A Minimal Solution to the Generalized Pose-and-Scale Problem

We propose a novel solution to the generalized camera pose problem which includes the internal scale of the gener-alized camera as an unknown parameter. This further gen-eralization of the well-known absolute camera pose prob-lem has applications in multi-frame loop closure. While a well-calibrated camera rig has a fixed and known scale, camera trajectories produced by monocular motion estima-tion necessarily lack a scale estimate. Thus, when perform-ing loop closure in monocular visual odometry, or regis-tering separate structure-from-motion reconstructions, we must estimate a seven degree-of-freedom similarity trans-form from corresponding observations. Existing approaches solve this problem, in specialized configurations, by aligning 3D triangulated points or indi-vidual camera pose estimates. Our approach handles gen-eral configurations of rays and points and directly estimates the full similarity transformation from the 2D-3D corre-spondences. Four correspondences are needed in the mini-mal case, which has eight possible solutions. The minimal solver can be used in a hypothesize-and-test architecture for robust transformation estimation. Our solver also produces a least-squares estimate in the overdetermined case. The approach is evaluated experimentally on synthetic and real datasets, and is shown to produce higher accu-racy solutions to multi-frame loop closure than existing ap-proaches. 1