A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm

Area of Science:

• Mathematics
• Linear Algebra
• Computational Geometry

Background:

• The constrained orthogonal Procrustes problem seeks an optimal rotation matrix to align two matrices.
• The Kabsch-Umeyama algorithm, based on singular value decomposition, is the standard solution.
• Existing justifications rely on Lagrange multipliers, lacking a purely algebraic foundation.

Purpose of the Study:

• To provide a transparent, purely algebraic justification for the Kabsch-Umeyama algorithm.
• To demonstrate the algorithm's solution using only fundamental linear algebra principles.
• To elucidate the connection between rigid motion alignment and the Procrustes problem.

Main Methods:

• Utilized basic concepts from linear algebra.
• Avoided calculus-based methods like Lagrange multipliers.
• Presented a novel algebraic derivation of the Kabsch-Umeyama algorithm.

Main Results:

• A purely algebraic proof of the Kabsch-Umeyama algorithm is presented.
• The paper confirms that orientation-preserving rigid motion problems reduce to the constrained orthogonal Procrustes problem.
• The derivation offers a more transparent understanding of the matrix alignment process.

Conclusions:

• The study successfully provides a purely algebraic justification for a widely used algorithm.
• This work enhances the theoretical understanding of matrix alignment and rigid motion problems.
• The findings offer a simplified and more accessible approach to solving the constrained orthogonal Procrustes problem.