A fast iterative shrinkage-thresholding algorithm for linear inverse problems

SIAM Journal on Imaging Sciences

Volumn 2, Issue 1, 2009, Pages 183-202

  Beck, Amir ^a   Teboulle, Marc ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Deconvolution; Global rate of convergence; Image deblurring; Iterative shrinkage thresholding algorithm; Least squares and l1 regularization problems; Linear inverse problem; Optimal gradient method; Two step iterative algorithms

Indexed keywords

APPROXIMATION THEORY; DECONVOLUTION; DIFFERENTIAL EQUATIONS; GRADIENT METHODS; IMAGE ENHANCEMENT; ITERATIVE METHODS; PROBLEM SOLVING; SHRINKAGE;

GLOBAL RATE OF CONVERGENCES; IMAGE DEBLURRING; ITERATIVE ALGORITHM; ITERATIVE SHRINKAGE-THRESHOLDING ALGORITHMS; LEAST SQUARE; LINEAR INVERSE PROBLEMS; OPTIMAL GRADIENT METHODS;

INVERSE PROBLEMS;

EID: 85014561619     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/080716542     Document Type: Article

References (35)

1. A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2001.
2. D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, MA, 1999.
3. J. Bioucas-Dias and M. Figueiredo, A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. Image Process., 16 (2007), pp. 2992-3004.
4. A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
5. K. Bredies and D. Lorenz, Iterative Soft-Thresholding Converges Linearly, Technical report, 2008; available online at http://arxiv.org/abs/0709.1598v3.
6. R. J. Bruck, On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl., 61 (1977), pp. 159-164.
7. A. Chambolle, R. A. DeVore, N. Y. Lee, and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), pp. 319-335.
8. S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), pp. 33-61.
9. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168-1200.
10. I. Daubechies, M. Defrise, and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), pp. 1413-1457.
11. D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Statist. Assoc., 90 (1995), pp. 1200-1224.
12. M. Elad, B. Matalon, and M. Zibulevsky, Subspace optimization methods for linear least squares with non-quadratic regularization, Appl. Comput. Harmon. Anal., 23 (2007), pp. 346-367.
13. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996.
14. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. II, Springer Ser. Oper. Res., Springer-Verlag, New York, 2003.
15. M. A. T. Figueiredo and R. D. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), pp. 906-916.
16. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1 (2007), pp. 586-597.
17. G. H. Golub, P. C. Hansen, and D. P. O’Leary, Tikhonov regularization and total least squares, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185-194.
18. 1-Regularized Minimization with Applications to Compressed Sensing, CAAM Technical report TR07-07, Rice University, Houston, TX, 2007.
19. P. C. Hansen and D. P. O’Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), pp. 1487-1503.
20. P. C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), pp. 1-35.
21. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1997.
22. P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering, Fundam. Algorithms 3, SIAM, Philadelphia, 2006.
23. E. S. Levitin and B. T. Polyak, Constrained minimization methods, Comput. Math. Math. Phys., 6 (1966), pp. 787-823.
24. B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), pp. 154-158.
25. P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors, IEEE Trans. Inform. Theory, 45 (1999), pp. 909-919.
26. A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, New York, 1983.
27. 2), Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543-547 (in Russian).
28. Y. E. Nesterov, Gradient Methods for Minimizing Composite Objective Function, CORE report, 2007; available at http://www.ecore.be/DPs/dp 1191313936.pdf.
29. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics Appl. Math. 30, SIAM, Philadelphia, 2000.
30. G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J.Math. Anal. Appl., 72 (1979), pp. 383-390.
31. B. T. Polyak, Introduction to Optimization, Translations Series in Mathematics and Engineering, Optimization Software, Publications Division, New York, 1987.
32. J. L. Starck, D. L. Donoho, and E. J. Candès, Astronomical image representation by the curevelet transform, Astron. Astrophys., 398 (2003), pp. 785-800.
33. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems, V. H. Winston, Washington, DC, 1977.
34. C. Vonesch and M. Unser, Fast iterative thresholding algorithm for wavelet-regularized deconvolution, in Proceedings of the SPIE Optics and Photonics 2007 Conference on Mathematical Methods: Wavelet XII, Vol. 6701, San Diego, CA, 2007, pp. 1-5.
35. S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2008), 2008, pp. 3373-3376.