Gaussian Mixture Implementations of Probability Hypothesis Density Filters for Non-linear Dynamical Models Daniel Clark ∗ , Ba-Tuong Vo † , Ba-Ngu Vo ‡ , and Simon Godsill § Abstract The Probability Hypothesis Density (PHD) filter is a multiple- target filter for recursively estimating the number of targets and their state vectors from sets of observations. The filter is able to operate in environments with false alarms and missed detec- tions. Two distinct algorithmic implementations of this tech- nique have been developed. The first of which, called the Parti- cle PHD filter, requires clustering techniques to provide target state estimates which can lead to inaccurate estimates and is computationally expensive. The second algorithm, called the Gaussian Mixture PHD (GM-PHD) filter does not require clus- tering algorithms but is restricted to linear-Gaussian target dy- namics, since it uses the Kalman filter to estimate the means and covariances of the Gaussians. This article provides a re- view of Gaussian filtering techniques for non-linear filtering and shows how these can be incorporated within the Gaussian mixture PHD filters. Finally, we show some simulated results of the different variants. 1 Introduction The multiple-target tracking framework based on random-sets was proposed to unify the problems of detecting, identify- ing, classifying and tracking targets within a unified Bayesian paradigm [1]. An optimal multi-target Bayes filter, analogous to the single target Bayes filter, can be derived which propa- gates a multi-target posterior density in time. The complexity of computing this recursion grows exponentially with the num- ber of targets and is thus not practical for more than a few tar- gets. The Probability Hypothesis Density (PHD) filter was pro- posed as a practical suboptimal alternative to computing the full multiple-target posterior distribution by propagating the first- order moment statistic [2]. This approach has led to efficient multiple target tracking algorithms for jointly estimating the number of targets and their states from a sequence of observa- tion sets with data association uncertainty, missed detections, false alarms and noisy measurements. A closed-form solution to the PHD filter was derived un- der linear assumptions on the system and observation equations ∗ Daniel Clark is in the Department of Electrical and Computing Engineer- ing at Heriot-Watt University. dec1@hw.ac.uk † Ba Tuong Vo is in the Department of Electrical and Computing Engineer- ing at University of Western Australia. vob@ ‡ Ba-Ngu Vo is in the Department of Electrical, Electronic Engineering at the University of Melbourne. bv@ee.unimelb.edu.au § Simon Godsill is in the Department of Engineering at the University of Cambridge sjg30@cam.ac.uk and Gaussian process and observation noises, called the Gaus- sian Mixture PHD filter [3]. The PHD is approximated at each stage with a mixture of Gaussians, where the means and co- variances of the Gaussian components are calculated according to the Kalman filter equations, and the weights are calculated according to the PHD filter equations. It is a common misperception that target trajectories can not be maintained with the PHD filter since in the original formu- lation methods for target state estimation and track continuity were not explicitly defined [2]. In the Gaussian mixture im- plementation, the multiple target states are estimated by taking the Gaussian components with highest weights and tracks can be maintained by labelling the Gaussian components [4]. More complex methods for dealing with target resolution uncertainty have been developed using this approach as a basis [5]. These techniques are also directly applicable to other Gaussian mix- ture intensity based filters such as the Cardinalized PHD fil- ter [6] and to the methods described in this paper. However, since the main focus of the paper is on different Gaussian filter- ing techniques and not track continuity, we do not exploit these techniques here. In this paper, we investigate strategies for calculating the Gaussian mixture recursion with non-linear motion and obser- vation models. If the posterior distribution can be approximated by a Gaussian density, then the core objective is to find tech- niques for Gaussian quadrature. Approaches for this include sigma-point filters such as the unscented Kalman filter, cen- tral difference, and divided difference filters. Other approaches which have been studied include Gauss-Hermite quadrature, which computes the integral exactly, and Quasi-Monte filter- ing, which attempts to deterministically choose sample points to calculate the density more efficiently than standards Monte Carlo integration. These techniques are based on approximat- ing the means and covariances in the Kalman filter recursion. An alternative approach uses Monte Carlo integration and im- portance sampling to approximate the integrals in the Bayes fil- ter directly, called the Gaussian Particle filter [7]. Based on the Gaussian mixture framework, we compare different strategies for approximating the PHD recursion under non-linear target dynamics. 2 Gaussian Filtering The single-target filtering problem is to estimate recursively in time, the probability distribution p(x k |Z k ) of the signal, where x 0:t := {x 0 , ..., x k } be an unobserved signal process of dimen- sion n that we wish to estimate, and Z k := σ({z 1 , ..., z k }) be the σ-algebra generated by noisy observations of dimension m ≤ n 1