Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Tracking via Poisson likelihoods <s> Much of the literature on target tracking is based on the assumption that the observing sensor produces a single point measurement of the target. In practice, this is often not the case. For example, a high resolution sensor may be able to resolve features on an extended target. An analogous problem is that of tracking a group of point targets moving in formation. In both cases, there is a strong interdependency between the individual sensor measurements. In this paper, we handle both of these cases via the same model of individual motion superposed on a common “bulk” effect. A common approach to tracking groups of (dependent) points is to perform the update operation in two distinct steps. The first stage is to find the “best” match or registration between the set of measurements and the predicted pattern, while the second step is to generate an estimate based on the assumption that the selected association is correct. The complexity of managing multiple hypotheses with strongly interdependent groups is usually avoided. However, particularly in stressing scenarios, where the measurement-object association is highly uncertain, a multiple hypothesis approach should have significant advantages. We present a Bayesian multiple hypothesis solution using the bootstrap or particle filter technique in which the probability distribution of the problem state vector is represented by a set of random samples or “particles”. The great advantage of this approach is that the sample set implicitly includes information on previous hypotheses and so awkward hypothesis management is avoided. Only the feasible association hypotheses for the current update of the filter need to be considered for construction of the measurement likelihood function. (4 pages) <s> BIB001 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Tracking via Poisson likelihoods <s> It is common practice to represent a target group (or an extended target) as set of point sources and attempt to formulate a tracking filter by constructing possible assignments between measurements and the sources. We suggest an alternative approach that produces a measurement model (likelihood) in terms of the spatial density of measurements over the sensor observation region. In particular, the measurements are modelled as a Poisson process with a spatially dependent rate parameter. This representation allows us to model extended targets as an intensity distribution rather than a set of points and, for a target formation, it gives the option of modelling part of the group as a spatial distribution of target density. Furthermore, as a direct consequence of the Poisson model, the measurement likelihood may be evaluated without constructing explicit association hypotheses. This considerably simplifies the filter and gives a substantial computational saving in a particle filter implementation. The Poisson target-measurement model will be described and its relationship to other filters will be discussed. Illustrative simulation examples will be presented. <s> BIB002 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Tracking via Poisson likelihoods <s> A Bayesian filter has been developed for tracking an extended object in clutter based on two simple axioms: (i) the numbers of received target and clutter measurements in a frame are Poisson distributed (so several measurements may originate from the target) and (ii) target extent is modelled by a spatial probability distribution and each target-related measurement is an independent 'random draw' from this spatial distribution (convolved with a sensor model). Diffuse spatial models of target extent are of particular interest. This model is especially suitable for a particle filter implementation, and examples are presented for a Gaussian mixture model and for a uniform stick target convolved with a Gaussian error. A rather restrictive special case that admits a solution in the form of a multiple hypothesis Kalman filter is also discussed and demonstrated. <s> BIB003 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Tracking via Poisson likelihoods <s> The authors consider the problem of extended object tracking. An extended object is modelled as a set of point features in a target reference frame. The dynamics of the extended object are formulated in terms of the translation and rotation of the target reference frame relative to a fixed reference frame. This leads to realistic, yet simple, models for the object motion. It is assumed that the measurements of the point features are unlabelled, and contaminated with a high level of clutter, leading to measurement association uncertainty. Marginalising over all the association hypotheses may be computationally prohibitive for realistic numbers of point features and clutter measurements. The authors present an alternative approach within the context of particle filtering, where they augment the state with the unknown association hypothesis, and sample candidate values from an efficiently designed proposal distribution. This proposal elegantly captures the notion of a soft gating function. The performance of the algorithm is demonstrated on a challenging synthetic tracking problem, where the ground truth is known, in order to compare between different algorithms. <s> BIB004 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Tracking via Poisson likelihoods <s> We present a novel filtering algorithm for tracking multiple clusters of coordinated objects. Based on a Markov chain Monte Carlo (MCMC) mechanism, the new algorithm propagates a discrete approximation of the underlying filtering density. A dynamic Gaussian mixture model is utilized for representing the time-varying clustering structure. This involves point process formulations of typical behavioral moves such as birth and death of clusters as well as merging and splitting. For handling complex, possibly large scale scenarios, the sampling efficiency of the basic MCMC scheme is enhanced via the use of a Metropolis within Gibbs particle refinement step. As the proposed methodology essentially involves random set representations, a new type of estimator, termed the probability hypothesis density surface (PHDS), is derived for computing point estimates. It is further proved that this estimator is optimal in the sense of the mean relative entropy. Finally, the algorithm's performance is assessed and demonstrated in both synthetic and realistic tracking scenarios. <s> BIB005
Groups and extended objects give rise to multiple measurements. These can be useful for determining the shape, size, orientation and other characteristics of the groups/extended targets. However, multiple measurements require associating them, with the objects on the scene, a problem known as data association. In general, data association is an intractable combinatorial problem. The presence of environmental interferences, including unwanted returned echoes, called clutter (e.g. from ground, sea, rain) adds an extra level of difficulty. This, in turn, implies that the amount of observations per time step may be very large, and hence care must be taken for regulating these potentially massive data sets. Additionally, in a typical scenario, measurements often originate from closely spaced 3 objects, and in such cases standard data association techniques fail to disambiguate individual groups and group constituents. One of the most appealing solutions BIB002 BIB003 BIB001 allowing to avoid the data association problem are based on the assumptions: (i) the numbers of received target and clutter measurements in a time step are Poisson distributed (so several measurements may originate from the target), (ii) target extent is modeled by a spatial probability distribution and each target-related measurement is an independent 'random draw' from this spatial distribution (convolved with a sensor model). At each time step a set of sensor measurements is recorded. These measurements could come from either targets or clutter. The basic idea is to figure out how to assigns each measurements to the right target or identify it as a false alarm. The likelihood can then be calculated as derived in BIB002 BIB003 BIB001 : where ρ is the clutter density and λ T is the mean value of the number of measurements originated from the targets, m k is the overall number of measurements received at time k and X k is the state of interest. The clutter distribution is typically assumed to be uniform, although non-uniform clutter can be incorporated (see BIB004 ). This Poisson likelihood model (10) was further extended in BIB005 and below we describe this approach. Assume that at time k there are l k groups, or extended objects at unknown locations. Each group may produce more than one observation yielding the measurement set , where typically m k l k . At this point we assume that the observation concentrations can be adequately represented by a parametric statistical model. Letting Z 0:k = {Z 0 , . . . , Z k } be the measurement history up to time k, the group tracking problem can be defined as follows. We are concerned with estimating the posterior distribution of the random set of unknown θ k , i.e. p(θ k |Z 0:k ), from which point estimates for θ k and posterior confidence intervals can be extracted. If we restrict ourselves to groups in which the shape can be modeled via a Gaussian pdf, then only the first two moments (the mean and covariance) need to be specified for each group. Under these assumptions, the group tracking problem is equivalent to that of estimating an evolving Gaussian mixture model with a variable number of components. Thus the unknown vector θ k = {θ j k } n j=1 to be estimated is in the form  Under the assumption that a single observation z k (i) is conditionally independent on the extent vector parameters (θ k , e k ), the likelihood can be represented in the form where the pdf p(z k (i)|θ k , e k ) describes the statistical relation between a single observation and the cluster parameters. An explicit expression of (13) is derived in BIB002 assuming a spatial Poisson distribution for the expected number of observations. Following BIB002 , the number of observations within the jth group can be assumed with Poisson distribution while the likelihoods for each of the measurement z k (i) are modeled via Gaussian 