MLEM and OSEM Deviate From the Cramer-Rao Bound at Low Counts

Maximum Likelihood (ML) reconstruction estimators are non biased and achieve the lowest variance, called the Cramer-Rao lower bound (CRLB), in the asymptotic regime, which in positron emission tomography (PET) or in single photon emission tomography (SPECT) corresponds to measuring an infinite number of counts. At finite number of counts or iterations, practical reconstruction algorithms are however biased, and nothing guarantees that the minimum variance expressed by the CRLB is achieved. We study the two dimensional Ordered Subsets Expectation Maximization (2D OSEM) algorithm with a finite number of counts and iterations, and investigate the question: given its bias, does this algorithm achieve the minimum variance predicted by the biased Cramer-Rao lower bound? We found a threshold separating two regimes: an asymptotic regime at large counts, where the variance almost equals the biased CRLB, even for a finite number of iterations and in cold regions, and a low counts regime where the variance significatively exceeds the biased CRLB. We extended our analysis to a realistic image by introducing the neighborhood method that evaluates the Cramer-Rao lower bound by inverting a submatrix of the Fisher matrix. We finally showed, both with a simulation and a theoretical toy example, that MLEM shares with OSEM the existence of a threshold in number of counts. A further analysis is needed to investigate the reason of the difference at low counts, which might indicate that there exists an algorithm with a smaller variance than OSEM for the same bias, or that a higher bound could be found.