A perturbation analysis and performance bound for the estimation of 3-D camera translation direction

Abstract

The estimation of the 3-D displacement of a moving camera relative to a scene is critical for applications in robotics as well as object-based video compression. By estimating the relative motion within a scene, one can accurately predict object positions for the future, thereby reducing the amount of information which must be coded. This problem has an inherently nonlinear nature and can therefore be computationally expensive. However, several related linear methods which use optical flow to recover the translation direction of a moving camera have been previously presented. These algorithms all depend on solving a constrained least-squares minimization problem. In this paper, we investigate the effects of noise perturbations on eigenvector and eigenvalue estimates for least-squares solutions, including a necessary modification to the Cramér-Rao lower bound (CRLB), which specifies the minimum possible covariance matrix for constrained estimators. We propose a computationally simple nonlinear algorithm which produces near-optimal translation-direction estimates (in the sense that the mean vector is unbiased and the covariance matrix approaches the CRLB). Sample numerical results are used to compare the various techniques and illustrate the agreement between predicted and observed values. Finally, some remaining problems meriting further investigation are discussed.