Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction
This work addresses pose estimation challenges in computer graphics, machine vision, and robotics, but it is incremental as it builds on existing methods with a new algebraic approach.
The paper tackles the problem of extracting quaternion-based poses from rotation matrices by using the adjugate matrix to model quaternions as a manifold, leading to an exact solution for 3D orthographic least squares pose extraction and improved results for perspective pose extraction.
Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the adjugate matrix of the characteristic equation of a related eigenvalue problem to obtain the manifold of the space of a quaternion eigenvector. We argue that quaternions parameterized by their corresponding rotation matrices cannot be expressed, for example, in machine learning tasks, as single-valued functions: the quaternion solution must instead be treated as a manifold, with different algebraic solutions for each of several single-valued sectors represented by the adjugate matrix. We conclude with novel constructions exploiting the quaternion adjugate variables to revisit several classic pose estimation applications: 2D point-cloud matching, 2D point-cloud-to-projection matching, 3D point-cloud matching, 3D orthographic point-cloud-to-projection matching, and 3D perspective point-cloud-to-projection matching. We find an exact solution to the 3D orthographic least squares pose extraction problem, and apply it successfully also to the perspective pose extraction problem with results that improve on existing methods.