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

This paper deals with the problem of determining motion and structure for a planar surface and the error estimation. Since the motion of a planar patch is a degenerate case for linear algorithms (algorithms that consist of solving mainly linear equations and give closed-form solution) for general surfaces, the motion of such a planar surface should be considered separately. First, a new algorithm is presented 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 precence of noise, than the existing ones. There are generally two solutions given two image frames. From three image frames the solution is generally unique. An approach is proposed to test whether the points are coplanar. Based on this algorithm, the errors in the motion parameters and surface structure can be estimated for each pair of images. Specifically, the standard deviation of the errors is calculated in terms of the variance of the errors in the image coordinates. The presented approach to estimating errors is applicable to leastsquares, pseudo-inverse and eigenvalue eigenvector problems.