Title: Extensions and Exact Solutions to the Quaternion-Based RMSD Problem,
Abstract: 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.