Extensions and Exact Solutions to the Quaternion-Based RMSD Problem.

We examine the problem of transforming matching collections of data points into optimal correspondence. The classic (root-mean-square deviation) method calculates a 3D rotation that minimizes the 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 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 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 data set to a reference set. We observe that composites of the and QRMSD measures, combined with problem-dependent parameters including scaling factors to make their incommensurate dimensions compatible, could be suitable for certain matching tasks.