Solving three-dimensional small-rotation motion equations: Uniqueness, algorithms, and numerical results

In the first part of this paper, a theorem on the uniqueness of the solution to a set of overdetermined nonlinear equations obtained by T. S. Huang and R. Y. Tsai [1] for approximately determining 3-D motion parameters when the rotation angle is small is presented. The main result is that if nine points which are not on a second-order surface passing through the viewing point, are correspondingly selected from two sequential images of a moving object, then the solution of the motion equations can be uniquely determined. In the second part, the practical aspects of solving these overdetermined nonlinear equations are discussed. A modified Newton method for solving nonlinear equations and a modified Levenberg-Marquardt method for solving the nonlinear least-squares problem, which are better than the original Newton and Levenberg-Marquardt methods when applied to the problem of motion estimation, are proposed. The effects on convergence and solution accuracy of the number of corresponding image point pairs, the geometrical configuration of the points in object space, the distance of the object from the image plane, the initial guess solution, and image resolution are also studied experimentally.