Abstract
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 \(p \geq 1\), and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.
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Johansen, A.M., Singh, S.S., Doucet, A. et al. Convergence of the SMC Implementation of the PHD Filte. Methodol Comput Appl Probab 8, 265–291 (2006). https://doi.org/10.1007/s11009-006-8552-y
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DOI: https://doi.org/10.1007/s11009-006-8552-y