Dual Quaternion SE(3) Synchronization with Recovery Guarantees

Synchronization over the special Euclidean group $\mathrm{SE}(3)$ aims to recover absolute poses from noisy pairwise relative transformations and is a core primitive in robotics and 3D vision. Standard approaches often require multi-step heuristic procedures to recover valid poses, which are difficult to analyze and typically lack theoretical guarantees. This paper adopts a dual quaternion representation and formulates $\mathrm{SE}(3)$ synchronization directly over the unit dual quaternion. A two-stage algorithm is developed: A spectral initializer computed via the power method on a Hermitian dual quaternion measurement matrix, followed by a dual quaternion generalized power method (DQGPM) that enforces feasibility through per-iteration projection. The estimation error bounds are established for spectral estimators, and DQGPM is shown to admits a finite-iteration error bound and achieves linear error contraction up to an explicit noise-dependent threshold. Experiments on synthetic benchmarks and real-world multi-scan point-set registration demonstrate that the proposed pipeline improves both accuracy and efficiency over representative matrix-based methods.

Robots, self-driving systems, and 3D reconstruction tools often need to combine many local camera or scanner views into one consistent 3D map. This is difficult because each pair of views only gives noisy information about how two local coordinate systems are related, and small errors can accumulate across the whole network. This paper studies a mathematical way to recover all poses—both rotations and translations—at once. We represent each 3D motion using a dual quaternion, which is a compact object that describes rotation and translation together. Based on this representation, we design a two-stage algorithm: first it computes a spectral initial estimate, and then it refines the estimate while ensuring that every intermediate output remains a valid 3D motion. The method comes with theoretical error guarantees and performs well on synthetic and real 3D registration tasks. This can make multi-view 3D alignment more reliable and efficient for robotics, mapping, and 3D vision applications.