Motion estimation in computer vision:: Optimization on Stiefel manifolds

Motion recovery from image correspondences is typically a problem of optimizing an objective function associated with the epipolar (or Longuet-Higgins) constraint. This objective function is defined on the so called essential manifold. In this paper, the intrinsic Riemannian structure of the essential manifold is thoroughly studied. Based on existing optimization techniques on Riemannian manifolds, in particular on Stiefel manifolds, we propose a Riemannian Newton algorithm to solve the motion recovery problem, making use of the natural geometric structure of the essential manifold. Although only the Newton algorithm is studied in detail, the same ideas also apply to other typical conjugate gradient algorithms. It is shown that the proposed nonlinear algorithms converge very rapidly (with quadratic rate of convergence) as long as the conventional SVD based eight-point linear algorithm has a unique solution. Such Riemannian algorithms have also been applied to the differential (or continuous) case where the velocities are recovered from optical flows.