An expectation-maximization algorithm for positron emission particle tracking

We develop a new algorithm for the tracking of radioactive particles using Positron Emission Particle Tracking (PEPT). The algorithm relies on the maximization of the likelihood of a simple Gaussian mixture model of the lines of response associated with positron annihilation. The model includes a component that accounts for spurious lines caused by scattering and random coincidence, and it 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 the 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.