Geometric algorithms for least squares estimation of 3-D information from monocular image

Estimating the three-dimensional (3-D) information of an object from a sequence of projections is of paramount importance in diverse domains such as autonomous navigation, robot vision, object recognition, world modeling, and human-computer interaction. We present geometric optimization-based algorithms for simultaneous 3-D shape (depth) and motion (linear and angular velocities) recovery that minimizes the squared distance between the observed and predicted optical flow measurements. A principal advantage of our approach is that, using a separable nonlinear least-squares approach in various ways, the search space dimension in the numerical optimization is reduced. For improvement of performance in the numerical optimization, we also develop various methods to take advantage of the separable nonlinear least-squares approach effectively. A second advantage is that, by explicitly accounting for the existence of an one-parameter family of solutions, a geometric optimization algorithms on the Cartesian product manifold can be applied. The Hessian is straightforwardly evaluated as a by-product of the algorithm, making sensitivity analysis straightforward.