Representation issues in the ML estimation of camera motion

The computation of camera motion from image measurements is a parameter estimation problem. We show that for the analysis of the problem's sensitivity, the parameterization must enjoy the property of fairness, which makes sensitivity results invariant to changes of coordinates. We prove that Cartesian unit norm vectors and quaternions are fair parameterizations of rotations and translations, respectively, and that spherical coordinates and Euler angles are not. We extend the Gauss-Markov theorem to implicit formulations with constrained parameters, a necessary step in order to take advantage of fair parameterizations. We show how maximum likelihood (ML) estimation problems whose sensitivity depends on a large number of parameters, such as coordinates of points in the scene, can be partitioned into equivalence classes, with problems in the same class exhibiting the same sensitivity.