Synchronization of rotations via Riemannian trust-regions
We estimate unknown rotation matrices $R_i\in\textrm{SO}(n=2,3)$ from a set of measurements of relative rotations $R_i^{}R_j^T$. Each measurement is either slightly noisy, or an outlier bearing no information. We study the case where most measurements are outliers. In (A.~Singer, \emph{Angular Synchronization by Eigenvectors and Semidefinite Programming}, ACHA~30(1), pp.~20--36, 2011), an estimator is computed from a dominant subspace of a matrix. We observe this essentially results in least-squares estimation, and propose instead a Maximum Likelihood Estimator, explicitly acknowledging outliers. We compute the MLE via trust-region optimization on a matrix manifold. Comparison of our estimator with Riemannian Cram\'er-Rao bounds suggests efficiency.