Convergence of the SMC Implementation of the PHD Filter

The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order p ≥ 1, and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation and show that the variance is finite under similar assumptions. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.