A row-action alternative to the EM algorithm for maximizing likelihoods in emission tomography

The maximum likelihood (ML) approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography (ECT) since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection (FBP) algorithm. The expectation maximization (EM) algorithm is an often-used iterative approach for maximizing the Poisson likelihood in ECT because of its attractive theoretical and practical properties. Its major disadvantage is that, due to its slow rate of convergence, a large amount of computation is often required to achieve an acceptable image. In this paper we present a row-action maximum likelihood algorithm (RAMLA) as an alternative to the EM algorithm for maximizing the Poisson likelihood in ECT. We deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of RAMLA increase the Poisson likelihood in ECT at an order of magnitude faster that the standard EM algorithm. Specifically, we show that, from the point of view of measuring total radionuclide uptake in simulated brain phantoms, iterations 1, 2, 3, and 4 of RAMLA perform at least as well as iterations 45, 60, 70, and 80, respectively, of EM. Moreover, we show that iterations 1, 2, 3, and 4 of RAMLA achieve comparable likelihood values as iterations 45, 60, 70, and 80, respectively, of EM. We also present a modified version of a recent fast ordered subsets EM (OS-EM) algorithm and show that RAMLA is a special case of this modified OS-EM. Furthermore, we show that our modification converges to a ML solution whereas the standard OS-EM does not.