Some convergence results of OS-EM algorithm for ill-posed integral equations

References

• Shepp LA, Vardi Y. Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging. 1982;1:113–122. doi: 10.1109/TMI.1982.4307558
   PubMedGoogle Scholar
• Mülthei HN, Schorr B. On an iterative method for a class of integral equations of the first kind. Math Methods Appl Sci. 1987;9:137–168. doi: 10.1002/mma.1670090112
   Web of Science ®Google Scholar
• Mülthei HN, Schorr B. On properties of the iterative maximum likelihood reconstruction method. Math Methods Appl Sci. 1989;11:331–342. doi: 10.1002/mma.1670110303
   Web of Science ®Google Scholar
• Mülthei HN. Iterative continuous maximum likelihood reconstruction method. Math Methods Appl Sci. 1993;15:275–286. doi: 10.1002/mma.1670150405
   Web of Science ®Google Scholar
• Kondor A. Method of convergent weights—an iterative procedure of solving Fredholm's integral equations of the first kind. Nuclear Instrum Methods Phys Res. 1983;216:177–181. doi: 10.1016/0167-5087(83)90348-4
   Google Scholar
• Lange K, Carson R. EM reconstruction algorithms for emission and transmission tomography. J Comput Assist Tomogr. 1984;8:306–316.
   PubMed Web of Science ®Google Scholar
• Hudson HM, Larkin RS. Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans Med Imaging. 1994;13:601–609. doi: 10.1109/42.363108
   PubMed Web of Science ®Google Scholar
• Kaufman L. Implenenting and accelerating the EM algorithm for positron emission tomography. IEEE Trans Med Imaging. 1987;6:37–51. doi: 10.1109/TMI.1987.4307796
   PubMed Web of Science ®Google Scholar
• Lewitt RM, Muehllehner G. Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation. IEEE Trans Med Imaging. 1986;5:16–22. doi: 10.1109/TMI.1986.4307734
   PubMed Web of Science ®Google Scholar
• Haltmeier M, Leitao A, Resmerita E. On the regularization Methods of EM-Kaczmarz type. Inverse Probl. 2009;25:1–17. doi: 10.1088/0266-5611/25/7/075008
   Web of Science ®Google Scholar
• Vishampayan S, Stamos J, Mayans R, Koral K, Clinthorne N, Roger WL. Maximum likelihood image reconstruction for SPECT. J. Nucl. Med. 1985;1:620–635.
   Google Scholar
• Tanaka E, Nohara N, Tomitani T, Yamamoto M. Utilization of non-negativity constraints in reconstruction of emission tomograms. Proceedings of the 9th Conference of Information Processing in Medical Imaging, Washington DC; 2014. p. 379–393.
   Google Scholar
• Hwang D, Zeng GL. Convergence study of an accelerated ML-EM algorithm using bigger step size. Phys Med Biol. 2006;51:237–252. doi: 10.1088/0031-9155/51/2/004
   PubMed Web of Science ®Google Scholar
• Geng C, Wang J. Convergence analysis of an accelerated expectation-maximization algorithm for ill-posed integral equations. Math Meth Appl Sci. 2016;39:668–675. doi: 10.1002/mma.3508
   Web of Science ®Google Scholar
• Engl H, Hanke M, Neubauer A. Regularization of inverse problems. Dordrecht: Kluwer; 1996.
   Google Scholar
• Vardi Y, Shepp LA, Kanfman L. A statistical model for positron emission tomography. J Amer Statist Assoc. 1985;80:8–20. doi: 10.1080/01621459.1985.10477119
   Web of Science ®Google Scholar
• Resmerita E, Anderssen RS. Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems. Math Methods Appl Sci. 2007;30:1527–1544. doi: 10.1002/mma.855
   Web of Science ®Google Scholar
• Resmerita E, Engl HW, Insem AN. The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis. Inverse Probl. 2007;23:2575–2588. doi: 10.1088/0266-5611/23/6/019
   Web of Science ®Google Scholar
• Borwein JM, Lewis AS. Convergence of best entropy estimates. SIAM J Optim. 1991;I(2):191–205. doi: 10.1137/0801014
   Google Scholar
• Lindblad G. Entropy, information and quantum measurements. Commun. Math. Phys.. 1973;33:305–322. doi: 10.1007/BF01646743
   Web of Science ®Google Scholar
• Perlman MD. Jensen's inequality for a convex vector-valued function on an infinite-dimensional space. J Multivar Anal. 1974;4:52–65. doi: 10.1016/0047-259X(74)90005-0
   Google Scholar
• Iusem AN. A short convergence proof of the EM algorithm for a specific Poisson model. REBRAPE. 1992;6(1):57–67.
   Google Scholar