Optimization criteria and geometric algorithms for motion and structure estimation

Prevailing efforts to study the standard formulation of motion and structur e recovery have recently been focused on issues of sensitivity and robustne ss of existing techniques. While many cogent observations have been made an d verified experimentally, many statements do not hold in general settings and make a comparison of existing techniques difficult. With an ultimate go al of clarifying these issues, we study the main aspects of motion and stru cture recovery: the choice of objective function, optimization techniques a nd sensitivity and robustness issues in the presence of noise. We clearly reveal the relationship among different objective functions, suc h as "(normalized) epipolar constraints," "reprojection error" or "triangul ation," all of which can be unified in a new "optimal triangulation" proced ure. Regardless of various choices of the objective function, the optimizat ion problems all inherit the same unknown parameter space, the so-called "e ssential manifold." Based on recent developments of optimization techniques on Riemannian manifolds, in particular on Stiefel or Grassmann manifolds, we propose a Riemannian Newton algorithm to solve the motion and structure recovery problem, making use of the natural differential geometric structur e of the essential manifold. We provide a clear account of sensitivity and robustness of the proposed li near and nonlinear optimization techniques and study the analytical and pra ctical equivalence of different objective functions. The geometric characte rization of critical points and the simulation results clarify the differen ce between the effect of bas-relief ambiguity, rotation and translation con founding and other types of local minima. This leads to consistent interpre tations of simulation results over a large range of signal-to-noise ratio a nd variety of configurations.