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Dual Quaternion SE(3) Synchronization with Recovery Guarantees

Jianing Zhao, Linglingzhi Zhu, Anthony Man-Cho So

TL;DR

This work introduces a unit dual quaternion (UDQ) formulation for SE(3) synchronization, addressing fundamental representational gaps in matrix embeddings by operating directly on unit dual quaternions. It develops a two-stage pipeline: a spectral initializer obtained from the dominant eigenpair of a Hermitian dual-quaternion measurement matrix $\mathbf{C}$ via a power method, followed by a dual-quaternion generalized power method (DQGPM) that enforces feasibility at every iteration through per-entry projection to $\mathrm{UDQ}^n$. The authors establish nonasymptotic error bounds for the spectral estimator and a finite-iteration recovery guarantee for DQGPM, with linear contraction up to a noise-dependent floor, and demonstrate through synthetic and real-data experiments that the proposed method outperforms matrix-based approaches in both accuracy and efficiency. The results offer rigorous recovery guarantees for SE(3) synchronization in the dual-quaternion setting and show practical benefits for multi-scan point-set registration tasks. Overall, the approach delivers principled initialization, provable convergence, and computational advantages critical to robust 3D spatial perception pipelines.

Abstract

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.

Dual Quaternion SE(3) Synchronization with Recovery Guarantees

TL;DR

This work introduces a unit dual quaternion (UDQ) formulation for SE(3) synchronization, addressing fundamental representational gaps in matrix embeddings by operating directly on unit dual quaternions. It develops a two-stage pipeline: a spectral initializer obtained from the dominant eigenpair of a Hermitian dual-quaternion measurement matrix via a power method, followed by a dual-quaternion generalized power method (DQGPM) that enforces feasibility at every iteration through per-entry projection to . The authors establish nonasymptotic error bounds for the spectral estimator and a finite-iteration recovery guarantee for DQGPM, with linear contraction up to a noise-dependent floor, and demonstrate through synthetic and real-data experiments that the proposed method outperforms matrix-based approaches in both accuracy and efficiency. The results offer rigorous recovery guarantees for SE(3) synchronization in the dual-quaternion setting and show practical benefits for multi-scan point-set registration tasks. Overall, the approach delivers principled initialization, provable convergence, and computational advantages critical to robust 3D spatial perception pipelines.

Abstract

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.
Paper Structure (27 sections, 13 theorems, 149 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 27 sections, 13 theorems, 149 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Spectral Estimator
  4. Iterative Estimation Performance
  5. Numerical Experiments
  6. Data Generation and Evaluation Metrics
  7. Data Generation
  8. Evaluation Metrics
  9. Recovery Performance and Computational Time
  10. Efficiency and Accuracy on Real Data
  11. Algebras
  12. Dual Numbers
  13. Quaternions
  14. Dual Quaternion
  15. Representation of SO(3) and SE(3)
  16. ...and 12 more sections

Key Result

Proposition 2.1

For a Hermitian dual quaternion matrix $\bm{C}\in\mathbb{DH}^{n\times n}$, Problem (opt: LS2) is equivalent to and their optimal objective values differ only by a multiplicative factor of $2$.

Figures (3)

  • Figure 1: Error decay over iterations: (a) rotation error, (b) translation error, with $(\sigma_t,\sigma_r) = (0.1,10^{\circ})$ and $p=0.3$.
  • Figure 2: 3D reconstructions obtained by DQGPM on four real datasets. Colors indicate individual point clouds.
  • Figure 3: Point-cloud alignment over DQGPM iterations.

Theorems & Definitions (18)

  • Proposition 2.1: Equivalent QPQC Formulation
  • Remark 2.3: Well-definedness of the Inverse
  • Proposition 2.4: Eigenvector Estimation Error
  • Lemma 2.5: Lipschitz Property of $\mathcal{N}$
  • Remark 2.6
  • Proposition 2.7: Projection onto $\mathrm{UDQ}^n$
  • Theorem 2.8: Rounded Spectral Estimation Error
  • Remark 3.1: Algebraic Order vs. Estimation Metric
  • Theorem 3.2: Estimation Error of DQGPM
  • Remark 3.3
  • ...and 8 more