Optimization Criteria, Sensitivity and Robustness of Motion and Structure Estimation

The prevailing efforts to study the standard formulation of motion and structure recovery have been recently focused on issues of sensitivity and 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 the problem: the choice of objective functions, optimization techniques and the 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”, which can all be be unified in a new “ optimal triangulation” procedure formulated as a constrained optimization problem. Regardless of various choices of the objective function, the optimization problems all inherit the same unknown parameter space, the so called “essential manifold”, making the new optimization techniques on Riemanian manifolds directly applicable. Using these analytical results 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 of a function defined on essential manifold and the simulation results clarify the difference between the effect of bas relief ambiguity and other types of local minima leading to a consistent interpretations of simulation results over large range of signal-to-noise ratio and variety of configurations. Further more we justify the choice of the linear techniques for (re)initialization and for detection of incorrect local minima.