First-moment multi-object forward-backward smoothing

The optimal solution to the problem of detecting, tracking and identifying multiple targets can be found through a direct generalisation of the Bayes filter to multi-object systems using Mahler's Finite Set Statistics. Due to the inherent complexity of the multi-object Bayes filter, Mahler proposed to propagate the first-order multi-object moment density, known as the Probability Hypothesis Density (PHD), instead of the multi-object posterior. This was derived using the concept of the probability generating functional (p.g.fl.) from point process theory. In this paper, I derive multi-object first-moment smoothers for forward-backward smoothing through a new formulation of the p.g.fl. smoother which takes advantage of the p.g.fl. Bayes update. This formulation permits the straightforward derivation of first-moment multi-object smoothers, including the PHD smoother.