Solving for Relative Pose with a Partially Known Rotation is a Quadratic Eigenvalue Problem

We propose a novel formulation of minimal case solutions for determining the relative pose of perspective and generalized cameras given a partially known rotation, namely, a known axis of rotation. An axis of rotation may be easily obtained by detecting vertical vanishing points with computer vision techniques, or with the aid of sensor measurements from a smart phone. Given a known axis of rotation, our algorithms solve for the angle of rotation around the known axis along with the unknown translation. We formulate these relative pose problems as Quadratic Eigen value Problems which are very simple to construct. We run several experiments on synthetic and real data to compare our methods to the current state-of-the-art algorithms. Our methods provide several advantages over alternatives methods, including efficiency and accuracy, particularly in the presence of image and sensor noise as is often the case for mobile devices.