A simple convergence proof of the ML-EM algorithm in the presence of background emission. (English) Zbl 1498.78022
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 K. Salvo and M. Defrise [“A convergence proof of MLEM and MLEM-3 with fixed background”, IEEE Trans. Med. Imaging. 38, No. 3, 721–729 (2019; doi:10.1109/TMI.2018.2870968)] 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.
MSC:
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 68U10 | Computing methodologies for image processing |
| 92C55 | Biomedical imaging and signal processing |
| 85-08 | Computational methods for problems pertaining to astronomy and astrophysics |
| 65K10 | Numerical optimization and variational techniques |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
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