The inherent ambiguities in recovering 3-D motion information from a single optical flow field are studied using a statistical model. The ambiguities are quantified using the Cramer-Rao lower bound. 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 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 the results support S. Adiv's (1989) analysis of the inherent ambiguities of motion parameters. For the general motion of an arbitrary surface. 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.>

The inherent ambiguities in recovering 3-D motion information of a planar surface 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. Motion-estimation algorithms which give unbiased estimates of motion parameters are considered. The bound is independent of the rotational motion parameters. It is studied for the motion of 3-D rigid planar surfaces. The dependence of the bound on several factors (such as the underlying motion, surface position, surface orientation, the field of view, and the density of available pixels) is derived in the form of closed-form expressions and can be computed by inverting a 5*5 positive-definite matrix.>

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.>

A kinematic model-based approach for the estimation of 3-D motion parameters from a sequence of noisy stereo images is discussed. The approach is based on representing the constant acceleration translational motion and constant precession rotational motion in the form of a bilinear state-space model using standard rectilinear states for translation and quaternions for rotation. Closed-form solutions of the state transition equations are obtained to propagate the quaternions. The measurements are noisy perturbations of 3-D feature points represented in an inertial coordinate system. It is assumed that the 3-D feature points are extracted from the stereo images and matched over the frames. Owing to the nonlinearity in the state model, nonlinear filters are designed for the estimation of motion parameters. Simulation results are included. The Cramer-Rao performance bounds for motion parameter estimates are computed. A constructive proof for the uniqueness of motion parameters is given. It is shown that with uniform sampling in time, three noncollinear feature points in five consecutive binocular image pairs contain all the spatial and temporal information. Both nondegenerate and degenerate motions are analyzed. A deterministic algorithm to recover motion parameters from a stereo image sequence is summarized from the constructive proof.>