Achieving globally optimal point cloud registration under partial overlaps and large misalignments remains a fundamental challenge. While simultaneous transformation ($\boldsymbolθ$) and correspondence ($\mathbf{P}$) estimation has the advantage of being robust to nonrigid deformation, its non-convex coupled objective often leads to local minima for heuristic methods and prohibitive convergence times for existing global solvers due to loose lower bounds. To address this, we propose DC-Reg, a robust globally optimal framework that significantly tightens the Branch-and-Bound (BnB) search. Our core innovation is the derivation of a holistic concave underestimator for the coupled transformation-assignment objective, grounded in the Difference of Convex (DC) programming paradigm. Unlike prior works that rely on term-wise relaxations (e.g., McCormick envelopes) which neglect variable interplay, our holistic DC decomposition captures the joint structural interaction between $\boldsymbolθ$ and $\mathbf{P}$. This formulation enables the computation of remarkably tight lower bounds via efficient Linear Assignment Problems (LAP) evaluated at the vertices of the search boxes. We validate our framework on 2D similarity and 3D rigid registration, utilizing rotation-invariant features for the latter to achieve high efficiency without sacrificing optimality. Experimental results on synthetic data and the 3DMatch benchmark demonstrate that DC-Reg achieves significantly faster convergence and superior robustness to extreme noise and outliers compared to state-of-the-art global techniques.

翻译：在部分重叠与大尺度错位条件下实现全局最优的点云配准仍是一项基础性挑战。尽管同步变换（$\boldsymbolθ$）与对应关系（$\mathbf{P}$）估计具备对非刚性形变的鲁棒性优势，但其非凸耦合目标函数常导致启发式方法陷入局部极小值，且现有全局求解器因下界松弛度过高而导致收敛时间无法接受。针对该问题，我们提出DC-Reg——一种通过显著紧缩分支定界（BnB）搜索实现鲁棒全局最优的框架。核心创新在于：基于凸差（DC）规划范式，为耦合的变换-赋值目标函数推导出整体凹下界估计器。与依赖逐项松弛（如McCormick包络）而忽略变量交互的既往工作不同，本研究的整体凸差分解捕捉了$\boldsymbolθ$与$\mathbf{P}$的联合结构相互作用。该公式可在搜索盒顶点处通过高效线性赋值问题（LAP）计算极为紧致的下界。我们分别对二维相似变换与三维刚体配准验证了该框架——后者利用旋转不变特征在保证最优性的同时实现高效率。在合成数据与3DMatch基准上的实验结果表明，DC-Reg相较于最先进全局技术，实现了显著更快的收敛速度及对极端噪声与离群点的更强鲁棒性。