  In Positron Emission Tomography, movement leads to blurry reconstructions
when not accounted for. Whether known a priori or estimated jointly to
reconstruction, motion models are increasingly defined in continuum rather that
in discrete, for example by means of diffeomorphisms. The present work provides
both a statistical and functional analytic framework suitable for handling such
models. It is based on time-space Poisson point processes as well as regarding
images as measures, and allows to compute the maximum likelihood problem for
line-of-response data with a known movement model. Solving the resulting
optimisation problem, we derive an Maximum Likelihood Expectation Maximisation
(ML-EM) type algorithm which recovers the classical ML-EM algorithm as a
particular case for a static phantom. The algorithm is proved to be monotone
and convergent in the low-noise regime. Simulations confirm that it correctly
removes the blur that would have occurred if movement were neglected.
