A simple convergence proof of the ML-EM algorithm in the presence of background emission
For data obeying a Poisson statistics, the ML-EM (“Maximum Likelihood - Expectation Maximization”), also known as the Richardson-Lucy algorithm, is frequently used and its convergence properties are well known since several decades. To take into account the ubiquitous presence of background emission...
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| Veröffentlicht in: | Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche Jg. 68; H. 2; S. 259 - 275 |
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| Abstract | For data obeying a Poisson statistics, the ML-EM (“Maximum Likelihood - Expectation Maximization”), also known as the Richardson-Lucy algorithm, is frequently used and its convergence properties are well known since several decades. To take into account the ubiquitous presence of background emission in several important applications, e.g. in astronomy and medical imaging, a modified algorithm is used in practice. However, despite of its popularity, the convergence of this modified algorithm with background has been established only recently by Salvo and Defrise (IEEE Trans. Med. Imaging. 38:721–729, 2019) in the usual probabilistic context of EM (Expectation Maximization) methods. We present in this paper an alternative convergence proof, which we deem simpler, in a deterministic framework and using only basic tools from convex analysis and optimization theory. |
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| AbstractList | For data obeying a Poisson statistics, the ML-EM (“Maximum Likelihood - Expectation Maximization”), also known as the Richardson-Lucy algorithm, is frequently used and its convergence properties are well known since several decades. To take into account the ubiquitous presence of background emission in several important applications, e.g. in astronomy and medical imaging, a modified algorithm is used in practice. However, despite of its popularity, the convergence of this modified algorithm with background has been established only recently by Salvo and Defrise (IEEE Trans. Med. Imaging. 38:721–729, 2019) in the usual probabilistic context of EM (Expectation Maximization) methods. We present in this paper an alternative convergence proof, which we deem simpler, in a deterministic framework and using only basic tools from convex analysis and optimization theory. |
| Author | Bertero, Mario De Mol, Christine |
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| References | Natterer, Wübbeling (CR11) 2001 Lange, Carson (CR7) 1984; 8 De Pierro (CR17) 1995; 14 Byrne (CR18) 2014 Bertero, Boccacci, De Mol (CR2) 2022 Fessler, Hero (CR13) 1995; 4 Lange, Hunter, Yang (CR15) 2000; 9 Shepp, Vardi (CR5) 1982; 1 Mülthei, Schorr, Törnig (CR9) 1987; 9 Vardi, Shepp, Kaufman (CR8) 1985; 80 Salvo, Defrise (CR1) 2019; 38 Politte, Snyder (CR12) 1991; 10 Richardson (CR3) 1972; 62 Dempster, Laird, Rubin (CR6) 1977; 39 De Pierro (CR16) 1993; 12 Bertero, Boccacci, Ruggiero (CR14) 2018 Lucy (CR4) 1974; 79 Iusem (CR10) 1992; 6 JA Fessler (414_CR13) 1995; 4 AP Dempster (414_CR6) 1977; 39 WH Richardson (414_CR3) 1972; 62 K Lange (414_CR7) 1984; 8 DG Politte (414_CR12) 1991; 10 AR De Pierro (414_CR16) 1993; 12 CL Byrne (414_CR18) 2014 AR De Pierro (414_CR17) 1995; 14 AN Iusem (414_CR10) 1992; 6 K Lange (414_CR15) 2000; 9 M Bertero (414_CR2) 2022 K Salvo (414_CR1) 2019; 38 M Bertero (414_CR14) 2018 F Natterer (414_CR11) 2001 LA Shepp (414_CR5) 1982; 1 Y Vardi (414_CR8) 1985; 80 LB Lucy (414_CR4) 1974; 79 HN Mülthei (414_CR9) 1987; 9 |
| References_xml | – volume: 12 start-page: 328 issue: 2 year: 1993 end-page: 333 ident: CR16 article-title: On the Relation Between the ISRA and the EM Algorithm for Positron Emission Tomography publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.232263 – volume: 14 start-page: 132 issue: 1 year: 1995 end-page: 137 ident: CR17 article-title: A Modified Expectation Maximization Algorithm for Penalized Likelihood Estimation in Emission Tomography publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.370409 – year: 2014 ident: CR18 publication-title: Iterative Optimization in Inverse Problems doi: 10.1201/b16485 – year: 2001 ident: CR11 publication-title: Mathematical Methods in Image Reconstruction doi: 10.1137/1.9780898718324 – volume: 62 start-page: 55 issue: 1 year: 1972 end-page: 59 ident: CR3 article-title: Bayesian-Based Iterative Method of Image Restoration publication-title: J. Opt. Soc. Am. doi: 10.1364/JOSA.62.000055 – volume: 4 start-page: 1417 issue: 10 year: 1995 end-page: 1429 ident: CR13 article-title: Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms publication-title: IEEE Trans. Image Process. doi: 10.1109/83.465106 – volume: 9 start-page: 1 issue: 1 year: 2000 end-page: 20 ident: CR15 article-title: Optimization Transfer Using Surrogate Objective Functions publication-title: J. Comput. Graph. Stat. doi: 10.2307/1390605 – volume: 38 start-page: 721 issue: 3 year: 2019 end-page: 729 ident: CR1 article-title: A Convergence Proof of MLEM and MLEM-3 with Fixed Background publication-title: IEEE Trans. Med. Imaging doi: 10.1109/TMI.2018.2870968 – volume: 80 start-page: 8 issue: 389 year: 1985 end-page: 20 ident: CR8 article-title: A Statistical Model for Positron Emission Tomography publication-title: J. Am. Stat. Assoc. doi: 10.1080/01621459.1985.10477119 – volume: 8 start-page: 306 issue: 2 year: 1984 end-page: 316 ident: CR7 article-title: EM Reconstruction Algorithms for Emission and Transmission Tomography publication-title: J. Comput. Assist. Tomogr. – volume: 6 start-page: 57 year: 1992 end-page: 67 ident: CR10 article-title: A Short Convergence Proof of the EM Algorithm for a Specific Poisson Model publication-title: Revista Brasileira de Probabilidade e Estatistica – year: 2018 ident: CR14 publication-title: Inverse Imaging with Poisson Data - From cells to galaxies doi: 10.1088/2053-2563/aae109 – volume: 10 start-page: 82 issue: 1 year: 1991 end-page: 89 ident: CR12 article-title: Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.75614 – volume: 79 start-page: 745 year: 1974 end-page: 754 ident: CR4 article-title: An iterative technique for the rectification of observed distributions publication-title: Astron. J. doi: 10.1086/111605 – volume: 39 start-page: 1 issue: 1 year: 1977 end-page: 38 ident: CR6 article-title: Maximum Likelihood for Incomplete Data via the EM Algorithm publication-title: J. Roy. Stat. Soc. B – volume: 9 start-page: 137 issue: 1 year: 1987 end-page: 168 ident: CR9 article-title: On an iterative method for a class of integral equations of the first kind publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.1670090112 – volume: 1 start-page: 113 issue: 2 year: 1982 end-page: 122 ident: CR5 article-title: Maximum Likelihood Reconstruction for Emission Tomography publication-title: IEEE Trans. Med. Imaging doi: 10.1109/TMI.1982.4307558 – year: 2022 ident: CR2 publication-title: Introduction to Inverse Problems in Imaging doi: 10.1201/9781003032755 – volume: 39 start-page: 1 issue: 1 year: 1977 ident: 414_CR6 publication-title: J. Roy. Stat. Soc. B doi: 10.1111/j.2517-6161.1977.tb01600.x – volume-title: Inverse Imaging with Poisson Data - From cells to galaxies year: 2018 ident: 414_CR14 doi: 10.1088/2053-2563/aae109 – volume-title: Introduction to Inverse Problems in Imaging year: 2022 ident: 414_CR2 doi: 10.1201/9781003032755 – volume: 38 start-page: 721 issue: 3 year: 2019 ident: 414_CR1 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/TMI.2018.2870968 – volume: 12 start-page: 328 issue: 2 year: 1993 ident: 414_CR16 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.232263 – volume-title: Iterative Optimization in Inverse Problems year: 2014 ident: 414_CR18 doi: 10.1201/b16485 – volume: 79 start-page: 745 year: 1974 ident: 414_CR4 publication-title: Astron. J. doi: 10.1086/111605 – volume: 9 start-page: 1 issue: 1 year: 2000 ident: 414_CR15 publication-title: J. Comput. Graph. Stat. doi: 10.2307/1390605 – volume: 6 start-page: 57 year: 1992 ident: 414_CR10 publication-title: Revista Brasileira de Probabilidade e Estatistica – volume: 14 start-page: 132 issue: 1 year: 1995 ident: 414_CR17 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.370409 – volume: 10 start-page: 82 issue: 1 year: 1991 ident: 414_CR12 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/42.75614 – volume: 62 start-page: 55 issue: 1 year: 1972 ident: 414_CR3 publication-title: J. Opt. Soc. Am. doi: 10.1364/JOSA.62.000055 – volume: 1 start-page: 113 issue: 2 year: 1982 ident: 414_CR5 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/TMI.1982.4307558 – volume: 8 start-page: 306 issue: 2 year: 1984 ident: 414_CR7 publication-title: J. Comput. Assist. Tomogr. – volume: 9 start-page: 137 issue: 1 year: 1987 ident: 414_CR9 publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.1670090112 – volume: 4 start-page: 1417 issue: 10 year: 1995 ident: 414_CR13 publication-title: IEEE Trans. Image Process. doi: 10.1109/83.465106 – volume-title: Mathematical Methods in Image Reconstruction year: 2001 ident: 414_CR11 doi: 10.1137/1.9780898718324 – volume: 80 start-page: 8 issue: 389 year: 1985 ident: 414_CR8 publication-title: J. Am. Stat. Assoc. doi: 10.1080/01621459.1985.10477119 |
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| Title | A simple convergence proof of the ML-EM algorithm in the presence of background emission |
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