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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Bibliographic Details
Published in:Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche Vol. 68; no. 2; pp. 259 - 275
Main Authors: Bertero, Mario, De Mol, Christine
Format: Journal Article
Language:English
Published: Milan Springer Milan 01.11.2022
Springer Nature B.V
Subjects:
Algebraic Geometry
Algorithms
Analysis
Astronomy
Convergence
Emission
Geometry
History of Mathematical Sciences
Mathematics
Mathematics and Statistics
Maximization
Medical imaging
Numerical Analysis
Optimization
92C55
68U10
Inverse problems
78A46
85-08
Imaging
Background emission
65K10
Poisson data
EM algorithm
ISSN:0430-3202, 1827-1510
Online Access:Get full text
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Summary: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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ISSN:0430-3202
1827-1510
DOI:10.1007/s11565-022-00414-9