. Theory of Computing, Montréal, QC, Canada, May 2002, pp. 152-161.
13. E. J. Candès, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information.," IEEE Trans. Inf. Theory, to be published.
14. E. J. Candès, "Robust Uncertainty Principles and Signal Recovery," presented at the 2nd Int. Conf. Computational Harmonic Anaysis, Nashville, TN, May 2004.
15. C. A. Micchelli and T. J. Rivlin, "A survey of optimal recovery," in Optimal Estimation in Approximation Theory, C. A. Micchelli and T. J. Rivlin, Eds: Plenum, 1977, pp. 1-54.
16. J. Peetre and G. Sparr, "Interpolation of normed abelian groups," Ann. Math. Pure Appl., ser. 4, vol. 92, pp. 217-262, 1972.
17. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. Berlin, Germany: Springer-Verlag, 1976.
18. B. Carl, "Entropy numbers s-numbers, and eigenvalue problems," J. Funct. Anal., vol. 41, pp. 290-306, 1981.
19. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge, U.K.: Cambridge Univ. Press, 1989.
20. C. Schütt, "Entropy numbers of diagonal operators between symmetric Banach spaces," J. Approx. Theory, vol. 40, pp. 121-128, 1984.
21. T. Kuhn, "A lower estimate for entropy numbers," J. Approx. Theory, vol. 110, pp. 120-124, 2001.
22. S. Gal and C. Micchelli, "Optimal sequential and nonsequential procedures for evaluating a functional," Appl. Anal., vol. 10, pp. 105-120, 1980.
23. A. A. Melkman and C. A. Micchelli, "Optimal estimation of linear operators from inaccurate data," SIAM J. Numer. Anal., vol. 16, pp. 87-105, 1979.
24. M. A. Kon and E. Novak, "The adaption problem for approximating linear operators," Bull. Amer. Math. Soc., vol. 23, pp. 159-165, 1990.
25. E. Novak, "On the power of adaption," J. Complexity, vol. 12, pp. 199-237, 1996.
26. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci Comp., vol. 20, no. 1, pp. 33-61, 1999.
27. D. L. Donoho and X. Huo, "Uncertainty principles and ideal atomic decomposition," IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2845-62, Nov. 2001.
28. M. Elad and A. M. Bruckstein, "A generalized uncertainty principle and sparse representations in pairs of bases," IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2558-2567, Sep. 2002.
29. 1 norm minimization," Proc. Natl. Acad. Sci. USA, vol. 100, no. 5, pp. 2197-2002, Mar. 2002.
30. R. Gribonval and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans Inf Theory, vol. 49, no. 12, pp. 3320-3325, Dec. 2003.
31. J. J. Fuchs, "On sparse representation in arbitrary redundant bases," IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1341-1344, Jun. 2002.
32. J. A. Tropp, "Greed is good: Algorithmic results for sparse approximation," IEEE Trans Inf. Theory, vol. 50, no. 10, pp. 2231-2242, Oct. 2004.
33. J. A. Tropp, "Just relax: Convex programming methods for identifying sparse signals in noise," IEEE Trans Inf. Theory, vol. 52, no. 3, pp. 1030-1051, Mar. 2006.
34. D. L. Donoho, M. Elad, and V. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory, vol. 52, no. 1, pp. 6-18, Jan. 2006.
35. 1-norm near-solution approximates the sparsest near-solution," Commun. Pure Appl. Math., to be published.
36. I. C. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992.
37. S. Mallat, A Wavelet Tour of Signal Processing. San Diego, CA: Academic, 1998.
38. 2)," Amer. J. Math., vol. 121, pp. 587-628, 1999.
39. E. J. Candès and D. L. Donoho, "Curvelets - A surprisingly effective nonadaptive representation for objects with edges," in Curves and Surfaces, C. Rabut, A. Cohen, and L. L. Schumaker, Eds. Nashville, TN: Vanderbilt Univ. Press, 2000.
40. 2 singularities," Comm. Pure Appl. Math., vol. LVII, pp. 219-266, 2004.
41. n and random matrices," Amer. J. Math., vol. 112, pp. 819-842, 1990.
42. S. J. Szarek, "Condition numbers of random matrices," J. Complexity, vol. 7, pp. 131-149, 1991.
43. A. Dvoretsky, "Some results on convex bodies and banach spaces," in Proc. Symp. Linear Spaces, Jerusalem, Israel, 1961, pp. 123-160.
44. T. Figiel, J. Lindenstrauss, and V. D. Milman, "The dimension of almost-spherical sections of convex bodies," Acta Math., vol. 139, pp. 53-94, 1977.
45. V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces (Leecture Notes in Mathematics). Berlin, Germany: Springer-Verlag, 1986, vol. 1200.
46. D. Pollard, Empirical Processes: Theory and Applications. Hayward, CA: Inst. Math. Statist., vol. 2, NSF-CBMS Regional Conference Series in Probability and Statistics.
47. E. J. Candès and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies," Applied and Computational Mathematics, Calif. Inst. Technol., Tech. Rep., 2004.