Constrained object tracking on compact one-dimensional manifolds based on directional statistics

In this paper, we present a novel approach for tracking objects whose movement is constrained to a compact one-dimensional manifold, for example a conveyer belt or a mobile robot whose movement is restricted to tracks. Standard approaches either ignore the constraint at first and retroactively move the estimate to lie on the manifold, or consider the tracking problem on a manifold but falsely assume a Gaussian distribution. Our method explicitly takes the actual topology into account from the beginning and relies on special types of probability distributions defined on the proper manifold. In particular, we consider objects moving along a closed one-dimensional track, for example an ellipse, a polygon, or similar closed shapes. This shape is transformed to a circle with a homeomorphism. Thus, we can apply a recursive circular filtering algorithm to the constrained tracking problem. Finally, the estimate is transformed back to the original manifold. We evaluate the proposed method in an experiment by tracking a toy train moving along a track and comparing the results to those of traditional approaches for this problem.