Sufficient Conditions for Double or Unique Solution of Motion and Structure

Abstract

This paper presents several sufficient conditions for a double or unique solution of the problem of motion and structure estimation of a rigid surface from pairs of monocular images. These conditions further the understanding of the uniqueness problem of rigid motion solution. We will show that five correspondences of noncolinear points that do not lie on a special type of quadratic curve, called a Maybank curve, in the image plane suffice to determine a pure rotation uniquely, and six correspondences of points that do not correspond to space points lying on a Maybank quadric suffice to determine a motion with nonzero translation uniquely. We will show that each Maybank quadric can sustain at most two physically acceptable motion solutions and surface interpretations, provided that a sufficient number of correspondences are present. In particular, we will show that in the plane motion case, six correspondences of points that do not lie on a quadratic curve in the image plane will admit only the true motion and structure and their duals as solutions. We will discuss how noise affects the uniqueness of solution and present a nonlinear algorithm for estimation of motion parameters. We will list several properties of the essential matrix T × R and the plane motion matrix R + TN^τ, both of which are frequently used in the motion and structure estimation problem. Simulation results are provided for verifying the theorems in this paper.