  Positron Emission Particle Tracking (PEPT) is an imaging method that tracks
individual radioactive particles. PEPT relies on the detection of back-to-back
photon pairs emitted by positron annihilation. It requires an algorithm to
locate the radioactive particles based on the set of lines defined by
successive photon-pair detections. We propose and test a new algorithm for this
task. The algorithm relies on the maximization of a likelihood arising from a
simple Gaussian-mixture model defined in the space of lines. The model includes
a component that accounts for spurious lines caused by scattering and random
coincidence, and treats the relative activity of particles as well as their
positions as parameters to be inferred. Values of these parameters that
approximately maximize the likelihood are computed by application of an
expectation-maximization algorithm. A generalization of the model that includes
the particle velocities and accelerations as additional parameters takes
advantage of the information contained in the exact timing of positron
annihilations to reconstruct pieces of trajectories rather than fixed
positions, with clear benefits.
  We test the algorithm on both simulated and experimental data. The results
show the algorithm to be highly effective for the simultaneous tracking of many
particles (up to 80 in one test). It provides estimates of particle positions
that are easily mapped to entire trajectories and handles a variable number of
particles in the field of view. The ability to track a large number of
particles robustly offers the possibility of a dramatic expansion of the scope
of PEPT
