2ND-ORDER OPTIC FLOW

We investigate the structure of the second-order spatial differential structure of optic flow for the case of a rigid surface in parallel projection. We disregard such problems as that of noise and that of how to obtain optic flow from image flow in the first place. The focus of the investigation is on the essential geometric structures. It is shown that the second-order structure is conveniently represented as a tensor that allows one to obtain the second-order directional derivative of the flow for arbitrary directions in the field of view. This second-order directional derivative in an arbitrary direction turns out to be a vector in the epipolar direction with a magnitude that is proportional to the normal curvature of the surface in the direction of derivation. Our analysis involves only affine concepts and constructions; metric concepts do not enter into the analysis. The second-order structure immediately reveals the projection of Dupin's indicatrix, which captures important affine aspects of shape. When the second-order structure is combined with the first-order differential structure and after the introduction of a metric, only one degree of ambiguity is left: The rate of turn about an axis in the frontoparallel plane can be traded against the height of relief. Finally we construct hypothetical neural implementations for the second-order flow analysis. A simple, center-surround organization suffices, although one may not pool over orientations.