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.