Optimal Track Fusion Using Bayes Factors

In deciding whether to associate and fuse a group of tracks sent from independent local trackers, use should be made of all the data supporting them during theircommon history. This paper provides a closed form expression for the relevant Bayes factor, which is the ratio of the probability that the tracks are caused by a single target to the probability that they are each caused by a different target. The expression is recursive, and it applies to processes which may include a linear Gaussian process and several discrete Markov processes. Two variants are provided, one that requires the data to be sent from local trackers to the fusion centre, and one that only requires discrete probabilities state estimates and covariance matrices. 1. BAYES FACTORS The track association problem we consider concerns whether tracks estimated independently by different trackers using different data are caused by the same target. If so, the tracks should be fused. This paper provides techniques for making such decisions which use all the data from the period of their shared existence. The equations can also be formulated so that only the track estimates and covariances are needed, and the data can be thrown away. Our line of attack is to evaluate the Bayes factor [1] for the fusion hypothesis pair H0 : Target states are equal; (1) H1 : Target states are different: (2) A Bayes factor is the ratio of the probability of two hypotheses given relevant data Z . Frequently, the parameters have to be integrated out. The Bayes factor for this hypothesis pair isP (H0jZ) P (H1jZ) = P (H0) R p(Zj ;H0) d P (H1) R p(Zj ;H1) d : (3) Bayes factors behave differently from generalised likelihood ratios (GLRs), in which the integrations, above, are substituted by maximisations over the parameters. GLRs may not represent the relative validity of the hypotheses because although the likelihood is maximised at the maximum likelihood estimate, the peak may be narrow, giving a small integrated likelihood p(ZjH). Bayes factors do not rely on such estimates and are therefore more robust; but they are more difficult to obtain. 2. THE TRACKING MODEL It is possible to treat the trackers for mixed linear Gaussian and discrete Markov processes in a unified way. Analytical association results can be found for such generalised tracks [3, 4, 5]. The joint state X comprises the Gaussian kinematic state XG and the discrete states, XD. We assume that the classification states and the kinematic states are independent and that the data can be partitioned into subsetsZ and R; Z provides information about XG, and R provides information about XD. Under these circumstances, finite sufficient statistics exist for the states, and the posterior probabilities of XG and XD factorise: p(XG;XDjZ;R) = p(XGjZ)P (XDjR); (4) and the sequential updates of the Kalman and Markov process filters are independent [4]. The value of joint tracking and classification under this model derives from the association process, which becomes more resistant to clutter and measurement noise [3]. Each sensor suite drives a local tracker. Periodically, the local trackers send information to the fusion centre (e.g. state estimates, covariance estimates and, in the case of discrete variables, the entire distribution). The track fusion and association equations we derive are applicable to both feedback and independent distributed tracking frameworks [6, p. 434 et. seq.]. In the feedback framework, the fusion centre sends fused estimates back to the local trackers. The detailed modifications needed for asynchronous updating can readily be worked out and are not described here. Previous work by Chang, Bar-Shalom and Li [7, 6] has provided algorithms for fusing and associating tracks of Gaussian processes using the current track state estimates and fusing fixed discrete feature states. 1The Bayes factor is the posterior odds of the two hypotheses. In the control literature it recently been referred to the marginalised likelihood ratio [2].