A closed-form solution for the probability hypothesis density filter

The problem of dynamically estimating a time-varying set of targets can be cast as a filtering problem using the random finite set (or point process) framework. The probability hypothesis density (PHD) filter is a recursion that propagates the posterior intensity function-a 1st-order moment-of the random set of multiple targets in time. Like the Bayesian single-target filter, the PHD recursion also suffers from the curse of dimensionality. Although sequential Monte Carlo implementations have demonstrated the potential of the PHD filter, so far no closed-form solutions have yet been developed. In this paper, an analytic solution to the PHD recursion is proposed for linear Gaussian target dynamics with Gaussian births. This result is analogous to the Kalman recursion in Bayesian single-target filtering. Extension to nonlinear dynamics is also discussed.