Motion from Optic Flow: Multiplicity of Solutions

Abstract

One of the long-standing problems in dynamic computer vision has been the determination of three-dimensional motion and structure of rigid bodies given their optic flow. Since the equations relating optic flow to motion and structure are nonlinear and many-to-one, multiple solutions are often obtained. In this paper, we give an alternative proof for a previously known result. Given optic flow at five image points, there are in general at most 10 solutions for motion, or values for rotation and translation. Our method of proof is based on some new techniques from algebraic geometry and is used to prove several new results. We show that if optic flow is known at six or more image points, then the solution is almost always unique. If the motion is known to be purely rotational, then it can be uniquely determined from the optic flow at just two distinct points. Finally, we show that if the moving surface is planar, then the optic flow at four or more points almost always results in exactly two motion values. Since a large class of problems in dynamic computer vision requires solution of multivariate polynomial equations, we believe that our approach will have general utility in this field.