Characterization and Computation of Matrices of Maximal Trace over Rotations

TL;DR

This paper characterizes matrices of maximal trace over rotations in terms of eigenvalues and provides methods for identifying such matrices, offering alternatives to SVD for solving related alignment problems in 2D and 3D.

Contribution

It introduces a new eigenvalue-based characterization of matrices of maximal trace over rotations and proposes alternative solution methods for 2D and 3D cases.

Findings

Eigenvalue characterization of maximal trace matrices

Criteria for identifying maximal trace matrices in 2D and 3D

Alternative methods to SVD for solving the problem

Abstract

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space. This problem generalizes to the so-called Wahba's problem which is the same problem with nonnegative weights. Given a dxd matrix M, solutions to these problems are intimately related to the problem of finding a dxd rotation matrix U that maximizes the trace of UM, i.e., that makes UM a matrix of maximal trace over rotations, and it is well known this can be achieved with a method based on the computation of the singular value decomposition (SVD) of M. As the main goal of this paper, we characterize dxd matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for d = 2, 3, we show how this characterization can be used to determine whether a matrix is of maximal trace over rotation matrices. Finally, although depending only slightly on the characterization, as a secondary goal of the paper, for d = 2, 3, we identify alternative ways, other than the SVD, of obtaining solutions to the aforementioned problems.