Subspace correction methods for total variation and l1- minimization

SIAM Journal on Numerical Analysis

Volumn 47, Issue 5, 2009, Pages 3397-3428

  Fornasier, Massimo ^a   Schönlieb, Carola Bibiane ^b  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

11 Minimization; Convex optimization; Degenerate elliptic PDEs; Discontinuous solutions; Domain decomposition method; Image and signal processing; Parallel computation; Subspace corrections; Total variation minimization

Indexed keywords

11-MINIMIZATION; DEGENERATE ELLIPTIC PDES; DISCONTINUOUS SOLUTIONS; IMAGE AND SIGNAL PROCESSING; PARALLEL COMPUTATION; SUBSPACE CORRECTIONS; TOTAL VARIATION MINIMIZATION;

AIRCRAFT ENGINES; CONVERGENCE OF NUMERICAL METHODS; CONVEX OPTIMIZATION; GEOMETRY; HILBERT SPACES; IMAGE PROCESSING; NONLINEAR CONTROL SYSTEMS; OPERATIONS RESEARCH; OPTIMIZATION; PARALLEL ALGORITHMS; SIGNAL PROCESSING;

DOMAIN DECOMPOSITION METHODS;

EID: 70350159240     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070710779     Document Type: Article

References (62)

1. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press. New York, 2000.
2. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Partial differential Equations and the Calculus of Variation, Springer, New York, 2002.
3. G. Aubert and L. Vese, A variational method in image recovery, SIAM J. Numer. Anal., 34 (1997) 1948-1979.
4. R. Baraniuk, Compressive sensing, Lecture Notes in IEEE Signal Processing Magazine, 24 (2007) 118-120.
5. V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 2nd ed., D. Reidel, Dordrecht, The Netherlands, 1986.
6. H. H. Bauschke, F. Deutsch, H. Hundal, and S.-H. Park, Accelerating the convergence of the method of alternating projections, Trans. Amer. Math. Soc., 355 (2003) 3433-3461.
7. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009) 183-202.
8. J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57 (1991) 1-21.
9. H. H. Bauschke, J. M. Borwein, and A. S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space, in Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995), Contemp. Math. 204, AMS, Providence, RI, 1997 1-38.
10. K. Bredies and D. Lorenz, Linear convergence of iterative soft-thresholding, J. Fourier Anal. Appl., 14 (2008) 813-837.
11. R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math., 3 (1977) 459-470.
12. E. J. Candes, Compressive sampling, in International Congress of Mathematicians Vol. III, Eur. Math. Soc., Zürich, 2006 1433-1452.
13. E. J. Candes, J. Romberg, and T. Tao, Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006) 489-509.
14. E. J. Candes and T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies?, IEEE Trans. Inform. Theory, 52 (2006) 5406-5425.
15. E. J. Candes, M. Wakin, and S. Boyd, Enhancing sparsity by reweighted-1 minimization, J. Fourier Anal. Appl., 14 (2007) 877-905.
16. C. Carstensen, Domain decomposition for a non-smooth convex minimalization problems and its application to plasticity, Numer. Linear Algebra Appl., 4 (1998) 177-190.
17. A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004) 89-97.
18. A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, UCLA CAM report 08-19, 2008.
19. A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997) 167-188.
20. T. F. Chan and T. P. Mathew, Domain decomposition algorithms, in Acta Numerica 1994, Acta Numer., Cambridge University Press, Cambridge, UK, 1994 61-143.
21. S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998) 33-61.
22. J. F. Claerbout and F. Muir, Robust modeling with erratic data, Geophysics, 38 (1973) 826-844.
23. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005) 1168-1200.
24. J. Darbon and M. Sigelle, A fast and exact algorithm for total variation minimization, in Proceedings of the 2nd Iberian Conference on Pattern Recognition and Image Analysis (IbPRIA 2005), Lecture Notes in Comput. Sci. 3522, Springer-Verlag, New York, 2005 351-359.
25. I. Daubechies, M. Defrise, and C. DeMol, An iterative thresholding algorithm for linear inverse problems, Comm. Pure Appl. Math., 57 (2004) 1413-1457.
26. I. Daubechies, M. Fornasier, and I. Loris, Acceleration of the projected gradient method for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14 (2008) 764-792.
27. I. Daubechies, G. Teschke, and L. Vese, Iteratively solving linear inverse problems under general convex constraints, Inverse Probl. Imaging, 1 (2007) 29-46.
28. D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal., 34 (1997) 1779-1791.
29. D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006) 1289-1306.
30. D. L. Donoho and B. F. Logan, Signal recovery and the large sieve, SIAM J. Appl. Math., 52 (1992) 577-591.
31. D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49 (1989) 906-931.
32. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. Translated by Minerva Translations, Ltd., London, Studies inMathematics and Its Applications 1, North-Holland, Amsterdam; Elsevier, New York, 1976.
33. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer, Dordrecht, The Netherlands, 1996.
34. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
35. M. Figueiredo, R. Nowak, and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Selected Topics in Signal Process., 4 (2007) 586-597.
36. M. Fornasier, Domain decomposition methods for linear inverse problems with sparsity constraints, Inverse Problems, 23 (2007) 2505-2526.
37. M. Fornasier, A. Langer, and C.-B. Schönlieb, A convergent overlapping domain decomposition method for total variation minimization, preprint, http://arxiv.org/abs/0905.2404, 2009.
38. M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008) 577-613.
39. M. Fornasier and H. Rauhut, Iterative thresholding algorithms, Appl. Comput. Harmonic Anal., 25 (2008) 187-208.
40. T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, UCLA CAM Report 08-29, 2008.
41. S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, An interior-point method for large-scale-1-regularized least squares, IEEE J. Selected Topics in Signal Process., 1 (2007) 606-617.
42. Y-J. Lee, J. Xu, and L. Zikatanov, Successive subspace correction method for singular system of equations, in Fourteenth International Conference on Domain Decomposition Methods, I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, eds., UNAM, Mexico City, Mexico, 2003 315-321.
43. I. Loris, On the performance of algorithms for the minimization of 1-penalized functionals, Inverse Problems, 25 (2009), 035008.
44. P.-L. Lions, On the Schwarz alternating method, in Proceedings of the First International Symposium on Domain Decomposition Methods for Partial differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Philadelphia, 1988 1-42.
45. R. Nabben and D. B. Szyld, Schwarz iterations for symmetric positive semidefinite problems, Research Report 05-11-103, Department of Mathematics, Temple University, Philadelphia, PA, 2005.
46. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005) 460-489.
47. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial differential Equations, Numer. Math. Sci. Comput., The Clarendon Press, New York, 1999.
48. R. T. Rockafellar and R. J. B. Wets, Variational analysis, Grundlehren Math. Wiss. 317, Springer-Verlag, Berlin, 1998.
49. L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992) 259-268.
50. F. Santosa and W. W. Symes, Linear inversion of band-limited re.ection seismograms, SIAM J. Sci. Statist. Comput., 7 (1986) 1307-1330.
51. J.-L. Starck, E. J. Candes, and D. L. Donoho, The curvelet transform for image denoising, IEEE Trans. Image Process., 11 (2002) 670-684.
52. X.-C. Tai and P. Tseng, Convergence rate analysis of an asynchronous space decomposition method for convex minimization, Math. Comp., 71 (2001) 1105-1135.
53. X.-C. Tai and J. Xu, Global convergence of subspace correction methods for convex optimization problems, Math. Comp., 71 (2002) 105-124.
54. H. L. Taylor, S. C. Banks, and J. F. McCoy, Deconvolution with the-1 norm, Geophys., 44 (1979) 39-52.
55. R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996) 267-288.
56. L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001) 131-161.
57. C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996) 227-238.
58. C. Vonesch and M. Unser, A fast multilevel algorithm for wavelet-regularized image restoration, IEEE Trans. Image Process., 18 (2009) 509-523.
59. P. Weiss, L. Blanc-Féraud, and G. Aubert, effcient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009) 2047-2080.
60. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992) 581-613.
61. J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc., 15 (2002) 573-597.
62. http://homepage.univie.ac.at/carola.schoenlieb/webpagetvdode/ tvdodenumerics.htm, 2008.