用对偶四元数实现SE(3)同步，理论保证精度与效率。
Dual Quaternion SE(3) Synchronization with Recovery Guarantees
- 基于对偶四元数构建直接优化框架，避免传统多步启发式流程。
- 提出两阶段算法，可证明在噪声阈值内线性收敛，误差有界。
- 适用于机器人定位与三维重建，尤其适合高精度点云配准任务。
SE(3) 同步旨在从含噪的相对位姿测量中恢复绝对位姿，是机器人学与三维视觉的核心问题。传统方法常依赖多步启发式流程，难以分析且缺乏理论保障。本文采用对偶四元数表示，将 SE(3) 同步直接建模于单位对偶四元数空间。提出两阶段算法：第一阶段通过赫米特对偶四元数测量矩阵的幂法计算谱初始值；第二阶段采用对偶四元数广义幂法（DQGPM），每轮迭代进行可行性投影以保证解的有效性。建立了谱估计器的误差界，证明 DQGPM 具有有限次迭代误差界，并在噪声相关阈值内实现线性误差收缩。在合成基准与真实多扫描点云配准实验中，该方法相较代表性矩阵基方法显著提升精度与效率。
原文摘要 · Abstract (English)
Synchronization over the special Euclidean group 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 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 admit 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.
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