POSITRON EMISSION TOMOGRAPHY, BOREL MEASURES AND WEAK-CONVERGENCE

In this paper, we develop a refined version of the usual Poisson model for positron emission tomography (PET), in which the data space is fi nite dimensional, but the unknown emission intensity is represented by a Borel measure on the region of interest. We demonstrate that maximu m likelihood (ML) estimators exist in the space of Borel measures and analyse an extension of the finite dimensional EM algorithm for recons tructing the emission intensity. We present evidence that convergence of this functional iteration should be considered in the weak topology and obtain partial convergence results, which contain all the known c onvergence results to date as special cases. General conditions are ob tained under which an ML estimator can be represented by a bounded fun ction. In particular, we show that the regularity of ML estimators dep ends heavily on properties of the probabilities governing the PET math ematical model. We also show that, in some cases, no ML estimator can be represented by a bounded function. Although this paper is motivated by PET, the results apply to general inverse problems in which the un known measure, and the kernel representing the blurring operator are a ll positive.