Uncertainty Modeling for Optimal Structure from Motion

The parameters estimated by Structure from Motion contain inherent indeterminacies. In particular, shape and motion parameters are only recovered up to a similarity transformation. Past work on uncertainty modeling has imposed gauge constraints on the coordinates and scale to reduce the number of parameters and eliminate these indeterminacies. In this paper our approach is instead to maintain these indeterminacies, or gauge freedoms, in our covariance-based uncertainty representation. The solution is not unique so we define the normal form for a covariance matrix which captures the essential geometric uncertainty in the parameters along with the indeterminacies, and derive an equivalence relationship between covariances that capture the same geometric uncertainty. We then propose an efficient method for estimating the shape and motion covariances. The indeterminacies imply that Structure from Motion over-parametrizes the problem and that the parameters do not have direct physical meani..