Statistical analysis of inherent ambiguities in recovering 3-D motion from a noisy flow field

The inherent ambiguities in recovering 3-D motion information from a single optical flow field are studied using a statistical model. These ambiguities are quantified using the Cramer-Rao lower bound, which is a lower bound for the error variances of motion parameter estimates. This performance bound is independent of the motion estimation algorithms, and can always be computed for any arbitrary 3-D motion of a rigid surface by inverting a 5×5 matrix. For the general motion of an arbitrary surface, it turns out that not every pixel gives information regarding 3-D motion estimation. It is shown that the aperture problem in computing the optical flow restricts the nontrivial information about the 3-D motion to a sparse set of pixels at which both components of the flow velocity are observable. Computer simulations are used to study the dependence of the inherent ambiguities on the underlying motion, the field of view, and the number of feature points for the motion in front of a nonplanar environment. It is shown that introducing a smoothness constraint by fitting local patches gives even lower bounds and thus is a justified technique for stabilizing the ill-posed motion estimation problem