# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 1 Introduction  
For multi-target tracking (MTT) task, the number and the states of targets are jointly estimated from a sequence of measurements. MTT is widely used in many applications, such as radar [1, 2], sonar [3], computer vision [4], robotics [5], etc. Compared with single target tracking, the number of targets changes with time in MTT because of target birth and target death. The simplest method for MTT is the global nearest neighbor (GNN) tracker [6]. However, since a unique association between targets and measurements is obtained, this method cannot work well for close-space targets. Joint probabilistic data association (JPDA) filter [7, 8], multiple hypotheses tracking (MHT) [9, 10], and random finite set (RFS) framework [11, 12] are three major works for MTT. JPDA filter and MHT consist of two steps: data association and (single-target) filtering. However, data association is integrated into filtering step in RFS framework with explicit data association being avoided. JPDA filter is widely used to track multiple targets under the assumption that the number of targets is already known and fixed. In JPDA, joint association probabilities rather than a unique association result are obtained toavoid conflicting association. Several extensions and related works of JPDA can be found in [13; 14; 15; 16]. MHT attempts to take all of the association hypotheses into account and eliminates association hypotheses with low posterior probability to achieve tractable computational complexity. Related works and extensions of MHT can be found in [17; 18; 19]. In RFS framework, multi-target state is modeled as a finite set [11; 12], and the posterior density of the multi-target state is propagated recursively based on Bayesian formulation. There are many works based on RFS framework, such as the probability hypothesis density (PHD) [20], cardinalized PHD (CPHD) [21], multi-Bernoulli filters [22; 23], generalized labeled multi-Bernoulli filter (GLMB) [24; 25] and labeled multi-Bernoulli filter (LMB) [26]. In many scenarios, targets travel in group, such as a group of airplanes moving in formation and the scattering centers on a 2D rigid target [27; 28]. The scattering centers on a 2D rigid target share common motions and maintain relative order in certain time range [29], these scattering centers can be considered as group of targets, each target in group is almost rigidly positioned with respect to each other, thus the structure information of these scattering centers can be used to enhance tracking performance. For example, the states of undetected targets can be inferred through other detected targets. Although the structure information is beneficial to MTT, the targets in group are treated independently in JPDA filter, MHT, LMB, etc. In these filters, the measurement is associated with the closest target's state, the reason is that the smallest distance corresponds to the highest likelihood under the assumption that these targets share common motion. Therefore, incorrect data association results can be obtained when actual motion model is not coincided with the preset dynamic motion model. In this paper, the structure information are used to enhance the tracking performance of rigid targets. In order to make better use of structure information of group targets, a representation of hypergraph is introduced in this paper. Hypergraph representation is widely used in many applications, such as target tracking [30; 31], feature association [32], etc. Considering measurements and predicted target states as the vertices of the hypergraph, the problem of data association can be formulated as a hypergraph matching problem. Hypergraph matching problem has been widely investigated with numerous algorithms being developed. [33] has presented a probabilistic model for soft hypergraph matching, the hypergraph matching problem is derived in a probabilistic setting represented by a convex optimization. [34] introduces a random walk view on the hypergraph matching problem.