Optimal Gaussian Filtering for Polynomial Systems Applied to Association-free Multi-Target Tracking Marcus Baum, Benjamin Noack, Frederik Beutler, Dominik Itte, and Uwe D. Hanebeck Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics, Karlsruhe Institute of Technology (KIT), Germany. Email: {marcus.baum, noack, beutler}@kit.edu, dominik.itte@student.kit.edu, uwe.hanebeck@ieee.org Abstract—This paper is about tracking multiple targets with the so-called Symmetric Measurement Equation (SME) filter. The SME filter uses symmetric functions, e.g., symmetric polynomials, in order to remove the data association uncertainty from the measurement equation. By this means, the data association problem is converted to a nonlinear state estimation problem. In this work, an efficient optimal Gaussian filter based on analytic moment calculation for discrete-time multi-dimensional polynomial systems corrupted with Gaussian noise is derived, and then applied to the polynomial system resulting from the SME filter. The performance of the new method is compared to an UKF implementation by means of typical multiple target tracking scenarios. Keywords: Gaussian filtering, polynomial systems, SME filter, multi-target tracking. I. I NTRODUCTION Tracking multiple targets based on noisy measurements is a frequently occuring problem in many applications such as surveillance [1], [2]. A major problem in multi-target tracking is that the measurement-to-target association is unknown, i.e., the target from which a measurement originates is not given. Many different solutions and methodologies for dealing with this data association uncertainty can be found in literature [1]–[3]. For instance, the Multi-Hypothesis-Tracker (MHT) [4] maintains all feasible data association hypotheses over a finite time horizon. The well-known Joint Probabilistic Data Association Filter (JPDAF) [3] combines all single target estimates according to association probabilities. Association-free methods such as the PHD-filter [5], [6] do not explicitly evaluate association hypotheses. This paper is about an association-free method called Symmetric Mea- surement Equation (SME) filter [7]–[10]. The SME filter uses symmetric functions in order to remove the data association uncertainty from the measurement equation. A function is called symmetric if a permutation of its arguments does not change the result. Usually, the SME filter is based on symmetric polynomials, and hence, the resulting measurement equation is a polynomial as well. As a consequence, the data association problem is converted to a nonlinear estimation problem with a polynomial measurement equation. In this work, we aim at using a Gaussian state estimator for the arising polynomial equations of the SME filter. Examples for a Gaussian state estimator are the well-known Extended Kalman filter (EKF) [11] or deterministic sampling approaches such as the UKF [12] or [13]. The Divided Difference Fil- ter (DDF) performs a derivative-free approximation of the system functions based on Stirling’s formula [14], [15]. The Polynomial Extended Kalman filter [16] employes a Carleman linearization in order to obtain a bilinear system. A well-known and widely-used techniques is the analytic moment calculation of nonlinear transformed random vari- ables. For instance, the Divided Difference Filter (DDF) [14] performs analytic moment calculation for a second- order polynomial approximation of the system functions. In [17]–[19], the moments of a second-order polynomial system function are directly be computed in closed form und used for optimal estimation. Recently, analytic methods have been combined with approximate methods [20]–[23]. In [24], an optimal Kalman filter for one-dimensional polynomial systems is derived and in [25], a quasi-Gaussian Kalman filter for continuous dynamic systems is presented. Analytic moment calculation has been used in a variety of applications such as localization [17] or tracking [18], [19] and proven to be a promising alternative to approximate solutions as it gives the optimal solution in closed form and no parameter tuning is required. To the best of our knowledge, analytic moment calculation has not been applied to the SME filter yet. The contributions of this work are the followings. First, we describe an efficient black-box Gaussian-assumed Bayesian filter for multi-dimensional polynomial systems based on analytic moment calculation (see Section II). For this pur- pose, an automatic efficient method for computing the first two moments of polynomially transformed Gaussian random variables is derived based on a formula for the expectation of products of Gaussian random variables introduced in [26]. The filter is then applied for tracking multiple targets with the SME approach [7], [9], [10] using symmetric polynomial functions. Simulation results show that the introduced filter is feasible even for high-dimensional polynomial systems and outperforms an UKF implementation in typical tracking scenarios (see Section III). II. GAUSSIAN ASSUMED BAYESIAN FILTER FOR POLYNOMIAL SYSTEMS In this work, we consider a discrete time nonlinear dynamic system of the form x k+1 = a k (x k ,u k , w k ) , (1)