STATISTICAL-ANALYSIS OF INHERENT AMBIGUITIES IN RECOVERING 3-D MOTION FROM A NOISY FLOW FIELD

In this paper, 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 x 5 matrix. As a special case, the performance bound for the motion of 3-D rigid planar surfaces is studied in detail. The dependence of the bound on several factors, such as the underlying motion, surface position, surface orientation, field of view, and density of available pixels are derived as closed-form expressions. A subset of our results support Adiv's analysis of the inherent ambiguities of motion parameters. For the general motion of an arbitrary surface, it turns out that not every pixel gives information regarding 3-D motion estimation. We show 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 a smoothness constraint introduced by fitting local patches to 3-D depths gives lower bounds. However, this reduction is very small. Further, fitting local patches also relaxes the aperture problem since the motion information is not restricted to points at which both optical flow components are observable.