Number of Solutions for Motion and Structure from Multiple Frame Correspondence

Much of the dynamic computer vision literature deals with the determination of motion and structure by observing two frames captured at two instants of time. Motion prediction and understanding can be improved significantly, particularly in the presence of noise, by analyzing an image sequence containing more than two frames. In this paper, we assume knowledge of correspondence of points on the surface of an object which is moving with constant motion, i.e., constant translation and constant rotation around an unknown center. We give a new formulation of the problem and prove that the following results hold in general for the number of solutions to motion and structure values (i.e., values of translation, rotation, and depth):(a) For three point correspondences over three views, there are at most two solutions, only one of which has all positive depth values;(b) For two point correspondences over four views, there is a unique solution;(c) For one point correspondence over five views, there can be up to ten solutions;(d) For one point correspondence over six views, there is a unique solution.The method of solution for each of the above formulations requires the solving of a system of multivariate polynomials, whose coefficients are functions of the observed data. In order to determine the number of solutions to these systems, we use theorems from algebraic geometry which imply that under a few mild conditions, the number of solutions at one set of data points provides an upper bound on the number of solutions for almost all sets of data points.Thus a bound on the number of solutions is obtained when a single system is solved by a method such as homotopy continuation, which we use here.