对偶四元数 SE(3) 同步及其恢复保证
Dual Quaternion SE(3) Synchronization with Recovery Guarantees
AI总结：
采用对偶四元数表示，通过谱初始化和对偶四元数广义幂法实现 SE(3) 同步，并给出误差界和线性收敛保证。
摘要：
特殊欧几里得群 SE(3) 上的同步旨在从含噪的成对相对变换中恢复绝对位姿，是机器人和 3D 视觉中的核心基本操作。标准方法通常需要多步启发式程序来恢复有效位姿，这些程序难以分析且通常缺乏理论保证。本文采用对偶四元数表示，并直接在对偶四元数单位上制定 SE(3) 同步。开发了一个两阶段算法：通过 Hermitian 对偶四元数测量矩阵上的幂法计算谱初始化，随后是对偶四元数广义幂法 (DQGPM)，通过每次迭代投影来强制执行可行性。建立了谱估计器的估计误差界，并证明 DQGPM 具有有限迭代误差界，并实现线性误差收缩直至显式的噪声相关阈值。在合成基准和真实多扫描点集配准上的实验表明，所提出的流程在准确性和效率上均优于代表性的基于矩阵的方法。
英文摘要：
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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