Uncertainty Modeling for Optimal Structure from Motion

The parameters estimated by Structure from Motion (SFM) contain inherent indeterminacies which we call gauge freedoms. Under a perspective camera, shape and motion parameters are only recovered up to an unknown similarity transformation. In this paper we investigate how covariance-based uncertainty can be represented under these gauge freedoms. Past work on uncertainty modeling has implicitly imposed gauge constraints on the solution before considering covariance estimation. Here we examine the effect of selecting a particular gauge on the uncertainty of parameters. We show potentially dramatic effects of gauge choice on parameter uncertainties. However the inherent geometric uncertainty remains the same irrespective of gauge choice. We derive a Geometric Equivalence Relationship with which covariances under different parametrizations and gauges can be compared, based on their true geometric uncertainty. We show that the uncertainty of gauge invariants exactly captures the geometric uncertainty of the solution, and hence provides useful measures for evaluating the uncertainty of the solution. Finally we propose a fast method for covariance estimation and show its correctness using the Geometric Equivalence Relationship.