Abstract

We analyze the least–squares error for structure from mo- tion (SFM) with a single infinitesimal motion (“structure from opti- cal flow”). We present approximations to the noiseless error over two, complementary regions of motion estimates: roughly forward and non– forward translations. Experiments show that these capture the error’s detailed behavior over the entire motion range. They can be used to derive new error properties, including generalizations of the bas–relief ambiguity. As examples, we explain the error’s complexity for epipoles near the field of view; for planar scenes, we derive a new, double bas– relief ambiguity and prove the absence of local minima. For nonplanar scenes, our approximations simplify under reasonable assumptions. We show that our analysis applies even for large noise, and that the pro jec- tive error has less information for estimating motion than the calibrated error. Our results make possible a comprehensive error analysis of SFM.