A wide-sense estimation theory on the unit sphere

Online estimation of the direction of feature points moving in space from noisy projections on a plane is a classical problem occuring in computer vision which has traditionally been treated by ad hoc statistical methods in the literature. Picci (1997) formulated it as a Bayesian estimation problem on the unit sphere. A natural probabilistic structure which makes this estimation problem tractable has been introduced. Within this structure, exact recursive solutions can be given for sequential observations of a fixed target point, while for a moving object in general one has to resort to approximations. In this paper an approximate (“wide-sense”) solution is proposed which leads to very simple recursions similar to the Kalman filter. In certain situations this solution may provide a substantial improvement over the traditional EKF. As an example, we discuss estimation of the direction of points whose motion is described by a simple dynamic model of the random walk type. This model is of interest in practical situations when dealing with slowly time-varying observed feature points