Visual tracking of points as estimation on the unit sphere

Abstract

In this paper we consider the problem of estimating the direction of points moving in space from noisy projections. This problem occurs in computer vision and has traditionally been treated by ad hoc statistical methods in the literature. We formulate it as a Bayesian estimation problem on the unit sphere and we discuss a natural probabilistic structure which makes this estimation problem tractable. Exact recursive solutions are given for sequential observations of a fixed target point, while for a moving object we provide optimal approximate solutions which are very simple and similar to the Kalman Filter recursions. We believe that the proposed method has a potential for generalization to more complicated situations. These include situations where the observed object is formed by a set of rigidly connected feature points of a scene in relative motion with respect to the observer or the case where we may want to track a moving straight line, a moving plane or points constrained on a plane, or, more generally, points belonging to some smooth curve or surface moving in ℝ³. These problems have a more complicate geometric structure which we plan to analyze in future work. Here, rather than the geometry, we shall concentrate on the fundamental statistical aspects of the problem.