A conceptual Kalman filter for random fields is proposed for estimating multiple objects. The result exploits an approximation of point processes with Gaussian random fields. The motivation is to develop a solution for multi-object filtering problem in terms of the first two moments of a random field. Applications are discussed for a multi-target tracking model.

Multi-object filtering is a generalisation of stochastic filtering to deal with an unknown and time-varying number of targets, largely based on modelling with point processes. Some early works on this topic from the Soviet Union from 1960s-l980s appeared prior to well known results in the contemporary liter-ature. This article reviews some of these historical contributions.

The Cramér Rao bound provides a minimum achievable variance or covariance for a parameter for a univariate or vector-valued parameter. Point processes often have parameters that are described by functions and the variance and covariance for point processes are themselves functions with spatial variates. Consequently, the usual formulation of the Cramér Rao bound in these contexts is not applicable. The second-order derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér’s rate function, provides a description of the Cramér Rao bound. We follow this approach to develop a form of Cramér Rao bound for point processes and random measures derived from the second-order functional derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér’s rate functional for point processes and random measures.

Point processes are often described with functionals, such as the probability generating functional, the Laplace functional, and the factorial cumulant generating functional. These are used to facilitate modelling of different processes and to determine important statistics via functional differentiation. In information theory, generating functions have also been defined for probability densities to determine information quantities such as the Shannon information and Kullback-Leibler divergence, though as yet there are no such analogues for point processes. The purpose of this article is to exploit the advantages of both types of generating function to facilitate the derivation of information statistics for point processes. In particular, a generating functional for point processes is introduced for determining statistics related to entropy and relative entropy based on Golomb's information function and Moyal's probability generating functional. It is shown that the information generating functional permits the derivation of a suite of statistics, including localised Shannon entropy and Kullback-Leibler divergence calculations.

Multi-target tracking solutions with low computational complexity are required in order to address large-scale tracking problems. Solutions based on statistics determined from point processes, such as the PHD filter, CPHD filter, and newer second-order PHD filter are some examples of these algorithms. There are few solutions of linear complexity in the number of targets and number of measurements, with the PHD filter being one exception. However, the trade-off is that it is unable to propagate beyond first-order moment statistics. In this paper, a new filter is proposed with the same complexity as the PHD filter that also propagates second-order information via the second-order factorial cumulant. The results show that the algorithm is more robust than the PHD filter in challenging clutter environments.

Motivated by object tracking applications with networked sensors, we consider multi sensor state space models. Estimation of latent parameters in these models requires centralisation because the parameter likelihood depend on the measurement histories of all of the sensors. Consequently, joint processing of multiple histories pose difficulties in scaling with the number of sensors. We propose an approximation with a node-wise separable structure thereby removing the need for centralisation in likelihood computations. When leveraged with Markov random field models and message passing algorithms for inference, these likelihoods facilitate decentralised estimation in tracking networks as well as scalable computation schemes in centralised settings. We establish the connection between the approximation quality of the proposed separable likelihoods and the accuracy of state estimation based on individual sensor histories. We demonstrate this approach in a sensor network self-localisation example.

The paper formulates the problem of sequential Bayesian estimation of a compound state consisting of a multi-object dynamic state and a multi-sensor bias. The compound state is modelled by a doubly stochastic point process, where the multi-object bias is a parent, whereas the multi-object state is the offspring point process. The prediction and the update steps for the first-order moment of the posterior density of the doubly-stochastic point process can be expressed analytically. The implementation, however, in general has to be done numerically. The paper presents a particle filter implementation illustrated in the context of multi-target tracking using range-azimuth measuring sensors with unknown biases.

Recent work by Mullane, Vo, and Adams has re-examined the probabilistic foundations of feature-based Simultaneous Localization and Mapping (SLAM), casting the problem in terms of filtering with random finite sets. Algorithms were developed based on Probability Hypothesis Density (PHD) filtering techniques that provided superior performance to leading feature-based SLAM algorithms in challenging measurement scenarios with high false alarm rates, high missed detection rates, and high levels of measurement noise. We investigate this approach further by considering a hierarchical point process, or single-cluster multi-object, model, where we consider the state to consist of a map of landmarks conditioned on a vehicle state. Using Finite Set Statistics, we are able to find tractable formulae to approximate the joint vehicle-landmark state based on a single Poisson multi-object assumption on the predicted density. We describe the single-cluster PHD filter and the practical implementation developed based on a particle-system representation of the vehicle state and a Gaussian mixture approximation of the map for each particle. Synthetic simulation results are presented to compare the novel algorithm against the previous PHD filter SLAM algorithm. Results presented indicate a superior performance in vehicle and map landmark localization, and comparable performance in landmark cardinality estimation.

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