Algebraically Rigorous Quaternion Framework for the Neural Network Pose Estimation Problem

The 3D pose estimation problem -- aligning pairs of noisy 3D point clouds -- is a problem with a wide variety of real-world applications. Here we focus on the use of quaternion-based neural network approaches to this problem and apparent anomalies that have arisen in previous efforts to resolve them. In addressing these anomalies, we draw heavily from the extensive literature on closed-form methods to solve this problem. We suggest that the major concerns that have been put forward could be resolved using a simple multi-valued training target derived from rigorous theoretical properties of the rotation-to-quaternion map of Bar-Itzhack. This multi-valued training target is then demonstrated to have good performance for both simulated and ModelNet targets. We provide a comprehensive theoretical context, using the quaternion adjugate, to confirm and establish the necessity of replacing single-valued quaternion functions by quaternions treated in the extended domain of multiple-charted manifolds.