On closed-form solutions to the 4D nearest rotation matrix problem

In this paper, we address the problem of restoring the orthogonality of a numerically noisy 4D rotation matrix by finding its nearest (in Frobenius norm) correct rotation matrix. This problem can be straightforwardly solved using the Singular Value Decomposition (SVD). Nevertheless, to avoid numerical methods, we present two new closed-form methods. One relies on the direct minimization of the mentioned Frobenius norm, and the other on the passage to double quaternion representation. A comparison of these two methods with respect to the SVD reveals that the method based on a double quaternion representation is superior in all aspects.