Solutions to Minimal Generalized Relative Pose Problems

We present a method to obtain the solutions to the generalized 6-point relative pose problem. The problem is to find the relative positions of two generalized cameras so that six corresponding image rays meet in space. Here, a generalized camera is a camera that captures some arbitrary set of rays and does not adhere to the central perspective projection model. The cameras are assumed to be calibrated, which means that we know the image rays in Euclidean camera coordinate systems. Mathematically, the problem is therefore, given two Euclidean configurations consisting of six lines each, to find a rigid transformation of the first six lines so that each transformed line intersects its corresponding line from the second set. We show that the problem has 64 solutions in general and solve it by computing a matrix on closed form and then extracting its eigen-vectors. Hence we present a solver that corresponds to the intrinsic degree of difficulty of this minimal problem. Our numerical experiments show that the presented solver can be used in a RANSAC-implementation.