Abstract
The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses 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; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, 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. The article concludes 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. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.
Highlights
We explore the problem of finding global rotations that optimally align pairs of corresponding lists of spatial and/or orientation data
We introduce the measures that underlie the general spatial alignment problem, restrict our attention to the quaternion approach to the 3D problem, emphasizing a class of exact algebraic solutions that can be used as an alternative to the traditional numerical methods
We extend all of our 3D results to 4D space, exploiting quaternion pairs to formulate the 4D spatial-coordinate RMSD alignment and 4D orientation-based quaternion orientation-frame alignment problem (QFA) methods
Summary
Aligning matched sets of spatial point data is a universal problem that occurs in a wide variety of applications. General solutions may be found using singular-value-decomposition (SVD) methods, starting with the decomposition E 1⁄4 U Á S Á VT, where S is diagonal and U and V are orthogonal matrices, to give the result Ropt 1⁄4 V Á D Á UT, where D is the identity matrix up to a possible sign in one element [see, e.g., Kabsch (1976, 1978), Golub & van Loan (1983) and Markley (1988)] In addition to these general methods based on traditional linear algebra approaches, a significant literature exists for three dimensions that exploits the relationship between 3D rotation matrices and quaternions, and rephrases the task of finding Ropt as a quaternion eigensystem problem.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
