Solving the Discrete Euler-Arnold Equations for the Generalized Rigid Body Motion

We propose three iterative methods for solving the Moser-Veselov equation, which arises in the discretization of the Euler-Arnold differential equations governing the motion of a generalized rigid body. We start by formulating the problem as an optimization problem with orthogonal constraints and proving that the objective function is convex. Then, using techniques from optimization on Riemannian manifolds, the three feasible algorithms are designed. The first one splits the orthogonal constraints using the Bregman method, whereas the other two methods are of the steepest-descent type. The second method uses the Cayley-transform to preserve the constraints and a Barzilai-Borwein step size, while the third one involves geodesics, with the step size computed by Armijo's rule. Finally, a set of numerical experiments are carried out to compare the performance of the proposed algorithms, suggesting that the first algorithm has the best performance in terms of accuracy and number of iterations. An essential advantage of these iterative methods is that they work even when the conditions for applicability of the direct methods available in the literature are not satisfied.

POSEAMM: A Unified Framework for Solving Pose Problems using an Alternating Minimization Method

Pose estimation is one of the most important problems in computer vision. It can be divided in two different categories -- absolute and relative -- and may involve two different types of camera models: central and non-central. State-of-the-art methods have been designed to solve separately these problems. This paper presents a unified framework that is able to solve any pose problem by alternating optimization techniques between two set of parameters, rotation and translation. In order to make this possible, it is necessary to define an objective function that captures the problem at hand. Since the objective function will depend on the rotation and translation it is not possible to solve it as a simple minimization problem. Hence the use of Alternating Minimization methods, in which the function will be alternatively minimized with respect to the rotation and the translation. We show how to use our framework in three distinct pose problems. Our methods are then benchmarked with both synthetic and real data, showing their better balance between computational time and accuracy.