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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Acknowledgements
We are grateful to Michel Defrise and Giorgio Talenti for a critical reading of the manuscript and for suggestions to improve it. We also thank Michel Defrise for having shared with us a set of unpublished notes on the subject.
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Bertero, M., De Mol, C. A simple convergence proof of the ML-EM algorithm in the presence of background emission. Ann Univ Ferrara 68, 259–275 (2022). https://doi.org/10.1007/s11565-022-00414-9
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DOI: https://doi.org/10.1007/s11565-022-00414-9