A Minimal Solution to the Generalized 3-Point Pose Problem

Abstract

It is a well known classical result that given the image projections of three known world points it is possible to solve for the pose of a calibrated perspective camera to up to four pairs of solutions. We solve the generalised problem where the camera is allowed to sample rays in some arbitrary but known fashion and is not assumed to perform a central perspective projection. That is, given three back-projected rays that emanate from a camera or multi-camera rig in an arbitrary but known fashion, we seek the possible poses of the camera such that the three rays meet three known world points. We show that the generalised problem has up to eight solutions that can be found as the intersections between a circle and a ruled quartic surface. A minimal and efficient constructive numerical algorithm is given to find the solutions. The algorithm derives an octic polynomial whose roots correspond to the solutions. In the classical case, when the three rays are concurrent, the ruled quartic surface and the circle possess a reflection symmetry such that their intersections come in symmetric pairs. This manifests itself in that the odd order terms of the octic polynomial vanish. As a result, up to four pairs of solutions can be found in closed form. The proposed algorithm can be used to solve for the pose of any type of calibrated camera or camera rig. The intended use for the algorithm is in a hypothesise-and-test architecture.