Homography-Based Minimal-Case Relative Pose Estimation With Known Gravity Direction

In this paper, we propose a novel approach to two-view minimal-case relative pose problems based on homography with known gravity direction. This case is relevant to smart phones, tablets, and other camera-IMU (Inertial measurement unit) systems which have accelerometers to measure the gravity vector. We explore the rank-1 constraint on the difference between the euclidean homography matrix and the corresponding rotation, and propose an efficient two-step solution for solving both the calibrated and semi-calibrated (unknown focal length) problems. Based on the hidden variable technique, we convert the problems to the polynomial eigenvalue problems, and derive new 3.5-point, 3.5-point, 4-point solvers for two cameras such that the two focal lengths are unknown but equal, one of them is unknown, and both are unknown and possibly different, respectively. We present detailed analyses and comparisons with the existing 6- and 7-point solvers, including results with smart phone images.