Mathematical properties of the two-dimensional motion field: from singular points to motion parameters

Abstract

The motion field, that is, the two-dimensional vector field associated with the velocity of points on the image plane, can be seen as the flow vector of the solution to a planar system of differential equations. Therefore the theory of planar dynamical systems can be used to understand qualitative and quantitative properties of motion. In this paper it is shown that singular points of the motion field, which are the points where the field vanishes, and the time evolution of their local structure capture essential features of three-dimensional motion that make it possible to distinguish translation, rotation, and general motion and also make possible the computation of the relevant motion parameters. Singular points of the motion field are the perspective projection onto the image plane of the intersection between a curve called the characteristic curve, which depends on only motion parameters, and the surface of the moving object. In most cases, singular points of the motion field are left unchanged in location and spatial structure by small perturbations affecting the vector field. Therefore a description of motion based on singular points can be used even when the motion field of an image sequence has not been estimated with high accuracy.