An adaptive inverse scale space method for compressed sensing

Mathematics of Computation

Volumn 82, Issue 281, 2013, Pages 269-299

  Burger, Martin ^a   Möller, Michael ^a   Benning, Martin ^a   Osher, Stanley ^b  

^a UNIVERSITY OF MÜNSTER   (Germany)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Adaptivity; Compressed sensing; Greedy methods; Inverse scale space; Sparsity

Indexed keywords

EID: 84872179162     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-2012-02599-3     Document Type: Article

References (33)

1. M. Benning, T. K̈osters, F. Ẅubbeling, K. Scḧafers, and M. Burger, A nonlinear variational method for improved quantification of myocardial blood flow using dynamic H215O PET, IEEE Nuclear Science Symposium Conference Record, 2008, pp. 4472-4477.
2. T. Blumensath and M. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmonic Anal. 27 (2009), 265-274.
3. M. Burger, A note on sparse reconstruction methods, J. Phys. Conference Series, no. 012002, 2008.
4. M. Burger, K. Frick, S. Osher, and O. Scherzer, Inverse total variation flow, SIAMMultiscale Modelling and Simulation 6 (2007), 366-395.
5. M. Burger, G. Gilboa, S. Osher, and J. Xu, Nonlinear inverse scale space methods, Comm. Math. Sci. 4 (2006), 179-212.
6. M. Burger and S. Osher, A guide to TV zoo, preprint.
7. M. Burger, E. Resmerita, and L. He, Error estimation for Bregman iterations and inverse scale space methods in image restoration, Computing 81 (2007), 109-135.
8. 1-norm minimization, Math. Comp. 78 (2009), 2127-2136.
9. J. Cai, S. Osher, and Z. Shen, Linearized Bregman iterations for compressed sensing, Math.Comp. 78 (2009), 1515-1536.
10. E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 51 (2004), 4203-4215.
11. E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies, IEEE Trans. Inform. Theory 52 (2004), 5406-5425.
12. A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet schemes for elliptic operator equations-convergence rates, Math. Comp. 70 (2001), 27-75.
13. A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods II-beyond the elliptic case, Found. Comput. Math. 2 (2002), 203-245.
14. S. Dahlke, W. Dahmen, and K. Urban, Adaptive wavelet methods for saddle point problems-optimal convergence rates, SIAM J. Numer. Anal. 40 (2002), 1230-1262.
15. I. Daubechies, M. Defrise, and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math. 57 (2004), 1413-1457.
16. D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), 1289-1306.
17. 1 minimization, Proc. Natl. Acad. Sci. USA 100 (2003), 2197-2202.
18. D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (stomp), Tech. report, Stanford Univ., 2006, Stat. Dept. Tech. Rep. 2006-02.
19. I. Ekeland and R. Temam, Convex analysis and variational problems, corrected reprint edition ed., SIAM, Philadelphia, 1999.
20. S. G. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Transactions on Signal Processing 12 (1993), 3397-3415.
21. Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations: The fifteenth Dean Jacqueline B. Lewis memorial lectures, American Mathematical Society, 2001.
22. D. Needell and J. A. Tropp, Cosamp: Iterative signal recovery from incomplete and inaccurate samples, Applied and Computational Harmonic Analysis 26 (2009), 301-321.
23. D. Needell, J. A. Tropp, and R. Vershynin, Greedy signal recovery review, Proc. 42nd Asilomar Conf. Signals, Systems and Computers, 2008.
24. D. Needell and R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit, IEEE J. Sel. Topics Signal Process 4 (2009), 310-316.
25. S. Osher, M. Burger, D. Goldfarb, and J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. Simul. 4 (2005), 460-489.
26. S. Osher, Y. Mao, B. Dong, and W. Yin, Fast linearized Bregman iteration for compressive sensing and sparse denoising, Commun. Math. Sci. 8 (2010), 93-111.
27. Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition, Proceedings of the 27th Annual Asilomar Conference on Signals, Systems, and Computers, 1993, pp. 40-44.
28. A. J. Reader, J. C. Matthews, F. C. Sureau, C. Comtat, R. Trébossen, and I. Buvat, Fully 4D image reconstruction by estimation of an input function and spectral coefficients, IEEE Nuclear Science Symposium Conference Record, 2007, pp. 3260-3267.
29. J. A. Tropp, Just relax: Convex programming methods for identifying sparse signals in noise, IEEE Trans. Inform. Theory 52 (2006), 1030-1051.
30. J. A. Tropp and A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory 53 (2007), 4655-4666.
31. Y. Wang and W. Yin, Sparse signal reconstruction via iterative support detection, SIAM J. Imaging Sci. 3 (2010), 462-491.
32. W. Yin, Analysis and generalizations of the linearized Bregman method, SIAM J. Imaging Sciences 3 (2010), no. 4, pp. 856-877.
33. 1-minimization with applications to compressed sensing, SIAM J. Imaging Sci. 1 (2008), 143-168.