On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

Computational Optimization and Applications

Volumn 51, Issue 3, 2012, Pages 1065-1088

  Censor, Yair ^a   Chen, Wei ^a   Combettes, Patrick L ^a   Davidi, Ran ^a   Herman, Gabor T ^a   Chen, W ^b   Davidi, R ^b   Herman, G T ^b   Combettes, P L ^c  

^a UNIVERSITY OF HAIFA   (Israel)

^b City College of New York   (United States)

^c UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Convex feasibility problems; Linear inequalities; Numerical evaluation; Optimization; Projection methods; Sparse matrices

Indexed keywords

CONVEX FEASIBILITY PROBLEM; LINEAR INEQUALITIES; NUMERICAL EVALUATIONS; PROJECTION METHOD; SPARSE MATRICES;

COMPUTER SIMULATION; CONSTRAINT THEORY; OPTIMIZATION;

PROBLEM SOLVING;

EID: 84862006524     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-011-9401-7     Document Type: Article

References (86)

1. Agmon, S.: The relaxation method for linear inequalities. Can. J. Math. 6, 382-392 (1954)
2. Aharoni, R., Censor, Y.: Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Linear Algebra Appl. 120, 165-175 (1989)
3. Andersen, E.D., Andersen, K.D.: The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 197-232. Kluwer, Boston (2000)
4. Andrews, H.C., Hunt, B.R.: Digital Image Restoration. Englewood Cliffs, Prentice-Hall (1977)
5. Auslender, A.: Méthodes Numériques pour la Résolution des Problèmes d'Optimisation avec Contraintes. Thèse, Faculté des Sciences, Grenoble (1969)
6. Auslender, A.: Optimisation - Méthodes Numériques. Masson, Paris (1976)
7. Bauschke, H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150-159 (1996) (Pubitemid 126163209)
8. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367-426 (1996) (Pubitemid 126599619)
9. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248-264 (2001) (Pubitemid 34050614)
10. Bauschke, H.H., Combettes, P.L., Kruk, S.G.: Extrapolation algorithm for affine-convex feasibility problems. Numer. Algorithms 41, 239-274 (2006) (Pubitemid 43550992)
11. Bauschke, H.H., Deutsch, F., Hundal, H., Park, S.-H.: Accelerating the convergence of the method of alternating projections. Trans. Am. Math. Soc. 355, 3433-3461 (2003) (Pubitemid 37052679)
12. Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56, 715-738 (2004)
13. Bilbao-Castro, J.R., Marabini, R., Sorzano, C.O.S., García, I., Carazo, J.M., Fernández, J.J.: Exploiting desktop supercomputing for three-dimensional electron microscopy reconstructions using ART with blobs. J. Struct. Biol. 65, 19-26 (2009)
14. Bioucas-Dias, J., Figueiredo, M.: A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans. Image Process. 16, 2992-3004 (2007) (Pubitemid 350201232)
15. Blatt, D., Hero, A.O., III: Energy based sensor network source localization via projection onto convex sets (POCS). IEEE Trans. Signal Process. 54, 3614-3619 (2006) (Pubitemid 46405397)
16. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
17. Brègman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688-692 (1965)
18. Butnariu, D., Censor, Y.: On the behavior of a block-iterative projection method for solving convex feasibility problems. Int. J. Comput. Math. 34, 79-94 (1990)
19. Butnariu, D., Censor, Y., Reich, S. (eds.): Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Elsevier, Amsterdam (2001)
20. Butnariu, D., Davidi, R., Herman, G.T., Kazansev, I.G.: Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems. IEEE J. Sel. Top. Signal Process. 1, 540-547 (2007) (Pubitemid 351266621)
21. Carazo, J.M., Herman, G.T., Sorzano, C.O.S., Marabini, R.: Algorithms for three-dimensional reconstruction from imperfect projection data provided by electron microscopy. In: Frank, J. (ed.) Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, 2nd edn., pp. 217-243. Springer, New York (2006)
22. Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23, 444-466 (1981)
23. Censor, Y., Altschuler, M.D., Powlis, W.D.: On the use of Cimmino's simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Probl. 4, 607-623 (1988)
24. Censor, Y., Davidi, R., Herman, G.T.: Perturbation resilience and superiorization of iterative algorithms. Inverse Probl. 26, 065008 (2010)
25. Censor, Y., Elfving, T., Herman, G.T.: Averaging strings of sequential iterations for convex feasibility problems. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 101-114. Elsevier, Amsterdam (2001)
26. Censor, Y., Gordon, D., Gordon, R.: BICAV: A block-iterative, parallel algorithm for sparse systems with pixel-related weighting. IEEE Trans. Med. Imaging 20, 1050-1060 (2001) (Pubitemid 33777722)
27. Censor, Y., Segal, A.: On the string averaging method for sparse common fixed points problems. Int. Trans. Oper. Res. 16, 481-494 (2009)
28. Censor, Y., Tom, E.: Convergence of string-averaging projection schemes for inconsistent convex feasibility problems. Optim. Methods Softw. 18, 543-554 (2003)
29. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York (1997)
30. Cetin, A.E., Ozaktas, H., Ozaktas, H.M.: Resolution enhancement of low resolution wavefields with POCS algorithm. Electron. Lett. 39, 1808-1810 (2003)
31. Chen, W., Herman, G.T.: Effcient controls for finitely convergent sequential algorithms. ACM Trans. Math. Softw. 37, Article No. 14 (2010)
32. Chen, W., Craft, D., Madden, T.M., Zhang, K., Kooy, H.M., Herman, G.T.: A fast optimization algorithm for multicriteria intensity modulated proton therapy planning. Med. Phys. 7, 4938-4945 (2010)
33. Chiang, S., Cardi, C., Matej, S., Zhuang, H., Newberg, A., Alavi, A., Karp, J.S.: Clinical validation of fully 3-D versus 2.5-D RAMLA reconstruction on the Phillips-ADAC CPET PET scanner. Nucl. Med. Commun. 25, 1103-1107 (2004) (Pubitemid 39488942)
34. Choi, H., Baraniuk, R.G.: Multiple wavelet basis image denoising using Besov ball projections. IEEE Signal Process. Lett. 11, 717-720 (2004)
35. Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. Ric. Sci. (Roma) 1, 326-333 (1938)
36. Combettes, P.L.: The foundations of set theoretic estimation. Proc. IEEE 81, 182-208 (1993)
37. Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155-270 (1996)
38. Combettes, P.L.: Hilbertian convex feasibility problem: Convergence of projection methods. Appl. Math. Optim. 35, 311-330 (1997) (Pubitemid 127715737)
39. Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6, 493-506 (1997) (Pubitemid 127768380)
40. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005)
41. Combettes, P.L., Trussell, H.J.: Methods for digital restoration of signals degraded by a stochastic impulse response. IEEE Trans. Acoust. Speech Signal Process. 37, 393-401 (1989)
42. Cottle, R.W., Pang, J.-S.: On solving linear complementarity problems as linear programs. Math. Program. Stud. 7, 88-107 (1978)
43. Crombez, G.: Improving the speed of convergence in the method of projections onto convex sets. Publ. Math. (Debr.) 58, 29-48 (2001) (Pubitemid 33573952)
44. Crombez, G.: Finding common fixed points of strict paracontractions by averaging strings of sequential iterations. J. Nonlinear Convex Anal. 3, 345-351 (2002)
45. Davidi, R., Herman, G.T., Censor, Y.: Perturbation-resilient block-iterative projection methods with application to image reconstruction from projections. Int. Trans. Oper. Res. 16, 505-524 (2009)
46. Davidi, R., Herman, G.T., Klukowska, J.: SNARK09: A programming system for the reconstruction of 2D images from 1D projections. http://www.dig.cs.gc. cuny.edu/software/snark09/ (2011). Accessed 25 February 2011
47. Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, New York (2001)
48. Eckstein, J., Svaiter, B.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Optim. 48, 787-811 (2009)
49. Eggermont, P.P.B., Herman, G.T., Lent, A.: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl. 40, 37-67 (1981)
50. Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323-343 (2009)
51. Goldstein, T., Osher, S.: Personal communication (2010)
52. Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
53. González-Castaño, F.J., García-Palomares, U.M., Alba-Castro, J.L., Pousada-Carballo, J.M.: Fast image recovery using dynamic load balancing in parallel architectures, by means of incomplete projections. IEEE Trans. Image Process. 10, 493-499 (2001) (Pubitemid 32371584)
54. Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29, 471-482 (1970)
55. Gould, N.I.M.: How good are projection methods for convex feasibility problems? Comput. Optim. Appl. 40, 1-12 (2008) (Pubitemid 351464335)
56. Gu, J., Stark, H., Yang, Y.: Wide-band smart antenna design using vector space projection methods. IEEE Trans. Antennas Propag. 52, 3228-3236 (2004) (Pubitemid 40025707)
57. Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. U.S.S.R. Comput. Math. Math. Phys. 7, 1-24 (1967)
58. Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, London (2009)
59. Herman, G.T., Chen, W.: A fast algorithm for solving a linear feasibility problem with application to intensity-modulated radiation therapy. Linear Algebra Appl. 428, 1207-1217 (2008)
60. Hounsfield, G.N.: A method and apparatus for examination of a body by radiation such as X or gamma radiation. UK Patent No. 1283915 (1968/72)
61. Isola, A.A., Ziegler, A., Koehler, T., Niessen, W.J., Grass, M.: Motion compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions. Phys. Med. Biol. 53, 6777-6797 (2008)
62. Kaczmarz, S.: Angenäherte Auflösung von Systemen linearer Gleichungen. Bull. Acad. Sci. Pol. A 35, 355-357 (1937)
63. Kazantsev, I.G., Schmidt, S., Poulsen, H.F.: A discrete spherical X-ray transform of orientation distribution functions using bounding cubes. Inverse Probl. 25, 105009 (2009)
64. Kiwiel, K.C.: Monotone Gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems. Linear Algebra Appl. 252, 27-33 (1997) (Pubitemid 127380434)
65. Kiwiel, K.C., łopuch, B.: Surrogate projection methods for finding fixed points of firmly nonexpan-sive mappings. SIAM J. Optim. 7, 1084-1102 (1997)
66. Lee, S.-H., Kwon, K.-R.: Mesh watermarking based on projection onto two convex sets. Multimed. Syst. 13, 323-330 (2008) (Pubitemid 351191231)
67. Lewitt, R.M.: Multidimensional digital image representation using generalized Kaiser-Bessel window functions. J. Opt. Soc. Am. A 7, 1834-1846 (1990)
68. Liew, A.W.-C., Yan, H., Law, N.-F.: POCS-based blocking artifacts suppression using a smoothness constraint set with explicit region modeling. IEEE Trans. Circuits Syst. Video Technol. 15, 795-800 (2005) (Pubitemid 40819647)
69. Lu, Y.M., Karzand, M., Vetterli, M.: Demosaicking by alternating projections: theory and fast one-step implementation. IEEE Trans. Image Process. 19, 2085-2098 (2010)
70. Merzlyakov, Y.I.: On a relaxation method of solving systems of linear inequalities. U.S.S.R. Comput. Math. Math. Phys. 2, 504-510 (1963)
71. Motzkin, T.S., Schoenberg, I.J.: The relaxation method for linear inequalities. Can. J. Math. 6, 393-404 (1954)
72. Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)
73. Ottavy, N.: Strong convergence of projection-like methods in Hilbert spaces. J. Optim. Theory Appl. 56, 433-461 (1988) (Pubitemid 18625010)
74. Penfold, S.N., Schulte, R.W., Censor, Y., Bashkirov, V., McAllister, S., Schubert, K.E., Rosenfeld, A.B.: Block-iterative and string-averaging projection algorithms in proton computed tomography image reconstruction. In: Censor, Y., Jiang, M., Wang, G. (eds.) Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, pp. 347-367. Medical Physics Publishing, Madison (2010)
75. Pierra, G.: Éclatement de contraintes en parallèle pour la minimisation d'une forme quadratique. In: Proc. 7th IFIP Conf. on Optimization Techniques: Modeling and Optimization in the Service of Man. Lecture Notes in Comput. Sci., vol. 41, pp. 200-218. Springer, London (1976)
76. Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96-115 (1984) (Pubitemid 14526382)
77. Rhee, H.: An application of the string averaging method to one-sided best simultaneous approximation. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10, 49-56 (2003)
78. Samsonov, A.A., Kholmovski, E.G., Parker, D.L., Johnson, C.R.: POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging. Magn. Reson. Med. 52, 1397-1406 (2004) (Pubitemid 39612829)
79. Shaked, N.T., Rosen, J.: Multiple-viewpoint projection holograms synthesized by spatially incoherent correlation with broadband functions. J. Opt. Soc. Am. A 25, 2129-2138 (2008)
80. Sharma, G.: Set theoretic estimation for problems in subtractive color. Color Res. Appl. 25, 333-348 (2000)
81. Stark, H., Yang, Y.: Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley-Interscience, New York (1998)
82. van Wyk, B.J., van Wyk, M.A.: A POCS-based graph matching algorithm. IEEE Trans. Pattern Anal. Mach. Intell. 26, 1526-1530 (2004)
83. Youla, D.C., Webb, H.: Image restoration by the method of convex projections: Part 1-theory. IEEE Trans. Med. Imaging 1, 81-94 (1982) (Pubitemid 13557766)
84. Yukawa, M., Yamada, I.: Pairwise optimal weight realization-acceleration technique for set-theoretic adaptive parallel subgradient projection algorithm. IEEE Trans. Signal Process. 54, 4557-4571 (2006) (Pubitemid 46405600)
85. Zhang, T., Hong, H.: Restoration algorithms for turbulence-degraded images based on optimized estimation of discrete values of overall point spread functions. Opt. Eng. 44, 017005 (2005)
86. Ziegler, A., Grass, M., Koehler, T.: Method and device for the iterative reconstruction of cardiac images. US Patent Number 7596204 (2009)