Ambiguity in Structure from Motion: Sphere versus Plane

If 3D rigid motion can be correctly estimated from image sequences, the structure of the scene can be correctly derived using the equations for image formation. However, an error in the estimation of 3D motion will result in the computation of a distorted version of the scene structure. Of computational interest are these regions in space where the distortions are such that the depths become negative, because in order for the scene to be visible it has to lie in front of the image, and thus the corresponding depth estimates have to be positive. The stability analysis for the structure from motion problem presented in this paper investigates the optimal relationship between the errors in the estimated translational and rotational parameters of a rigid motion that results in the estimation of a minimum number of negative depth values. The input used is the value of the flow along some direction, which is more general than optic flow or correspondence. For a planar retina it is shown that the optimal configuration is achieved when the projections of the translational and rotational errors on the image plane are perpendicular. Furthermore, the projection of the actual and the estimated translation lie on a line through the center. For a spherical retina, given a rotational error, the optimal translation is the correct one; given a translational error, the optimal rotational negative deptherror depends both in direction and value on the actual and estimated translation as well as the scene in view. The proofs, besides illuminating the confounding of translation and rotation in structure from motion, have an important application to ecological optics. The same analysis provides a computational explanation of why it is easier to estimate self-motion in the case of a spherical retina and why shape can be estimated easily in the case of a planar retina, thus suggesting that nature's design of compound eyes (or panoramic vision) for flying systems and camera-type eyes for primates (and other systems that perform manipulation) is optimal.