Closed-Form Solutions to Minimal Absolute Pose Problems with Known Vertical Direction

In this paper we provide new simple closed-form solutions to two minimal absolute pose problems for the case of known vertical direction. In the first problem we estimate absolute pose of a calibrated camera from two 2D-3D correspondences and a given vertical direction. In the second problem we assume camera with unknown focal length and radial distortion and estimate its pose together with the focal length and the radial distortion from three 2D-3D correspondences and a given vertical direction. The vertical direction can be obtained either by direct physical measurement by, e.g., gyroscopes and inertial measurement units or from vanishing points constructed in images. Both our problems result in solving one polynomial equation of degree two in one variable and one, respectively two, systems of linear equations and can be efficiently solved in a closed-form. By evaluating our algorithms on synthetic and real data we demonstrate that both our solutions are fast, efficient and numerically stabled .