Dual Quaternion SE(3) Synchronization with Recovery Guarantees

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.

💡 Research Summary

This paper tackles the problem of SE(3) synchronization—recovering absolute poses from noisy pairwise relative transformations—by representing each pose as a unit dual quaternion (UDQ) rather than the conventional 4×4 homogeneous matrix. The authors argue that the matrix representation introduces a large “eigenspace‑sync gap” because SE(3) is non‑compact, causing the relaxed spectral solution to lie far from the feasible manifold and necessitating complex, multi‑step rounding heuristics.

The proposed framework consists of two stages. First, a spectral initializer is obtained by applying a power‑iteration method to a Hermitian dual‑quaternion measurement matrix C∈DH^{n×n}. The dominant eigenvector u₁ is computed, normalized to have ‖u₁‖₂²=n, and then projected element‑wise onto the UDQ set using a closed‑form normalization operator N(·). The authors prove a Lipschitz property of N(·) (Lemma 2.5) and derive non‑asymptotic error bounds for both the raw eigenvector (Proposition 2.4) and its projected version (Theorem 2.8): the distance to the ground‑truth scales as O(‖Δ‖_{op}/√n), where Δ denotes the Hermitian noise matrix.

Second, the Dual Quaternion Generalized Power Method (DQGPM) refines the initializer while maintaining feasibility at every iteration. Each iteration computes y_k = C x_{k−1} and then projects y_k onto UDQⁿ, yielding x_k = Π(y_k). Under a mild noise regime and an initializer within the basin of attraction, Theorem 3.2 shows linear convergence of the error d(x_k, x̂) to a noise‑dependent floor, i.e., d(x_k, x̂) ≤ (1−γ)^k d(x₀, x̂) + O(‖Δ‖_{op}/√n). This is the first finite‑iteration recovery guarantee for SE(3) synchronization in the dual‑quaternion setting.

The paper also addresses several algebraic challenges unique to dual quaternions: (1) the presence of zero divisors prevents unrestricted division, requiring the assumption λ₁,st≠0 for the dominant eigenvalue; (2) the dual‑number seminorm does not bound individual components, allowing translations to grow arbitrarily large even when the overall norm is bounded; (3) eigenvalue ordering must be defined lexicographically on the dual numbers, while error analysis is performed in Euclidean metrics. The authors resolve these issues through careful normalization, component‑wise analysis, and leveraging the Hermitian structure of C.

Experimental validation includes synthetic graphs with up to 10⁴ nodes and real multi‑scan point‑cloud datasets. Compared against spectral relaxation, semidefinite relaxation (SDR), and Riemannian optimization baselines, the proposed method achieves 12–30 % lower rotation and translation errors while reducing runtime by a factor of two or more, especially on large‑scale problems. The DQGPM’s per‑iteration feasibility allows early stopping with a valid pose estimate, a practical advantage over methods that only produce feasible solutions after a final rounding step.

In summary, the contributions are: (i) a unified dual‑quaternion formulation of SE(3) synchronization that eliminates the eigenspace‑sync gap; (ii) a provably accurate spectral initializer with explicit error bounds; (iii) a feasible, linearly convergent refinement algorithm with finite‑iteration recovery guarantees; and (iv) extensive empirical evidence of superior accuracy and efficiency. Limitations include the need for non‑zero standard parts of the dominant eigenvalue and the focus on average‑norm noise; future work may extend robustness to outliers and integrate the pipeline into real‑time SLAM systems.