Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> We consider the problem of state estimation for a dynamic system driven by unobserved, correlated inputs. We model these inputs via an uncertain set of temporally correlated dynamic models, where this uncertainty includes the number of modes, their associated statistics, and the rate of mode transitions. The dynamic system is formulated via two interacting graphs: a hidden Markov model (HMM) and a linear-Gaussian state space model. The HMM's state space indexes system modes, while its outputs are the unobserved inputs to the linear dynamical system. This Markovian structure accounts for temporal persistence of input regimes, but avoids rigid assumptions about their detailed dynamics. Via a hierarchical Dirichlet process (HDP) prior, the complexity of our infinite state space robustly adapts to new observations. We present a learning algorithm and computational results that demonstrate the utility of the HDP for tracking, and show that it efficiently learns typical dynamics from noisy data. <s> BIB001 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> Collectively moving object clusters are of particular interest in certain applications and have to be tracked as separate aggregated entities consisting of an unknown number of individuals. Tracking of convoys or large vehicles in wide area ground surveillance are practically important examples (GMTI: Ground Moving Target Indicator). In standard tracking algorithms, the objects of interest are usually considered as point source objects; i. e. compared to the sensor resolution their spatial extension is neglected. Due to the increasing resolution capabilities of modern sensors, however, different scattering centers of an extended object cam cause distinct detections. In this sense also collectively moving object groups can be considered as extended objects. Due to the resulting data association and resolution conflicts. any attempt of tracking individual objects within the group is no longer reasonable. In this paper ellipsoidal object extensions are modeled by random matrices, which are treated as additional state variables to be estimated. An important aspect is the incorporation of context information into the Bayesian data processing formalism. We here consider kinematical constraints such as road maps and sensor specific characteristics such as Doppler-blindness. <s> BIB002 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> In algorithms for tracking and sensor data fusion the targets to be observed are usually considered as point source objects; i.e., compared with the sensor resolution their extension is neglected. Due to the increasing resolution capabilities of modern sensors, however, this assumption is often no longer valid as different scattering centers of an object can cause distinct detections when passing the signal processing chain. Examples of extended targets are found in short-range applications (littoral surveillance, autonomous weapons, or robotics). A collectively moving target group can also be considered as an extended target. This point of view is the more appropriate, the smaller the mutual distances between the individual targets are. Due to the resulting data association and resolution conflicts any attempt of tracking the individual objects within the group seems to be no longer reasonable. With simulated sensor data produced by a partly unresolvable aircraft formation the addressed phenomena are illustrated and an approximate Bayesian solution to the resulting tracking problem is proposed. Ellipsoidal object extensions are modeled by random matrices, which are treated as additional state variables to be estimated or tracked. We expect that the resulting tracking algorithms are also relevant for tracking large, collectively moving target swarms. <s> BIB003 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> The task of tracking extended objects or (partly) unresolvable group targets raises new challenges for both data association and track maintenance. Due to limited sensor resolution capabilities, group targets (i. e., a number of closely spaced targets moving in a coordinated fashion) may show a similar detection pattern as extended objects, namely a varying number of detections whose spread is determined by both the statistical sensor errors as well as the physical extension of the group or extended object. Different tracking approaches treating these situations have been proposed where physical extension is represented by a random symmetric positive definite matrix. This paper discusses some results that should give deeper insight into behavior and performance analysis of these approaches. Further improvements are presented. <s> BIB004 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <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 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany †Dept. Sensor Data and Information Fusion, Fraunhofer FKIE, Wachtberg, Germany ‡Data Fusion Algorithms & Software, EADS Deutschland GmbH, Ulm, Germany marcus.baum@kit.edu, {michael.feldmann |wolfgang.koch}@fkie.fraunhofer.de dietrich.fraenken@eads.com, uwe.hanebeck@ieee.org <s> BIB006 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> Conventional tracking algorithms rely on the assumption that the targets under observation are point source ::: objects. However, due to increasing resolution capabilities of modern sensors, the point source assumption is ::: often not suitable and estimating the target extension becomes a crucial aspect. Recently, a Bayesian approach ::: to extended target tracking using random matrices has been proposed. Within this approach, ellipsoidal object ::: extensions are modeled by random matrices and treated as additional state variables to be estimated. However, ::: only a single-target solution has been presented so far. In this work we present the multi-target extension of ::: this approach. We derive a new variant of Probabilistic Multi-Hypothesis Tracking (PMHT) that simultaneously ::: estimates the ellipsoidal shape and the kinematics of each target. For this purpose, the PMHT auxiliary function ::: is extended by random matrices representing the target ellipsoids. Both, the ellipsoids and the kinematic states ::: are iteratively optimized by specific Kalman filter formulae arising directly from the auxiliary function. The new ::: method is demonstrated and evaluated by simulative examples. <s> BIB007 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> The task of tracking extended objects or (partly) unresolvable group targets raises new challenges for both data association and track maintenance. Extended objects may give rise to more than one detection per opportunity where the scattering centers may vary from scan to scan. On the other end, group targets (i. e., a number of closely spaced targets moving in a coordinated fashion) often will not cause as many detections as there are individual targets in the group due to limited sensor resolution capabilities. In both cases, tracking and data association under the one target-one detection assumption are no longer applicable. This paper deals with the problem of maintaining a track for an extended object or group target with varying number of detections. Herein, object extension is represented by a random symmetric positive definite matrix. A recently published Bayesian approach to tackling this problem is analyzed and discussed. From there, a new approach is derived that is expected to overcome some of the weaknesses the Bayesian approach suffers from in certain applications. <s> BIB008 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> Conventional tracking approaches are based on the assumption that the targets to be tracked are point targets and that the measurements to be processed are point measurements. However, when a sensor provides image data of high resolution in which targets might be distributed over several display cells, neither assumption is suitable. In such applications the estimation of the target extent and the utilization of the complete image frame are crucial to achieve good tracking performance. Recently, a Bayesian filter for single extended object tracking based on random matrices has been proposed. In this approach ellipsoidal object extents are modeled by random matrices and treated as additional state variables. This article deals with the integration of random matrices into the Histogram Probabilistic Multi-Hypothesis Tracker. The novel approach tracks multiple extended targets directly in an image sequence without previous point measurement extraction. The superiority of the algorithm is proven by simulations. <s> BIB009 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> This paper presents a random set based approach to tracking of an unknown number of extended targets, in the presence of clutter measurements and missed detections, where the targets' extensions are modeled as random matrices. For this purpose, the random matrix framework developed recently by Koch is adapted into the extended target phd framework, resulting in the Gaussian inverse Wishart phd (giw-phd) filter. A suitable multiple target likelihood is derived, and the main filter recursion is presented along with the necessary assumptions and approximations. The particularly challenging case of close extended targets is addressed with practical measurement clustering algorithms. The capabilities and limitations of the resulting extended target tracking framework are illustrated both in simulations and in experiments based on laser scans. <s> BIB010 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> We present an algorithm for the estimation of the structure of a class of dynamic networks in which object interactions depend on some one-dimensional function of their joint state (e.g. inter-object distance). By using a non-parametric Gaussian process prior assumption on the inter-object relationship strength the algorithm is able to infer a wide range of relationship types. We demonstrate this on a physical object tracking problem. The algorithm is able to cope with a certain degree of noise and can deal with systems involving hundreds of objects on modest hardware. <s> BIB011 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> The approach of using a random matrix for extended object and group target tracking (EOT and GTT) is efficient. Designing and effectively applying this approach rely on modeling the extended object and group targets accurately. To describe complex dynamical variation and practical observation distortion of the extension in size, shape and orientation, two random matrix based models are proposed. True measurement noise can also be incorporated into the measurement model easily. Facilitated by special properties of the models, an approximate Bayesian approach is proposed to estimate the kinematic state and the extension jointly. For maneuvering EOT and GTT, a multiple-model approach is derived by moment matching. To evaluate what is proposed, simulation results of a scenario for maneuvering EOT are also given. The results illustrate the effectiveness of the proposed models and approach. <s> BIB012 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> For irregular extended object and group target tracking (IEOT and IGTT), using a random matrix to simplify the extension as an ellipsoid, although efficient, may not be accurate without losing useful information in shape and orientation. In view of this, we consider modeling an irregular extended object or target group as a combination of multiple regular sub-objects. Each sub-object can be described adequately by an ellipsoid represented by a random matrix. Different regular sub-objects are distinguished by different dynamic models. Based on such models, a Bayesian approach is then proposed to estimate the kinematic states and the extensions of the sub-objects. For maneuvering IEOT and IGTT, a multiple-model approach is proposed. Overall maneuver models are combined with the sub-object models to describe the common and the individual motions of the sub-objects. Two scenarios for maneuvering and non-maneuvering IEOT and IGTT are simulated. The results demonstrate the effectiveness of the proposed modeling and estimation approach. <s> BIB013 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> In extended/group target tracking, where the extensions of the targets are estimated, target spawning and combination events might have significant implications on the extensions. This paper investigates target spawning and combination events for the case that the target extensions are modeled in a random matrix framework. The paper proposes functions that should be provided by the tracking filter in such a scenario. The results, which are obtained by a gamma Gaussian inverse Wishart implementation of an extended target probability hypothesis density filter, confirms that the proposed functions improve the performance of the tracking filter for spawning and combination events. <s> BIB014 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> This paper presents a cardinalized probability hypothesis density (CPHD) filter for extended targets that can result in multiple measurements at each scan. The probability hypothesis density (PHD) filter for such targets has been derived by Mahler, and different implementations have been proposed recently. To achieve better estimation performance this work relaxes the Poisson assumptions of the extended target PHD filter in target and measurement numbers. A gamma Gaussian inverse Wishart mixture implementation, which is capable of estimating the target extents and measurement rates as well as the kinematic state of the target, is proposed, and it is compared to its PHD counterpart in a simulation study. The results clearly show that the CPHD filter has a more robust cardinality estimate leading to smaller OSPA errors, which confirms that the extended target CPHD filter inherits the properties of its point target counterpart. <s> BIB015 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> The Histogram Probabilistic Multi-Hypothesis Tracker (H-PMHT) is a parametric mixture-fitting approach to track-before-detect. The original implementations of H-PMHT dealt with Gaussian shaped targets with fixed or known extent. More recent applications have addressed other special cases of the target shape. This article reviews these recent extensions and consolidates them into a new unified framework for targets with arbitrary appearance. The framework adopts a stochastic appearance model that describes the sensor response to each target and describes filters and smoothers for several example models. The article also demonstrates that H-PMHT can be interpreted as the decomposition of multi-target track-before-detect into decoupled single target track-before-detect using the notion of associated images. <s> BIB016 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> The approach with random matrices <s> This paper presents a new prediction update for extended targets whose extensions are modeled as random matrices. The prediction is based on several minimizations of the Kullback-Leibler divergence (KL-div) and allows for a kinematic state dependent transformation of the target extension. The results show that the extension prediction is a significant improvement over the previous work carried out on the topic. <s> BIB017
A common approach consists in decomposing the joint filtering pdf of groups' state and extent. Group extent has been modeled by means of symmetric positive definite (SPD) random matrices in BIB006 BIB002 BIB003 BIB008 BIB004 BIB007 . Attributed to BIB003 BIB002 it is called the random matrix approach as it considers an augmented state containing the kinematic states and the extent parameters, where the extent is modeled as a random matrix. In BIB005 the extent parameters are merely the covariance matrices in a Gaussian mixture representation of the groups' formation. In BIB006 BIB002 BIB003 BIB008 BIB004 the positive definiteness of the extent is preserved between consecutive time steps by sampling from the inverse Wishart distribution for which the mean is prescribed by the preceding covariance matrix. In particular, the propagated pdf of the jth group center, μ j k , and extent (SPD matrix), Σ j k , is expressed as (21) where I W denotes the inverse Wishart density with parame- are, respectively, the propagated group's mean and extent estimates, and an appropriate measure of state uncertainty. Eq. (21) can be shown to be a ) which thereby allows deriving a rather elegant filtering recursion for both cluster mean and extent BIB003 BIB002 . Hence, (21) is known as Gaussian inverse Wishart (GIW) distribution BIB003 BIB002 . A combination of the random matrices approach BIB003 BIB008 BIB004 for the group extent and the PHD filter is reported in BIB010 BIB014  BIB017 . Considering also the Poisson clutter rate unknown, the likelihood of the filter proposed in BIB014  is gamma Gaussian inverse Wishart (GGIW) distributed. The gamma distribution is for the clutter term, 4 and similarly to BIB003 BIB002 , the Gaussian term is for the kinematic state and the inverse Wishart for the extent. In BIB015 Cardinality PHD filter for extended targets is proposed which relaxes the Poisson assumptions of the extended target PHD filter in target and measurement numbers. A GGIW mixture filter is implemented and its efficiency in estimating the target extent and measurement rates together with the kinematic state of the target, is demonstrated. Apart from conjugate priors, other distributions have been recently used in few Bayesian nonparametric tracking methods. A promising trend is to avoid the often restrictive assumptions of parametric models by designing nonparametric models on infinite-dimensional spaces of functions (e.g. with Gaussian processes BIB011 ) and/or probability distributions (e.g. Dirichlet processes BIB001 ). In BIB012 BIB013 134 ] irregular extended and group targets is considered by the random matrices approach. A different approach called Histogram Probabilistic MultiHypothesis Tracker with Random Matrices (H-PMHT-RM) is surveyed in BIB016 BIB009 . A Wishart prior is applied to the inverse of the appearance covariance matrix. The initially proposed H-PMHT approach deals with Gaussian shaped targets and fixed or known extent. In the next subsection, we present the Random Finite Set Statistics approach which has been shown to be very powerful for the considered problems. 4 Remind that in Bayesian inference, the conjugate prior for the rate parameter of the Poisson distribution is the gamma distribution.