A Bayesian look at the optimal track labelling problem

In multi-target tracking (MTT), the problem of assigning labels to tracks (track labelling) is vastly covered in literature, but its exact mathematical formulation, in terms of Bayesian statistics, has not been yet looked at in detail. Doing so, however, may help us to understand how (and when) Bayes-optimal track labelling should be performed or numerically approximated. Moreover, it can help us to better understand and tackle some practical difficulties associated with the MTT problem, in particular, the situation where targets move in close proximity with each other and thereafter separate. In this paper, we rigorously formulate the probabilistic track labelling problem using Finite Set Statistics (FISST), identifying when the problem is well- (or ill-) posed. We look in detail at the phenomenon of confusion on track labelling after targets have moved in close proximity ("mixed labelling"). We derive statistics associated with track labelling, with solid physical interpretation, that can be used to quantify mixed labelling, find the optimal track label assignment, and evaluate track labelling performance.