An analysis of the Bayesian track labelling problem

In multi-target tracking (MTT), the problem of assigning labels to tracks (track labelling) is vastly covered in literature, but its exact mathematical formulation, in terms of Bayesian statistics, has not been yet looked at in detail. Doing so, however, may help us to understand how Bayes-optimal track labelling should be performed or numerically approximated. Moreover, it can help us to better understand and tackle some practical difficulties associated with the MTT problem, in particular the so-called ``mixed labelling'' phenomenon that has been observed in MTT algorithms. In this memorandum, we rigorously formulate the optimal track labelling problem using Finite Set Statistics (FISST), and look in detail at the mixed labeling phenomenon. As practical contributions of the memorandum, we derive a new track extraction formulation with some nice properties and a statistic associated with track labelling with clear physical meaning. Additionally, we show how to calculate this statistic for two well-known MTT algorithms

A theoretical analysis of Bayes-optimal multi-target tracking and labelling

In multi-target tracking (MTT), we are often interested not only in finding the position of the multiple objects, but also allowing individual objects to be uniquely identified with the passage of time, by placing a label on each track. While there are many MTT algorithms that produce uniquely identified tracks as output, most of them make use of certain heuristics and/or unrealistic assumptions that makes the global result suboptimal of Bayesian sense. An innovative way of performing MTT is the so-called joint multi-target tracking, where the raw output of the algorithm, rather than being already the collection of output tracks, is a multi-target density calculated by approximating the Bayesian recursion that considers the entire system to have a single multidimensional state. The raw output, i.e. the calculated multi-target density, is thereafter processed to obtain output tracks to be displayed to the operator. This elegant approach, at least in theory, would allow us to precisely represent multi-target statistics. However, most joint MTT methods in the literature handle the problem of track labelling in an ad-hoc, i.e. non-Bayesian manner. A number of methods, however, have suggested that the multi-target density, calculated using the Bayesian recursion, should contain information not only about the location of the individual objects but also their identities. This approach, that we refer as joint MTTL (joint multi-target tracking and labelling), looks intuitively advantageous. It would allow us, at least in theory, to obtain an output consisting of labelled tracks that is optimal in Bayesian sense. Moreover, it would allow us to have statistical information about the assigned labels; for instance, we would know what is the probability that track swap may have occurred after some approximation of targets (or, in simpler words, we would know how much we can believe that a target is what the display says that it is). However, the methods proposed in the still emerging joint MTTL literature do not address some problems that may considerably reduce the usefulness of the approach. These problems include: track coalescence after targets move closely to each other, gradual loss of ambiguity information when particle filters or multiple hypotheses approaches are used, and dealing with unknown/varying number of targets. As we are going to see, each of the previously proposed methods handles only a subset of these problems. Moreover, while obtaining a Bayes-optimal output of labelled tracks is one of the main motivations for joint MTTL, how such output should be obtained is a matter of debate. This work will tackle the joint MTTL problem together with a companion memorandum. In this work, we look at the problem from a theoretical perspective, i.e. we aim to provide an accurate and algorithm-independent picture of the aforementioned problems. An algorithm that actually handles these problems will be proposed in the companion memorandum. As one of the contributions of the memorandum, we clearly characterize the so-called "mixed labelling" phenomenon that leads to track coalescence and other problems, and we verify that, unlike implied in previous literature, it is a physical phenomenon inherent of the MTTL problem rather than specific to a particular approach. We also show how mixed labelling leads to nontrivial issues in practical implementations of joint MTTL. As another of the contributions of the memorandum, we propose a conceptual, algorithm-independent track extraction method for joint MTTL estimators, that gives an output with clear physical interpretation for the user

A Bayesian solution to multi-target tracking problems with mixed labelling

In Multi-Target Tracking (MTT), the problem of assigning labels to tracks (track labelling) is vastly covered in literature and has been previously formulated using Bayesian recursion. However, the existing literature lacks an appropriate measure of uncertainty related to the assigned labels which has sound mathematical basis and clear practical meaning (to the user). This is especially important in a situation where targets move in close proximity with each other and thereafter separate again. Because, in such a situation it is well-known that there will be confusion on target identities, also known as “mixed labelling‿. In this paper, we provide a mathematical characterization of the labelling uncertainties present in Bayesian multi-target tracking and labelling (MTTL) problems and define measures of labelling uncertainties with clear physical interpretation. The introduced uncertainty measures can be used to find the optimal track label assignment, and evaluate track labelling performance. We also analyze in details the mixed labelling phenomenon in the presence of two targets. In addition, we propose a new Sequential Monte Carlo (SMC) algorithm, the Labelling Uncertainty Aware Particle Filter (LUA-PF), for the multi target tracking and labelling problem that can provide good estimates of the uncertainty measures. We validate this using simulation and show that the proposed method performs much better when compared with the performance of the SIR multi-target SMC filter

Characterization of uncertainty in Bayesian estimation using sequential Monte Carlo methods

In estimation problems, accuracy of the estimates of the quantities of interest cannot be taken for granted. This means that estimation errors are expected, and a good estimation algorithm should be able not only to compute estimates that are optimal in some sense, but also provide meaningful measures of uncertainty associated with those estimates. In some situations, we might also be able to reduce estimation uncertainty through the use of feedback on observations, an approach referred to as sensor management. Characterization of estimation uncertainty, as well as sensor management, are certainly difficult tasks for general partially observed processes, which might be non-linear, non-Gaussian, and/or have dependent process and observation noises. Sequential Monte Carlo (SMC) methods, also known as particle filters, are numerical Bayesian estimators which are, in principle, able to handle highly general estimation problems. However, SMC methods are known to suffer from a phenomenon called degeneracy, or self-resolving, which greatly impairs their usefulness against certain classes of problems. One of such classes, that we address in the first part of this thesis, is the joint state and parameter estimation problem, where there are unknown parameters to be estimated together with the timevarying state. Some SMC variants have been proposed to counter the degeneracy phenomenon for this problem, but these state-of-the-art techniques are either non-Bayesian or introduce biases on the system model, which might not be appropriate if proper characterization of estimation uncertainty is required. For this type of scenario, we propose using the Rao-Blackwellized Marginal Particle Filter (RBMPF), a combination of two SMC algorithm variants: the Rao-Blackwellized Particle Filter (RBPF) and the Marginal Particle Filter (MPF). We derive two new versions of the RBMPF: one for models with low dimensional parameter vectors, and another for more general models. We apply the proposed methods to two practical problems: the target tracking problem of turn rate estimation for a constant turn maneuver, and the econometrics problem of stochastic volatility estimation. Our proposed methods are shown to be effective solutions, both in terms of estimation accuracy and statistical consistency, i.e. characterization of estimation uncertainty. Another problem where standard particle filters suffer from degeneracy, addressed in the second part of this thesis, is the joint multi-target tracking and labelling problem. In comparison with the joint state and parameter estimation problem, this problem poses an additional challenge, namely, the fact that it has not been properly mathematically formulated in previous literature. Using Finite Set Statistics (FISST), we provide a sound theoretical formulation for the problem, and in order to actually solve the problem, we propose a novel Bayesian algorithm, the Labelling Uncertainty-Aware Particle Filter (LUA-PF) filter, essentially a combination of the RBMPF and the Multi-target Sequential Monte Carlo (M-SMC) filter techniques. We show that the new algorithm achieves significant improvements on both finding the correct track labelling and providing a meaningful measure of labelling uncertainty. In the last part of this thesis, we address the sensor management problem. Although we apply particle filters to the problem, they are not the main focus of this part of the work. Instead, we concentrate on a more fundamental question, namely, which sensor management criterion should be used in order to obtain the best results in terms of information gain and/or reduction of uncertainty. In order to answer this question, we perform an in-depth theoretical and empirical analysis on two popular sensor management criteria based on information theory – the Kullback-Leibler and R´enyi divergences. On the basis of this analysis, we are able to either confirm or reject some previous arguments used as theoretical justification for these two criteria

Efficient characterization of labeling uncertainty in closely-spaced targets tracking

In this paper we propose a novel solution to the labeled multi-target tracking problem. The method presented is specially effective in scenarios where the targets have once moved in close proximity. When this is the case, disregarding the labeling uncertainty present in a solution (after the targets split) may lead to a wrong decision by the end user. We take a closer look at the main cause of the labeling problem. By modeling the possible crosses between the targets, we define some relevant labeled point estimates. We extend the concept of crossing objects, which is obvious in one dimension, to scenarios where the objects move in multiple dimensions. Moreover, we provide a measure of uncertainty associated to the proposed solution to tackle the labeling problem. We develop a novel, scalable and modular framework in line with it. The proposed method is applied and analyzed on the basis of one-dimensional objects and two-dimensional objects simulation experiments

Spooky effect in optimal OSPA estimation and how GOSPA solves it

In this paper, we show the spooky effect at a distance that arises in optimal estimation of multiple targets with the optimal sub-pattern assignment (OSPA) metric. This effect refers to the fact that if we have several independent potential targets at distant locations, a change in the probability of existence of one of them can completely change the optimal estimation of the rest of the potential targets. As opposed to OSPA, the generalised OSPA (GOSPA) metric (α=2) penalises localisation errors for properly detected targets, false targets and missed targets. As a consequence, optimal GOSPA estimation aims to lower the number of false and missed targets, as well as the localisation error for properly detected targets, and avoids the spooky effect

Tracking and Fusion Methods for Extended Targets Parameterized by Center, Orientation, and Semi-axes

The improvements in sensor technology, e.g., the development of automotive Radio Detection and Ranging (RADAR) or Light Detection and Ranging (LIDAR), which are able to provide a higher detail of the sensor’s environment, have introduced new opportunities but also new challenges to target tracking. In classic target tracking, targets are assumed as points. However, this assumption is no longer valid if targets occupy more than one sensor resolution cell, creating the need for extended targets, modeling the shape in addition to the kinematic parameters. Different shape models are possible and this thesis focuses on an elliptical shape, parameterized with center, orientation, and semi-axes lengths. This parameterization can be used to model rectangles as well. Furthermore, this thesis is concerned with multi-sensor fusion for extended targets, which can be used to improve the target tracking by providing information gathered from different sensors or perspectives. We also consider estimation of extended targets, i.e., to account for uncertainties, the target is modeled by a probability density, so we need to find a so-called point estimate. Extended target tracking provides a variety of challenges due to the spatial extent, which need to be handled, even for basic shapes like ellipses and rectangles. Among these challenges are the choice of the target model, e.g., how the measurements are distributed across the shape. Additional challenges arise for sensor fusion, as it is unclear how to best consider the geometric properties when combining two extended targets. Finally, the extent needs to be involved in the estimation. Traditional methods often use simple uniform distributions across the shape, which do not properly portray reality, while more complex methods require the use of optimization techniques or large amounts of data. In addition, for traditional estimation, metrics such as the Euclidean distance between state vectors are used. However, they might no longer be valid because they do not consider the geometric properties of the targets’ shapes, e.g., rotating an ellipse by 180 degree results in the same ellipse, but the Euclidean distance between them is not 0. In multi-sensor fusion, the same holds, i.e., simply combining the corresponding elements of the state vectors can lead to counter-intuitive fusion results. In this work, we compare different elliptic trackers and discuss more complex measurement distributions across the shape’s surface or contour. Furthermore, we discuss the problems which can occur when fusing extended target estimates from different sensors and how to handle them by providing a transformation into a special density. We then proceed to discuss how a different metric, namely the Gaussian Wasserstein (GW) distance, can be used to improve target estimation. We define an estimator and propose an approximation based on an extension of the square root distance. It can be applied on the posterior densities of the aforementioned trackers to incorporate the unique properties of ellipses in the estimation process. We also discuss how this can be applied to rectangular targets as well. Finally, we evaluate and discuss our approaches. We show the benefits of more complex target models in simulations and on real data and we demonstrate our estimation and fusion approaches compared to classic methods on simulated data.2022-01-2