# Particle Probability Hypothesis Density Filter Based on Pairwise Markov Chains
## 1 Introduction  
Random Finite Set (RFS) theory has been widely used in the multi-target tracking field. Unlike traditional solutions of multi-target tracking 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 cannot 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. The HMC model assumes that the state of a given target is a Markov Chain (MC): the states of the current moment are determined only by the states of the previous moment, which have nothing to do with other moments, and the observations of the current moment is determined only by the states of the current moment. However, the Markovian and independence assumption implicit in the HMC model may not be satisfied in practical applications, such as situations of correlated process and measurement noises and situations of colored measurement noise [8]. In 2000, [PERSON] proposed the PMC model in order to relax the independence assumption of the HMC model [9; 10; 11]; the HMC model is a special PMC model and the PMC model is more universally suitable than HMC model. In 2013, [PERSON] and [PERSON] proposed a PHD filter for targets following the PMC model (PMC-PHD) [11] and proved that the tracking performance of theproposed PMC-PHD filter is better than the \"classical\" PHD filter based on the HMC model under the situation which relaxed the independence assumptions implicit in the HMC model. The PMC-PHD filter proposed by [PERSON] 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, [PERSON] proposed a Cardinalized Probability Hypothesis Density filter based on the PMC model (PMC-CPHD), which is developed from the PMC-PHD filter by propagating the cardinality distribution function of the target simultaneously [12].  
The GM implementation of the PMC-PHD filter proposed by [PERSON] and [PERSON] is only suitable for the linear Gaussian multi-target tracking system but not to a nonlinear system. GM implementations of the PHD filter use approximate methods to deal with non-linear problems, while particle implementations are not affected by linearity and non-linearity. In this paper, a set of weighted particles is used to approximate the probability hypothesis density of multi-targets in the framework of the PMC model, the probability hypothesis density of multi-targets is updated iteratively by predicting and updating the particles in real time to estimate the state of the targets, and a particle probability hypothesis density filter based on the PMC model (PF-PMC-PHD) is proposed for the nonlinear multi-target tracking system. The 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.  
The rest of the paper is organized as follows. A brief introduction to the PHD filter based on the PMC model is given in Section 2. The particle PMC-PHD (PF-PMC-PHD) is given for the nonlinear multi-target tracking system based on the PMC model in Section 3. The simulation results can be found in Section 4. The conclusions are in Section 5.