A Geometric Algebra Solution to the Absolute Orientation Problem

We show here an alternative solution to the fundamental photogrammetric problem of determining Absolute Orientation by reformulating it in Geometric Algebra. Our work is centered on expressing rotors using Characteristic Multivectors. We compare our algorithm under different point cloud geometries and Gaussian Noise levels with the standard least-squares, Singular Value Decomposition solution for fitting 3D point clouds that are related via 1–1 correspondence developed by Arun in 1987. As with Arun’s formulation, the proposed solution is based solely on the point cloud geometries and does not use any extra information that the data may have or any noise filtering mechanisms. The new algorithm effectively follows that of Arun but replaces the rotation matrix calculation step with a rotor-calculation step. The rotation matrix can easily be recovered from the rotor. We show that our algorithm has very similar performance to the SVD method, indicating that it does not suffer from adverse numerical effects. We also indicate how it might be used in the future, where having this closed-form, non-iterative solution might prove beneficial.