Implementation of an optimal first-order method for strongly convex total variation regularization

BIT Numerical Mathematics

Volumn 52, Issue 2, 2012, Pages 329-356

  Jensen, T L ^a   Jorgensen J H ^b   Hansen, P C ^b   Jensen, S H ^a  

^a AALBORG UNIVERSITY   (Denmark)

^b TECHNICAL UNIVERSITY OF DENMARK   (Denmark)

Author keywords

Optimal first order optimization methods; Strong convexity; Tomography; Total variation regularization

Indexed keywords

EID: 84862527511     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-011-0359-8     Document Type: Article

References (42)

1. Alter, F., Durand, S., Froment, J.: Adapted total variation for artifact free decompression of JPEG images. J. Math. Imaging Vis. 23, 199-211 (2005).
2. Barzilai, J., Borwein, J. M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141-148 (1988).
3. Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18, 2419-2434 (2009).
4. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183-202 (2009).
5. Becker, S., Bobin, J., Candès, E. J.: NESTA: a fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4(1), 1-39 (2011).
6. Becker, S., Candès, E. J., Grant, M.: Templates for convex cone problems with applications to sparse signal recovery. Math. Program. Comput. 3, 165-218 (2011).
7. Bioucas-Dias, J. M., Figueiredo, M. A. T.: A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans. Image Process. 16(12), 2992-3004 (2007).
8. Birgin, E. G., Martínez, J. M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196-1211 (2000).
9. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004).
10. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89-97 (2004).
11. Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B., Yuille, A. L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 3757, pp. 136-152. Springer, Berlin (2005).
12. 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).
13. Chan, T. F., Golub, G. H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20, 1964-1977 (1998).
14. Chan, T. F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005).
15. Combettes, P. L., Luo, J.: An adaptive level set method for nondifferentiable constrained image recovery. IEEE Trans. Image Process. 11, 1295-1304 (2002).
16. Dahl, J., Hansen, P. C., Jensen, S. H., Jensen, T. L.: Algorithms and software for total variation image reconstruction via first-order methods. Numer. Algorithms 53, 67-92 (2010).
17. Dai, Y. H., Liao, L. Z.: R-linear convergence of the Barzilai and Borwein gradient method. IMA J. Numer. Anal. 22, 1-10 (2002).
18. Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation-Part I: Fast and exact optimization. J. Math. Imaging Vis. 26, 261-276 (2006).
19. Fletcher, R.: Low storage methods for unconstrained optimization. In: Allgower, E. L., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations, pp. 165-179. Am. Math. Soc., Providence (1990).
20. Goldfarb, D., Yin, W.: Second-order cone programming methods for total variation-based image restoration. SIAM J. Sci. Comput. 27, 622-645 (2005).
21. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton's method. SIAM J. Numer. Anal. 23, 707-716 (1986).
22. Hansen, P. C.: Discrete Inverse Problems: Insight and Algorithms. SIAM, Philadelphia (2010).
23. Herman, G. T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, New York (2009).
24. Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for total bounded variation-based INF-convolution-type image restoration. SIAM J. Sci. Comput. 28, 1-23 (2006).
25. Jørgensen, J. H.: Tomobox (2010). www. mathworks. com/matlabcentral/fileexchange/28496-tomobox.
26. Kak, A. C., Slaney, M.: Principles of Computerized Tomographic Imaging. SIAM, Philadelphia (2001).
27. Lan, G., Lu, Z., Monteiro, R. D. C.: Primal-dual first-order methods with O(1/ε{lunate}) iteration-complexity for cone programming. Math. Program., Ser. A 126(1), 1-29 (2011).
28. Nemirovsky, A. S., Yudin, D. B.: Problem Complexity and Method Efficiency in Optimization. Wiley, New York (1983).
29. 2). Sov. Math. Dokl. 269, 543-547 (1983).
30. Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer Academic, Dordrecht (2004).
31. Nesterov, Y.: Smooth minimization of nonsmooth functions. Math. Program., Ser. A 103, 127-152 (2005).
32. Nesterov, Y.: Gradient methods for minimizing composite objective function (2007). CORE Discussion Paper No 2007076, www. ecore. be/DPs/dp_1191313936. pdf.
33. Nolet, G. (ed.): Seismic Tomography with Applications in Global Seismology and Exploration Geophysics. Reidel, Dordrecht (1987).
34. Parrish, R.: getLebedevSphere (2010). www. mathworks. com/matlabcentral/fileexchange/27097-getlebedevsphere.
35. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26-33 (1997).
36. Rudin, L. I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259-268 (1992).
37. Schabel, M.: 3D Shepp-Logan phantom (2006). www. mathworks. com/matlabcentral/fileexchange/9416-3d-shepp-logan-phantom.
38. Tseng, P.: On accelerated proximal gradient methods for convex-concave optimization. Manuscript (2008). www. math. washington. edu/~tseng/papers/apgm. pdf.
39. Vandenberghe, L.: Optimization methods for large-scale systems. Lecture Notes (2009). www. ee. ucla. edu/~vandenbe/ee236c. html.
40. Vogel, C. R., Oman, M. E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17, 227-238 (1996).
41. Weiss, P., Blanc-Féraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31, 2047-2080 (2009).
42. Zhu, M., Wright, S. J., Chan, T. F.: Duality-based algorithms for total-variation-regularized image restoration. Comput. Optim. Appl. (2008). doi: 10. 1007/s10589-008-9225-2.