×

On the convergence of general projection methods for solving convex feasibility problems with applications to the inverse problem of image recovery. (English) Zbl 1412.90164

Summary: For an arbitrary family of closed convex sets with nonempty intersection in a Hilbert space, we consider the classical convex feasibility problem. We study the convergence property of the recently introduced unified projection algorithm B-EMOPP for solving this problem. For this, a new general control strategy is proposed, which we call the ‘quasi-coercive control’. Under mild assumptions, we prove the convergence of B-EMOPP using these new control strategies as well as various other strategies. Several known results are extended and improved. The proposed algorithm is then applied to the inverse problem of image recovery.

MSC:

90C48 Programming in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Combettes, PL, The convex feasibility problem in image recovery, Adv Imag Electron Phys, 95, 1, 155-270, (1996) · doi:10.1016/S1076-5670(08)70157-5
[2] Deutsch, F.; Singh, SP, Approximation theory, spline functions and applications, The method of alternating orthogonal projections, 105-121, (1992), Netherlands, Dordrecht: Springer, Netherlands, Dordrecht · doi:10.1007/978-94-011-2634-2_5
[3] Carmi, A.; Censor, Y.; Gurfil, P., Convex feasibility modeling and projection methods for sparse signal recovery, J Comput Appl Math, 236, 17, 4318-4335, (2012) · Zbl 1256.94016 · doi:10.1016/j.cam.2012.03.021
[4] Gholami, MR; Tetruashvili, L.; Ström, EG; Censor, Y., Cooperative wireless sensor network positioning via implicit convex feasibility, IEEE Trans Signal Proc, 61, 23, 5830-5840, (2013) · Zbl 1394.94208 · doi:10.1109/TSP.2013.2279770
[5] Censor, Y.; Altschuler, MD; Powlis, WD, 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, 3, 607-623, (1998) · Zbl 0653.65049 · doi:10.1088/0266-5611/4/3/006
[6] Bauschke, HH, Projection algorithms and monotone operators [PhD thesis], (1996), Department of Mathematics: Simon Fraser University, Burnaby (BC), Canada, Department of Mathematics
[7] Bauschke, HH; Borwein, JM, On projection algorithms for solving convex feasibility problems, SIAM Rev, 38, 3, 367-426, (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[8] Borwein, JM; Li, GY; Yao, LJ, Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets, SIAM J Optim, 24, 1, 498-527, (2014) · Zbl 1296.41011 · doi:10.1137/130919052
[9] Combettes, PL, Hilbertian convex feasibility problem: convergence of projection methods, Appl Math Optim, 35, 1, 311-330, (1997) · Zbl 0872.90069 · doi:10.1007/BF02683333
[10] Censor, Y.; Cegielski, A., Projection methods: an annotated bibliography of books and reviews, Optimization, 64, 11, 2343-2358, (2014) · Zbl 1328.65001 · doi:10.1080/02331934.2014.957701
[11] Censor, Y.; Chen, W.; Combettes, PL; Davidi, R.; Herman, GT, On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints, Comput Optim Appl, 51, 3, 1065-1088, (2012) · Zbl 1244.90155 · doi:10.1007/s10589-011-9401-7
[12] Zhao, XP; Ng, KF; Li, C.; Yao, JC, Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems, Appl Math Optim, (2017) · doi:10.1007/s00245-017-9417-1
[13] Bauschke, HH; Borwein, JM, On the convergence of von Neumann’s alternating projection algorithm for two sets, Set-Valued Anal, 1, 2, 185-212, (1993) · Zbl 0801.47042 · doi:10.1007/BF01027691
[14] Borwein, JM; Li, GY; Tam, MK, Convergence rate analysis for averaged fixed point iterations in common fixed point problems, SIAM J Optim, 27, 1, 1-33, (2017) · Zbl 1361.90045 · doi:10.1137/15M1045223
[15] Phan, HM, Linear convergence of the Douglas-Rachford method for two closed sets, Optimization, 65, 2, 369-385, (2016) · Zbl 1334.49089 · doi:10.1080/02331934.2015.1051532
[16] Li, C.; Ng, KF; Pong, TK, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear spaces, SIAM J Optim, 18, 2, 643-665, (2007) · Zbl 1151.90054 · doi:10.1137/060652087
[17] Li, C.; Ng, KF, The dual normal CHIP and linear regularity for infinite systems of convex sets in Banach spaces, SIAM J Optim, 24, 3, 1075-1101, (2014) · Zbl 1330.90076 · doi:10.1137/130941493
[18] Li, C.; Zhao, XP; Hu, YH, Quasi-Slater and Farkas-Minkowski qualifications for semi-infinite programming with applications, SIAM J Optim, 23, 4, 2208-2230, (2013) · Zbl 1298.90119 · doi:10.1137/130911287
[19] Ng, KF; Yang, WH, Regularities and their relations to error bounds, Math Program Ser A, 99, 3, 521-538, (2004) · Zbl 1077.90050 · doi:10.1007/s10107-003-0464-9
[20] Zhao, XP, On constraint qualification for an infinite system of quasiconvex inequalities in normed linear space, Taiwanese J Math, 20, 3, 685-697, (2016) · Zbl 1357.90120 · doi:10.11650/tjm.20.2016.6713
[21] Zhao, XP, Constraint qualification for quasiconvex inequality system with applications in constraint optimization, J Nonlinear Convex Anal, 17, 5, 879-889, (2016)
[22] Clarke, F., Optimization and nonsmooth analysis, Society for industrial and applied mathematics, (1990) · Zbl 0696.49002 · doi:10.1137/1.9781611971309
[23] Zălinescu, C., Convex analysis in general vector spaces, (2002), River Edge (NJ): World Scientific, River Edge (NJ) · Zbl 1023.46003 · doi:10.1142/9789812777096
[24] Biggs, DSC; Andrews, M., Acceleration of iterative image restoration algorithms, Appl Opt, 36, 8, 1766-1775, (1997) · doi:10.1364/AO.36.001766
[25] Holmes, TJ; Bhattacharyya, S.; Cooper, JA; Hanzel, D.; Krishnamurthi, V.; Lin, W-C; Roysam, B.; Szarowski, DH; Turner, JN; Pawley, JB, Handbook of biological confocal microscopy, Light microscopic images reconstructed by maximum likelihood deconvolution, 389-402, (1995), NewYork (NY): Plenum Press/Springer, NewYork (NY) · doi:10.1007/978-1-4757-5348-6_24
[26] Sochen, N.; Zeevi, YY, Super-resolution of grey-level images by inverse diffusion processes, 91-95, (1998), Tel-Aviv, Israel · doi:10.1109/MELCON.1998.692346
[27] Lavielle, M., Detection of multiple changes in a sequence of dependent variables, Stochastic Process Appl, 83, 1, 79-102, (1999) · Zbl 0991.62014 · doi:10.1016/S0304-4149(99)00023-X
[28] Rudin, L.; Osher, SJ; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 1-4, 259-268, (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[29] Censor, Y.; Gibali, A.; Lenzen, F.; Schnörr, Chr, The implicit convex feasibility problem and its application to adaptive image denoising, J Comput Math, 34, 6, 608-623, (2016) · Zbl 1374.90305 · doi:10.4208/jcm.1606-m2016-0581
[30] Johnson, GM, Measuring images: differences, quality and appearance [PhD thesis], (2003), Rochester Institute of Technology, Rochester (NY): College of Science, Rochester Institute of Technology, Rochester (NY)
[31] O’Connor, D.; Vanderberghe, L., Total variation image deblurring with space-varying kernel, Comput Optim Appl, 67, 3, 521-541, (2017) · Zbl 1369.65035 · doi:10.1007/s10589-017-9901-1
[32] Shampine, LW; Reichelt, MW, The MATLAB ODE suite, SIAM J Sci Comput, 18, 1, 1-22, (1997) · Zbl 0868.65040 · doi:10.1137/S1064827594276424
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.