2023/07/14 by Ido Hadi, Hadi, Ido, Tamir Bendory +3 · 1 citation
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical Analysis and Transform Methods #Matrix Theory and Algorithms #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2307.07640
openalex publication_date 2023/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 SE(3), the group of rigid motions of ℝ3. We based our method on embedding SE(3) into the algebra of dual quaternions, which has deep algebraic connections with the group SE(3). These connections suggest a natural rounding procedure considerably more straightforward than the current state-of-the-art for spectral SE(3) synchronization, which uses a matrix embedding of SE(3). We show by numerical experiments that our approach yields comparable results to the current state-of-the-art in SE(3) synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of SE(3), while yielding estimators of similar quality.