A Convergence Proof of MLEM and MLEM-3 with Fixed Background.

Maximum Likelihood Expectation-Maximization (MLEM) is a popular algorithm to reconstruct the activity image in Positron Emission Tomography (PET). This paper introduces a 'fundamental equality' for the MLEM complete data from which two key properties easily follow that allows us to: (i) 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 (ii) generalize this proof for the MLEM-3 algorithm. Moreover we give necessary and sufficient conditions for the solution to be unique.