Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Bayesian problem formulation <s> In this paper, we describe models and algorithms for detection and tracking of group and individual targets. We develop two novel group dynamical models, within a continuous time setting, that aim to mimic behavioural properties of groups. We also describe two possible ways of modeling interactions between closely spaced targets using Markov Random Field (MRF) and repulsive forces. These can be combined together with a group structure transition model to create realistic evolving group models. We use a Markov Chain Monte Carlo (MCMC)-Particles Algorithm to perform sequential inference. Computer simulations demonstrate the ability of the algorithm to detect and track targets within groups, as well as infer the correct group structure over time. <s> BIB001 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Bayesian problem formulation <s> In this paper, we describe models and algorithms for detection and tracking of group and individual targets. We develop two novel group dynamical models within a continuous time setting using stochastic differential equations (SDE) that aim to mimic behavioural properties of groups. We also describe a possible way of modeling interactions between closely spaced targets using repulsive forces. These can be combined with a group structure transition model to create realistic evolving group models. We use a Markov chain Monte Carlo (MCMC)-particles algorithm to perform sequential inference. Computer simulations demonstrate the ability of the algorithm to detect and track targets within groups as well as to infer the correct group structure over time. The group tracking model is also applied to two sets of real ground moving target indicator (GMTI) radar data with group targets. The results show significant improvement in tracking accuracy over tracking without group models. <s> BIB002 </s> Overview of Bayesian sequential Monte Carlo methods for group and extended object tracking <s> Bayesian problem formulation <s> This paper proposes a technique for motion estimation of groups of targets based on evolving graph networks. The main novelty over alternative group tracking techniques stems from learning the network structure for the groups. Each node of the graph corresponds to a target within the group. The uncertainty of the group structure is estimated jointly with the group target states. New group structure evolving models are proposed for automatic graph structure initialization, incorporation of new nodes, unexisting nodes removal, and the edge update. Both the state and the graph structure are updated based on range and bearing measurements. This evolving graph model is propagated combined with a sequential Monte Carlo framework able to cope with measurement origin uncertainty. The effectiveness of the proposed approach is illustrated over scenarios for group motion estimation in urban environments. Results with challenging scenarios with merging, splitting, and crossing of groups are presented with high estimation accuracy. The performance of the algorithm is also evaluated and shown on real ground moving target indicator (GMTI) radar data and in the presence of data origin uncertainty. <s> BIB003
The problem of tracking groups with a small number of components consists in estimating an augmented state vector X k containing the states of all groups, at discrete time k, with their components, given a parameter vector G k characterizing the structure of all groups. At time k when a measurement vector z k is received, the measurement likelihood function p(z k |X k ) can be calculated, and hence the state pdf for each group of objects. Under the Markovian assumption for the state transition, the Bayesian prediction and filtering steps can be written as follows: Fig. 4 . Boids system and virtual leader local interaction rules. The blue and red arrows represent object velocities and restoring forces, respectively. The position of the virtual leader is illustrated by a yellow circle in each group. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) where z k and Z 0:k denote the measurement at time k and the measurement history up to time k, respectively. of the group structure can be calculated in two ways. In BIB002 a prior transition matrix between the discrete possible group structures while in BIB003 the group structure model based on graphical network models is used. In this second approach the nodes of the graph correspond to the components of the group. The presence of an edge between two nodes reflects a link between these two objects. Merging, splitting, spawning and birth of groups are all modeled within the framework of evolving graphs and by taking into account geometrical distances between groups and within group components. Prediction is performed by an evolution model for the group/graph structure BIB003 . Next, the transition pdf p( targets is calculated knowing the previous time target states and group structure. With the assumption of independence between group motions, the pdf p( where ) is the transition density of the set of targets from the g-th group. The group structure can be represented in different ways. In BIB001 , G k represents a set of group's labels for each target. The group structure can be represented also as a random graph as shown in BIB003 . Consider N targets constituting the set of vertices {v 1 , . . . , v N }. Each vertex v i is associated with the target state and with the target state's corresponding variance. The set of edges linking the set of vertices is denoted by E. The graph structure can then be denoted by G = ({v 1 , . . . , v N }, E). One edge, in E, between two nodes v i and v j is denoted by (v i , v j ). In order to characterize the presence or absence of a link (edge) between two nodes, the distance between these two considered nodes is calculated, e.g., by the Mahalanobis distance criterion. In the previous example discussed, with the graphical representation, one group structure is: 5 )}) and the groups correspond to the connected components of the graph G k . The Mahalanobis distance is computed from the estimated positions and from the velocities of the separate objects. This estimated distance is thresholded and a decision is made about the connections. In this representation a group corresponds to a connected component of the graph structure. Note that, two nodes are in the same connected component if and only if a path between them exists. The Matlab code for the algorithm presented in this section is available on Matlab Central  . Algorithms for group birth, death, splitting and merging are proposed in BIB003 by taking into account geometric distances and velocity distances between the groups and between the separate group components. The next subsection presents interaction models and results inspired by biological systems.