Optimization Criteria and Geometric Algorithms for Motion and Structure Estimation

Abstract

Prevailing efforts to study the standard formulation of motion and structure recovery have recently been focused on issues of sensitivity and robustness of existing techniques. While many cogent observations have been made and verified experimentally, many statements do not hold in general settings and make a comparison of existing techniques difficult. With an ultimate goal of clarifying these issues, we study the main aspects of motion and structure recovery: the choice of objective function, optimization techniques and sensitivity and robustness issues in the presence of noise. We clearly reveal the relationship among different objective functions, such as “(normalized) epipolar constraints,” “reprojection error” or “triangulation,” all of which can be unified in a new “optimal triangulation” procedure. Regardless of various choices of the objective function, the optimization problems all inherit the same unknown parameter space, the so-called “essential 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 structure of the essential manifold. We provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions. The geometric characterization of critical points and the simulation results clarify the difference between the effect of bas-relief ambiguity, rotation and translation confounding and other types of local minima. This leads to consistent interpretations of simulation results over a large range of signal-to-noise ratio and variety of configurations.