Algorithms for the multi-sensor assignment problem in the δ -generalized labeled multi-Bernoulli filter Jun Ye Yu, Augustin-Alexandru Saucan, Mark Coates, Michael Rabbat Electrical and Computer Engineering, McGill University, Montreal, Quebec Email: jun.y.yu, augustin.saucan@mail.mcgill.ca and mark.coates, michael.rabbat@mcgill.ca Abstract—Previous adaptations of the δ-generalized labeled multi-Bernoulli (δ-GLMB) filter to the multi-sensor case involve the sequential application of the update step for each sensor or Gibbs sampling for multi-sensor data association. The practical usage of the sequential δ-GLMB filter is limited due to the number of hypotheses growing with each additional sensor. Similarly, the Gibbs method requires a large number of samples for each hypothesis. In this paper, in the aim of finding the optimal or near-optimal multi-sensor assignments, we propose two novel methods, the combination and the cross entropy methods. Numerical simulations are conducted to evaluate the proposed multi-assignment methods together with the standard sequential processing method and a stochastic optimization algo- rithm based on Gibbs sampling. The combination method is able to significantly reduce running time with respect to the sequential method while yielding competitive performance across a wide range of test scenarios. I. I NTRODUCTION The objective of multi-target tracking is to infer the target tracks in addition to estimating the number of targets and their kinematic states; but non-uniform detection probability, measurement origin uncertainty, false detection and target birth/death are all difficult obstacles to solving the problem. Random finite set (RFS) filters [1] have emerged as a popular paradigm for solving the multi-target tracking problem in the Bayesian framework. Since the exact multi-target Bayes filter is computationally intractable, the probability hypothesis density (PHD) filter [2], cardinalized PHD (CPHD) [3] filter and multi-Bernoulli (MB) filter [4] have been proposed as tractable approximations, although they do not provide target tracks over time. Vo et al. [5] introduced the notion of a labeled RFS in which unique labels are appended to the elements of the RFS to identify their targets (and their estimates) across time and hence infer target tracks. They subsequently developed the δ-generalized multi-Bernoulli (δ-GLMB) density [5] and a single-sensor tracker based on δ-GLMB RFSs [6]. While all these filters have been initially developed for single-sensor tracking, multi-sensor extensions have also been proposed [7]–[14]. In the iterator-corrector PHD filter [7], [8], each sensor’s measurements are processed sequentially and the output from one sensor is used as input for the next sensor. The recent work by Papi [10] presents the multi-sensor extension of the δ-GLMB filter, and its implementation also involves iterating through each sensor. Liu et al. [11] use an extended association table to generate the most likely asso- ciations between targets and measurements from all sensors, but no simulations are provided to validate the algorithm’s performance. Both the multi-sensor CPHD filter [12] and the multi-sensor multi-Bernoulli filter [13] process all sensor measurements simultaneously by using a greedy algorithm to select the most likely associations. Vo et al. [14] have recently applied Gibbs sampling to find a number of likely multi-sensor assignments in the δ-GLMB filter. The exact implementation of the multi-sensor δ-GLMB filter requires enumerating all multi-sensor assignments to compute the posterior multi-target density. In practice, we look for a number of likely multi-sensor assignments to construct a truncated density. Although the problem of finding the T best single-sensor assignments can be solved efficiently using Murty’s algorithm [15], the multi-sensor counterpart is NP-hard [16]. In this paper, we present two approximation algorithms, the combination and the cross entropy methods, that yield a number of likely multi-sensor assignments without exhaustive enumeration. We compare their performance to the Gibbs method [14] and the sequential processing method [10]. The combination method first solves the assignment problem locally at each sensor and then combines the locally optimal solutions to form high-scoring multi-sensor assignments. The cross entropy method constructs a distribution on the space of all multi-sensor assignments with higher probability for more likely assignments. The algorithms’ performances are compared via simulations and it is shown that the combination method greatly reduces computational time with respect to the sequential method while yielding near-optimal assignments. II. BACKGROUND The study of finite set statistics [1] has led to the develop- ment of multi-target filters in the Bayesian framework where the target states are modeled as random finite sets. An RFS is a finite set with random cardinality and elements, and thus it conveniently captures the two unknown quantities of interest in multi-target tracking problems: the number of targets and their states. A labeled RFS appends a unique label to each element in the RFS. Elements with the same label correspond to the same target and allow the formulation of target tracks. For the rest of the paper, unlabeled single-object states are denoted by lower case letters (e.g., x) and multi-object states (realizations of an RFS) by upper case letters (e.g., X). Their labeled counterparts are bold letters (e.g., x =(x,l), X = {x 1 ,...x n }). The blackboard bold letters (e.g., X, L) denote the corresponding state space. The projection function L(X)= {l, (x,l) ∈ X} returns the labels of a labeled RFS. The distinct