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?