Particle Probability Hypothesis Density Filter based on Pairwise Markov Chains

Most multi-target tracking filters assume that one target and its observation follow a Hidden Markov Chain (HMC) model, but the implicit independence assumption of HMC model is invalid in many practical applications, and a Pairwise Markov Chain (PMC) model is more universally suitable than traditional HMC model. A particle probability hypothesis density filter based on PMC model (PF-PMC-PHD) is proposed for the nonlinear multi-target tracking system. Simulation results show the effectiveness of PF-PMC-PHD filter, and that the tracking performance of PF-PMC-PHD filter is superior to the particle PHD filter based on HMC model in a scenario where we kept the local physical properties of nonlinear and Gaussian HMC models while relaxing their independence assumption.


Introduction
Random Finite Set (RFS) theory has been widely used in multi-target tracking field. Unlike traditional solutions based on data association, RFS-based solutions provide a theoretical framework without data association [1,2]. Among RFS based solutions, the Probability Hypothesis Density (PHD) filter propagates the first order moment of the posterior multi-target density [3], which is now widely applied . RFS based solutions can not get the analytical solution directly, and the implementations mainly based on numerical approximations, such as Gauss Mixture (GM) PHD filter [4,5], and on Sequential Monte Carlo (SMC, i.e. particle filter) methods [6,7].
Most multi-target tracking filters, including the classical PHD filter, assume that the targets and the observations they produce follow the well known HMC model. HMC model assumes that the state of a given target is a Markov Chain (MC), however, the Markovian and independence assumption implicit in the HMC model may not be invalid in practical applications [8]. In 2000, Pieczynski proposed the PMC model in order to relax the independence assumption of the HMC model [9][10][11], HMC model is a special PMC model, and PMC model is more universally suitable than HMC model. In 2013, Petetin and Desbouvries proposed a PHD filter for targets follow the PMC model(PMC-PHD) [8], and prove that the tracking performance of proposed PMC-PHD filter is better than the "classical" PHD filter based on the HMC model under the relaxed independence assumptions. and the PMC-PHD filter proposed by Petetin only considers the first order information of the target state, neglecting its high order information, and leads to the instability of the target number estimation. In view of this problem, Mahler proposed a Cardinalized Probability Hypothesis Density filter based on PMC model (PMC-CPHD), which is developed from PMC-PHD filter by propagating cardinality distribution function of the target simultaneously [12].
The GM implementation of PMC-PHD filter proposed by Petetin and Desbouvries is only suitable for the linear Gaussian multi-target tracking system. In this article, A particle implementation of PMC-PHD (PF-PMC-PHD) filter is proposed for the nonlinear multi-target tracking system based on PMC model. The sampling importance density function of the proposed PF-PMC-PHD filter does not contain the latest measurement information, which may lead to problems such as filtering divergence and particle degradation. Simulation result verifies the effectiveness of the PF-PMC-PHD filter, and shows that the performance of PF-PMC-PHD is better than the particle implementation of the typical HMC-PHD filter (PF-HMC-PHD) in a scenario where we kept the local physical properties of nonlinear and Gaussian HMC models while relaxing their independence assumption. can be factorized as follows: where 00 ( , ) p xy is the state distribution at the initial time, and the following formula (2) can be true [13]: Define target motion model follows the PMC model as: where 1 ,, k ww L are independent zero-mean Gaussian noises, and 11 21  Assuming that there is no spawning (if there is spawning the extension is immediate), according to the multi-target Bayesian principle, the joint PHD can be propagated through the following prediction and update formula: , where , ()

PF-PMC-PHD filter
There are two main methods to implement PHD filter, one is particle implementation and the other is GM implementation. The GM implementation of PMC-PHD filter proposed by Petetin and Desbouvries is only suitable for linear Gaussian multi-target tracking system. The particle PMC-PHD (PF-PMC-PHD) is given for the nonlinear multi-target tracking system based on PMC model in this paper.
A set of weighted random samples   x y x y x y (8) where () i k w represents the expected value of pair whose state is ( ) ( )
PF-PMC-PHD filter can be summarized as follow.
Step 1: Initialization of particles. x y x y x y (9) Step 2: Particles prediction.
x y x y x y x y xy xy (10) Assuming that there is no spawning, x y x x y x y (11) Predicted joint PHD function 1 ( , ) kk v  xy of pair ( , ) xy writes as： x y x y (12) Step 3: Particles update.
Recalculating weights of particles using measurements x y x y (14)   Step 4: Resampling of the particles.
Estimating the number of targets: Step 5: Approximation of posterior probability density.