Exact two-image structure from motion

For two-image structure from motion, we present a simple, exact expression for a least-squares image-reprojection error for finite motion that depends only on the motion. Optimal estimates of the structure and motion can be computed by minimizing this expression just over the motion parameters. Also, we present a solution to the triangulation problem: an exact, explicit expression for the optimal structure estimate given the motion. We identify a new ambiguity in recovering the structure for known motion. We study the exact error's properties experimentally and demonstrate that it often has several local minima for forward or backward motion estimates. Our experiments also show that the "reflected" local minimum of Oliensis (2001) and Soatto et al. (1998) occurs for large translational motions. Our exact results assume that the camera is calibrated and use a least-squares image-reprojection error that applies most naturally to a spherical imaging surface. We approximately extend our approach to the standard least-squares error in the image plane and uncalibrated cameras. We present an improved version of the Sampson error which gives better results experimentally.