We examine the problem of transforming matching collections of data points into optimal correspondence. The classic RMSD (root-mean-square deviation) method calculates a 3D rotation that minimizes the RMSD of a set of test data points relative to a reference set of corresponding points. Similar literature in aeronautics, photogrammetry, and proteomics employs numerical methods to find the maximal eigenvalue of a particular 4\!\times\! 4 quaternion-based matrix, thus specifying the quaternion eigenvector corresponding to the optimal 3D rotation. Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods. We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases. In addition, we consider data that are themselves quaternions isomorphic to orthonormal triads describing 3 coordinate frames (amino acids in proteins possess such frames). Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from a quaternion data set to a reference set. We observe that composites of the RMSD and QRMSD measures, combined with problem-dependent parameters including scaling factors to make their incommensurate dimensions compatible, could be suitable for certain matching tasks .
We review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For 3D point data, this " orthogonal Procrustes problem " is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional 3D spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices, and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. Supplementary Material covers extensions of quaternion methods to the 4D problem .
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examine the problem of transforming matching collections of data points into optimal correspondence. The classic RMSD (root-mean-square deviation) method calculates a 3D rotation that minimizes the RMSD of a set of test data points relative to a reference set of corresponding points. Similar literature in aeronautics, photogrammetry, and proteomics employs numerical methods to find the maximal eigenvalue of a particular 4\!\times\! 4 quaternion-based matrix, thus specifying the quaternion eigenvector corresponding to the optimal
review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For
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rotation. Here we generalize this basic problem , sometimes referred to as the
point data, this
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Procrustes Problem,
orthogonal Procrustes problem
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and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods. We begin with the 4D data
is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this
R
, a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the
for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional
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and 2D data problems as special cases. In addition, we consider data that are themselves quaternions isomorphic to orthonormal triads describing 3 coordinate frames (amino acids in proteins possess such frames). Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from
spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to
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quaternion data set to a reference set. We observe that composites of the RMSD and QRMSD measures, combined with problem-dependent parameters including scaling factors to make their incommensurate dimensions compatible, could be suitable for certain matching tasks
tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices, and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. Supplementary Material covers extensions of quaternion methods to the 4D problem