Multi-Target Tracking on Riemannian Manifolds via Probabilistic Data Association

Abstract

Riemannian manifolds are attracting much interest in various technical disciplines, since generated data can often be naturally represented as points on a Riemannian manifold. Due to the non-Euclidean geometry of such manifolds, usual Euclidean methods yield inferior results, thus motivating development of tools adapted or specially tailored to the true underlying geometry. In this letter we propose a method for tracking multiple targets residing on smooth manifolds via probabilistic data association. By using tools of differential geometry, such as exponential and logarithmic mapping along with the parallel transport, we extend the Euclidean multi-target tracking techniques based on probabilistic data association to systems constrained to a Riemannian manifold. The performance of the proposed method was extensively tested in experiments simulating multi-target tracking on unit hyperspheres, where we compared our approach to the von Mises-Fisher and the Kalman filters in the embedding space that projects the estimated state back to the manifold. Obtained results show that the proposed method outperforms the competitive trackers in the optimal sub-pattern assignment metric for all the tested hypersphere dimensions. Although our use case geometry is that of a unit hypersphere, our approach is by no means limited to it and can be applied to any Riemannian manifold with closed-form expressions for exponential/logarithmic maps and parallel transport along the geodesic curve. The paper code is publicly available <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> .