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Convergence of the SMC Implementation of the PHD Filte
Authors:Adam M Johansen  Sumeetpal S Singh  Arnaud Doucet  Ba-Ngu Vo
Institution:(1) Department of Engineering, Trumpington Street, Cambridge, CB2 1PZ, UK;(2) Department of Computer Science, University of British Columbia, 201-2366 Main Mall, Vancouver, BC, V6T 1Z2, Canada;(3) Department of Statistics, University of British Columbia, 333-6356 Agricultural Road, Vancouver, BC, V6T 1Z2, Canada;(4) Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, VIC, 3010, Australia
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.
Keywords:Central limit theorem  Filtering  Sequential Monte Carlo  Finite random sets
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