A universal, closed-form approach for absolute pose problems

Abstract We propose a general approach for absolute pose problems including the well known perspective-n-point (PnP) problem, its generalized variant (GPnP) with and without scale, and the pose from 2D line correspondences (PnL). These have received a tremendous attention in the computer vision community during the last decades. However, it was only recently that efficient, globally optimal, closed-form solutions have been proposed, which can handle arbitrary numbers of correspondences including minimal configurations as well as over-constrained cases with linear complexity. We follow the general scheme by eliminating the linear parameters first, which results in a least squares error function that only depends on the non-linear rotation and a small symmetric coefficient matrix of fixed size. Then, in a second step the rotation is solved with algorithms which are derived using methods from algebraic geometry such as the Grobner basis method. We propose a unified formulation based on a representation with orthogonal complements which allows to combine different types of constraints elegantly in one single framework. We show that with our unified formulation existing polynomial solvers can be interchangeably applied to problem instances other than those they were originally proposed for. It becomes possible to compare them on various registrations problems with respect to accuracy, numerical stability, and computational speed. Our compression procedure not only preserves linear complexity, it is even faster than previous formulations. For the second step we also derive an own algebraic equation solver, which can additionally handle the registration from 3D point-to-point correspondences, where other rotation solvers fail. Finally, we also present a marker-based SLAM approach with automatic registration to a target coordinate system based on partial and distributed reference information. It represents an application example that goes beyond classical camera pose estimation from image measurements and also serves for evaluation on real data.