The Multiplicity of Solutions for Camera Pose Given Two Image Points and Gravity

A new mathematical analysis is given for the problem of recovering the position and orientation of a camera equipped with an accelerometer from sensor images of two labeled landmarks whose positions in a coordinate system aligned in a known way with gravity are known (P2PA). This is a variant on the much studied P n P problem of recovering camera position and orientation from n points without any gravitational data. It is proved that in three types of degenerate cases there are infinitely many solutions, in another type of case there is one, and in a final type of case there are two. A precise geometric characterization of each type of case is given. This completes the incomplete and at times incorrect characterizations in previous research. In particular, there is always a unique solution in the practically interesting case where the two landmarks are at the same altitude and the camera is at a different altitude. It is also proved that if the two landmarks are unlabeled, then apart from the same degenerate cases, there are still always one or two solutions.