Closed from solutions to image flow equations for planar surfaces in motion

Abstract In this paper we consider the recovery of the 3-dimensional motion and orientation of a rigid planar surface from its image flow field. Closed form solutions are derived for the image flow equations formulated by Waxman and Ullman [1]. Also we give two important results relating to the uniqueness of solutions for the image flow equations. The first result concerns resolving the duality of interpretations that are generally associated with the instantaneous image flow of an evolving image sequence. It is shown that the interpretation for orientation and motion of planar surfaces is unique when either two successive image flows of one planar surface patch are given or one image flow of two planar patches moving as a rigid body is given. We have proved this by deriving explicit expressions for the evolving solution of an image flow sequence with time. These expressions can be used to resolve this ambiguity of interpretation in practical problems. The second result is the proof of uniqueness for the velocity of approach which satisfies the image flow equations for planar surfaces derived in [1]. In addition, it is shown that this velocity can be computed as the middle root of a cubic equation. These two results together suggest a new method for solving the image flow problem for planar surfaces in motion. We also describe a scheme to use first-order time derivatives of the image flow field in place of the second-order spatial derivatives to solve for the orientation and motion.