Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction

TL;DR

This paper introduces a novel quaternion manifold approach using the adjugate matrix to improve pose estimation techniques in computer vision and robotics, achieving exact solutions and better accuracy than existing methods.

Contribution

The paper presents a new quaternion manifold formulation via the adjugate matrix, enabling exact pose estimation solutions in orthographic and perspective scenarios.

Findings

Exact solution for 3D orthographic pose extraction.

Improved accuracy in perspective pose estimation.

Revisiting classic pose problems with the adjugate matrix approach.

Abstract

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.