Daniel E. Clark

Data Association and Track Management for the Gaussian Mixture Probability Hypothesis Density Filter

The Gaussian mixture probability hypothesis density (GM-PHD) recursion is a closed-form solution to the probability hypothesis density (PHD) recursion, which was proposed for jointly estimating the time-varying number of targets and their states from a sequence of noisy measurement sets in the presence of data association uncertainty, clutter, and miss-detection. However the GM-PHD filter does not provide identities of individual target state estimates, that are needed to construct tracks of individual targets. In this paper, we propose a new multi-target tracker based on the GM-PHD filter, which gives the association amongst state estimates of targets over time and provides track labels. Various issues regarding initiating, propagating and terminating tracks are discussed. Furthermore, we also propose a technique for resolving identities of targets in close proximity, which the PHD filter is unable to do on its own.

A Metric for Performance Evaluation of Multi-Target Tracking Algorithms

Performance evaluation of multi-target tracking algorithms is of great practical importance in the design, parameter optimization and comparison of tracking systems. The goal of performance evaluation is to measure the distance between two sets of tracks: the ground truth tracks and the set of estimated tracks. This paper proposes a mathematically rigorous metric for this purpose. The basis of the proposed distance measure is the recently formulated consistent metric for performance evaluation of multi-target filters, referred to as the OSPA metric. Multi-target filters sequentially estimate the number of targets and their position in the state space. The OSPA metric is therefore defined on the space of finite sets of vectors. The distinction between filtering and tracking is that tracking algorithms output tracks and a track represents a labeled temporal sequence of state estimates, associated with the same target. The metric proposed in this paper is therefore defined on the space of finite sets of tracks and incorporates the labeling error. Numerical examples demonstrate that the proposed metric behaves in a manner consistent with our expectations.

Adaptive Target Birth Intensity for PHD and CPHD Filters

The standard formulation of the probability hypothesis density (PHD) and cardinalised PHD (CPHD) filters assumes that the target birth intensity is known a priori. In situations where the targets can appear anywhere in the surveillance volume this is clearly inefficient, since the target birth intensity needs to cover the entire state space. This paper presents a new extension of the PHD and CPHD filters, which distinguishes between the persistent and the newborn targets. This extension enables us to adaptively design the target birth intensity at each scan using the received measurements. Sequential Monte-Carlo (SMC) implementations of the resulting PHD and CPHD filters are presented and their performance studied numerically. The proposed measurement-driven birth intensity improves the estimation accuracy of both the number of targets and their spatial distribution.

Improved SMC implementation of the PHD filter

The paper makes two contributions. First, a new formulation of the PHD filter which distinguishes between persistent and newborn objects is presented. This formulation results in an efficient sequential Monte Carlo (SMC) implementation of the PHD filter, where the placement of newborn object particles is determined by the measurements. The second contribution is a novel method for the state and error estimation from an SMC implementation of the PHD filter. Instead of clustering the particles in an ad-hoc manner after the update step (which is the current approach), we perform state estimation and, if required, particle clustering, within the update step in an exact and principled manner. Numerical simulations indicate a significant improvement in the estimation accuracy of the proposed SMC-PHD filter.

Convergence Analysis of the Gaussian Mixture PHD Filter

The Gaussian mixture probability hypothesis density (PHD) filter was proposed recently for jointly estimating the time-varying number of targets and their states from a sequence of sets of observations without the need for measurement-to-track data association. It was shown that, under linear-Gaussian assumptions, the posterior intensity at any point in time is a Gaussian mixture. This paper proves uniform convergence of the errors in the algorithm and provides error bounds for the pruning and merging stages. In addition, uniform convergence results for the extended Kalman PHD Filter are given, and the unscented Kalman PHD Filter implementation is discussed

Multi-target state estimation and track continuity for the particle PHD filter

Particle filter approaches for approximating the first-order moment of a joint, or probability hypothesis density (PHD), have demonstrated a feasible suboptimal method for tracking a time-varying number of targets in real-time. We consider two techniques for estimating the target states at each iteration, namely k-means clustering and mixture modelling via the expectation-maximization (EM) algorithm. We present novel techniques for associating the targets between frames to enable track continuity.

Convergence results for the particle PHD filter

Bayesian single-target tracking techniques can be extended to a multiple-target environment by viewing the multiple-target state as a random finite set, but evaluating the multiple-target posterior distribution is currently computationally intractable for real-time applications. A practical alternative to the optimal Bayes multitarget filter is the probability hypothesis density (PHD) filter, which propagates the first-order moment of the multitarget posterior instead of the posterior distribution itself. It has been shown that the PHD is the best-fit approximation of the multitarget posterior in an information-theoretic sense. The method avoids the need for explicit data association, as the target states are viewed as a single global target state, and the identities of the targets are not part of the tracking framework. Sequential Monte Carlo approximations of the PHD using particle filter techniques have been implemented, showing the potential of this technique for real-time tracking applications. This paper presents mathematical proofs of convergence for the particle filtering algorithm and gives bounds for the mean-square error

The GM-PHD Filter Multiple Target Tracker

The Gaussian mixture probability hypothesis density filter (GM-PHD Filter) was proposed recently for jointly estimating the time-varying number of targets and their states from a noisy sequence of sets of measurements which may have missed detections and false alarms. The initial implementation of the GM-PHD filter provided estimates for the set of target states at each point in time but did not ensure continuity of the individual target tracks. It is shown here that the trajectories of the targets can be determined directly from the evolution of the Gaussian mixture and that single Gaussians within this mixture accurately track the correct targets. Furthermore, the technique is demonstrated to be successful in estimating the correct number of targets and their trajectories in high clutter density and shows better performance than the MHT filter

A Note on the Reward Function for PHD Filters with Sensor Control

The context is sensor control for multi-object Bayes filtering in the framework of partially observed Markov decision processes (POMDPs). The current information state is represented by the multi-object probability density function (pdf), while the reward function associated with each sensor control (action) is the information gain measured by the alpha or Rényi divergence. Assuming that both the predicted and updated state can be represented by independent identically distributed (IID) cluster random finite sets (RFSs) or, as a special case, the Poisson RFSs, this work derives the analytic expressions of the corresponding Rényi divergence based information gains. The implementation of Rényi divergence via the sequential Monte Carlo method is presented. The performance of the proposed reward function is demonstrated by a numerical example, where a moving range-only sensor is controlled to estimate the number and the states of several moving objects using the PHD filter.

Distributed Fusion of PHD Filters Via Exponential Mixture Densities

In this paper, we consider the problem of Distributed Multi-sensor Multi-target Tracking (DMMT) for networked fusion systems. Many existing approaches for DMMT use multiple hypothesis tracking and track-to-track fusion. However, there are two difficulties with these approaches. First, the computational costs of these algorithms can scale factorially with the number of hypotheses. Second, consistent optimal fusion, which does not double count information, can only be guaranteed for highly constrained network architectures which largely undermine the benefits of distributed fusion. In this paper, we develop a consistent approach for DMMT by combining a generalized version of Covariance Intersection, based on Exponential Mixture Densities (EMDs), with Random Finite Sets (RFS). We first derive explicit formulae for the use of EMDs with RFSs. From this, we develop expressions for the probability hypothesis density filters. This approach supports DMMT in arbitrary network topologies through local communications and computations. We implement this approach using Sequential Monte Carlo techniques and demonstrate its performance in simulations.