$ \mathrm{SE} (3) $ Synchronization by Eigenvectors of Dual Quaternion Matrices

In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or "rounded", onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $\mathrm{SE}(3)$, the group of rigid motions of $\mathbb{R}^3$. We based our method on embedding $\mathrm{SE}(3)$ into the algebra of dual quaternions, which has deep algebraic connections with the group $\mathrm{SE}(3)$. These connections suggest a natural rounding procedure considerably more straightforward than the current state-of-the-art for spectral $\mathrm{SE}(3)$ synchronization, which uses a matrix embedding of $\mathrm{SE}(3)$. We show by numerical experiments that our approach yields comparable results to the current state-of-the-art in $\mathrm{SE}(3)$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $\mathrm{SE}(3)$, while yielding estimators of similar quality.