Algorithm-Independent Stability Analysis of Structure from Motion.

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. No particular estimators are used and no specific assumptions about the scene are made. 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. For a spherical retina, given a rotational error, the optimal translation is the correct one, while given a translational error the optimal rotational error is normal to the translational one at an equal distance from the real and estimated translations. 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 much easier to estimate self-motion in the case of a spherical retina and why it is much easier to estimate shape 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.