The quaternion-based spatial-coordinate and orientation-frame alignment problems

Area of Science:

• Mathematics
• Computer Science
• Robotics
• Geophysics

Background:

• The orthogonal Procrustes problem seeks optimal global rotation for aligning spatial coordinate sets.
• Minimizing root-mean-square deviation (RMSD) is a common approach for 3D point data alignment.
• Quaternion eigensystem methods have been independently discovered and applied for decades.

Purpose of the Study:

• To review and consolidate quaternion eigensystem methods for spatial and orientation data alignment.
• To explore exact algebraic solutions for the 3D orthogonal Procrustes problem.
• To extend quaternion methods to 3D orientation-frame alignment and rotation averaging.

Main Methods:

• Focus on quaternion eigensystem methods for solving the orthogonal Procrustes problem.
• Utilize exact algebraic solutions derived from Cardano's quartic equation solution.
• Investigate extensions of quaternion methods for 3D quaternion orientation-frame alignment (QFA).

Main Results:

• Quaternion eigensystem methods provide exact algebraic solutions for 3D spatial alignment.
• The 3D QFA problem is equivalent to the rotation-averaging problem.
• Novel extensions of quaternion methods to 4D alignment problems are presented.

Conclusions:

• Quaternion methods offer a unified approach to 3D spatial and orientation data alignment.
• Exact algebraic solutions reveal the underlying structure of the eigensystem.
• The study provides a comprehensive review and extensions for quaternion-based alignment problems.