Convergence of the SMC Implementation of the PHD Filte

  • Adam M. JohansenEmail author
  • Sumeetpal S. Singh
  • Arnaud Doucet
  • Ba-Ngu Vo


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.


Central limit theorem Filtering Sequential Monte Carlo Finite random sets 

AMS 2000 Subject Classification

Primary 60F05 Secondary 60F25 62P30 93E11 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Adam M. Johansen
    • 1
    Email author
  • Sumeetpal S. Singh
    • 1
  • Arnaud Doucet
    • 2
    • 3
  • Ba-Ngu Vo
    • 4
  1. 1.Department of EngineeringCambridgeUK
  2. 2.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  3. 3.Department of StatisticsUniversity of British ColumbiaVancouverCanada
  4. 4.Department of Electrical and Electronic EngineeringUniversity of MelbourneMelbourneAustralia

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