Motion and structure from point correspondences: A robust algorithm for planar case with error estimation

The problem of determining motion and structure for a planar surface and the error estimation are discussed. Since the motion of a planar patch is a degenerate case for linear algorithms (algorithms that consist of solving mainly linear equations and give a closed-form solution) for general surfaces, the motion of such a planar surface is considered separately. An algorithm is introduced that gives a closed-form solution to motion parameters using monocular perspective images of the points on a planar surface. The algorithm is simpler and more reliable, in the presence of noise, than existing ones. There are generally two solutions for two image frames. For three image frames the solution is generally unique. An approach is proposed to test whether the points are coplanar. The errors in the motion parameters and surface structure can be estimated for each pair of images. Specificially, the standard deviation of the errors is calculated in terms of the variance of the errors in the image coordinates. This approach to estimating errors is applicable to least-squares, pseudo-inverse and eigenvalue--eigenvector problems.