First-order algorithm with three clusters of optical flow vectors

The first-order algorithm is an algorithm for recovering the motion parameters from a single optical flow field. It compares the spatial derivatives, to first order, of the optical flow field obtained from two different parts of the field of view and obtains a linear constraint on the direction of the translational velocity. We modify this algorithm to incorporate the spatial derivatives, to first order, of the optical flow field obtained from a third part of the field of view, and thereby obtain two further linear constraints on the direction of the translational velocity. Only two linear constraints are required to identify the direction of the translational velocity. Therefore, with three linear constraints, there are three ways of estimating the direction of the translational velocity. Although all three estimates are derived from the same set of information, we show that the three estimates are not, in general, equally stable. We assume that each cluster of flow vectors arises from a plane in the environment and show that a linear constraint is most stable when it emanates from a pair of parallel planes, with the camera between them. We also identify the relative orientations of the clusters of flow vectors that maximize and minimize the stability of the linear constraint that they give rise to. (C) 1996 John Wiley & Sons, Inc.