Unifying algebraic solvers for scaled euclidean registration from point, line and plane constraints

We investigate recent state-of-the-art algorithms for absolute pose problems (PnP and GPnP) and analyse their applicability to a more general type, namely the scaled Euclidean registration from point-to-point, point-to-line and point-to plane correspondences. Similar to previous formulations we first compress the original set of equations to a least squares error function that only depends on the non-linear rotation parameters 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 Gröbner basis method. In previous approaches the first compression step was usually tailored to a specific correspondence types and problem instances. Here, 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 solvers surprisingly fail.