Relative Pose Estimation of Calibrated Cameras with Known $$\mathrm {SE}(3)$$ Invariants

The \(\mathrm {SE}(3)\) invariants of a pose include its rotation angle and screw translation. In this paper, we present a complete comprehensive study of the relative pose estimation problem for a calibrated camera constrained by known \(\mathrm {SE}(3)\) invariant, which involves 5 minimal problems in total. These problems reduces the minimal number of point pairs for relative pose estimation and improves the estimation efficiency and robustness. The \(\mathrm {SE}(3)\) invariant constraints can come from extra sensor measurements or motion assumption. Unlike conventional relative pose estimation with extra constraints, no extrinsic calibration is required to transform the constraints to the camera frame. This advantage comes from the invariance of \(\mathrm {SE}(3)\) invariants cross different coordinate systems on a rigid body and makes the solvers more convenient and flexible in practical applications. In addition to the concept of relative pose estimation constrained by \(\mathrm {SE}(3)\) invariants, we also present a comprehensive study of existing polynomial formulations for relative pose estimation and discover their relationship. Different formulations are carefully chosen for each proposed problems to achieve best efficiency. Experiments on synthetic and real data shows performance improvement compared to conventional relative pose estimation methods. Our source code is available at: http://github.com/prclibo/relative_pose.