Motion and Structure From Point Correspondences with Error Estimation: Planar Surfaces

We investigate in this paper the determination of motion and structure of a planar scene from monocular image sequences. Since algorithms for curved surfaces are not applicable to planar scenes, algorithms for planar surfaces need to be developed. In this paper, we present a new and simpler linear algorithm that gives closed-form solutions for motion and structure parameters using point correspondences between two images, assuming that the coplanar points undergo a rigid motion in 3-D space. A series of new analytical results is established. It is proved in this paper that any 3x3 matrix with a rank of no less than two is decomposable in the sense defined in this paper. This implies that our linear algorithm does not neglect any necessary constraint in solving the linear equations, and it optimizes a direct nonlinear, nonquadratic objective function. From two images, the two candidate solutions of the existing algorithms are both valid interpretations, although one of them does not agree with what actually happened (illusive solution). The existence of four object points, among which no three are collinear, is not only a sufficient condition for our linear algorithm to give at most two solutions, it is in fact a necessary condition for any algorithm. In other words, if this condition is not satisfied, no algorithm can reach at most two solutions from two views. There exists a class of so-called plane-perceivable surfaces, which includes planes as a spatial case. If the points lie in a noncoplanar surface in this class, one can still interpret the two images as the projections of coplanar points undergoing a rigid motion. If three images are available, the solution is generally unique, but not always. An approach to assessing the accuracy of the solutions is applied to this problem. The estimated errors give quantitative assessment of the reliability of the solutions and indicate any degenerate or nearly degenerate point configurations that cause failure of the motion analysis algorithm. This method of error estimation is applicable to other general least squares or minimum-norm least squares problems. Experimental results for real images are presented with automatically computed point correspondences.