A Theoretical Foundation for the Stein-Winter "Probability Hypothesis Density (PHD)" Multitarget tracking Approach

In several unpublished manuscripts written from 1993 to 1995, Michael Stein, C.L. Winter, and Robert Tenney introduced a multitarget tracking and evidential-accumulation concept called a "Probability Hypothesis Surface" (PHS) .A PHS is the graph of a probability distribution-the Probability Hypothesis Density (PHD)-that, when integrated over a region in target state space, gives the expected number of targets in that region. The PHD is uniquely defined by this property: Any other density function that satisfies it must be the PHD. In particular , the PHD is the expected value of the point process of a random track-set-i.e. , of the density that, when integrated over a region in state space, gives the exact (random) number of targets in that region. In 1997 in the book Mathematics of Data Fusion I sketched the elements of a theoretical foundation for PHS/PHD. The purpose of this paper is to publish a full account of this material for the first time. We show that the PHD is a first-order moment statistic of the random multitarget process and, consequently that from a computational perspective it is a multitarget analog of single-target constant-gain Kalman filters such as the a-fJ-'Y filter.l