Abstract
Maximum likelihood expectation-maximization (MLEM) is a popular algorithm to reconstruct the activity image in positron emission tomography. This paper introduces a "fundamental equality" for the MLEM complete data from which two key properties easily follow that allows us to: 1) prove in an elegant and compact way the convergence of MLEM for a forward model with fixed background (i.e., counts such as random and scatter coincidences) and 2) generalize this proof for the MLEM-3 algorithm. Moreover, we give necessary and sufficient conditions for the solution to be unique.
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