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A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm.

Jim Lawrence1,2, Javier Bernal1, Christoph Witzgall1

  • 1National Institute of Standards and Technology, Gaithersburg, MD 20899.

Journal of Research of the National Institute of Standards and Technology
|December 8, 2021
PubMed
Summary
This summary is machine-generated.

This study offers a purely algebraic justification for the Kabsch-Umeyama algorithm, a key method for solving the constrained orthogonal Procrustes problem. It simplifies alignment of matrices using basic linear algebra concepts.

Keywords:
Procrustesconstrainedleast-squaresorthogonalrigid motionrotationsingular value decompositiontrace

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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.