Point cloud registration based on correspondences computes the rigid transformation that maximizes the number of inliers constrained within the noise threshold. Current state-of-the-art (SOTA) methods employing spatial compatibility graphs or branch-and-bound (BnB) search mainly focus on registration under high outlier ratios. However, graph-based methods require at least quadratic space and time complexity for graph construction, while multi-stage BnB search methods often suffer from inaccuracy due to local optima between decomposed stages. This paper proposes a geometric maximum overlapping registration framework via rotation-only BnB search. The rigid transformation is decomposed using Chasles' theorem into a translation along rotation axis and a 2D rigid transformation. The optimal rotation axis and angle are searched via BnB, with residual parameters formulated as range maximum query (RMQ) problems. Firstly, the top-k candidate rotation axes are searched within a hemisphere parameterized by cube mapping, and the translation along each axis is estimated through interval stabbing of the correspondences projected onto that axis. Secondly, the 2D registration is relaxed to 1D rotation angle search with 2D RMQ of geometric overlapping for axis-aligned rectangles, which is solved deterministically in polynomial time using sweep line algorithm with segment tree. Experimental results on indoor 3DMatch/3DLoMatch scanning and outdoor KITTI LiDAR datasets demonstrate superior accuracy and efficiency over SOTA methods, while the time complexity is polynomial and the space complexity increases linearly with the number of points, even in the worst case.

翻译：基于对应关系的点云配准通过计算在噪声阈值约束下最大化内点数量的刚体变换来实现。当前最先进的（SOTA）方法采用空间兼容性图或分支定界（BnB）搜索，主要关注高外点率下的配准。然而，基于图的方法在图构建过程中至少需要二次空间和时间复杂度，而多阶段BnB搜索方法则常因分解阶段间的局部最优解导致精度不足。本文提出一种通过纯旋转BnB搜索实现的几何最大重叠配准框架。利用沙勒定理将刚体变换分解为沿旋转轴的平移和一个二维刚体变换。最优旋转轴和角度通过BnB搜索得到，剩余参数则被建模为区间最大值查询（RMQ）问题。首先，在通过立方体贴图参数化的半球内搜索前k个候选旋转轴，并通过投影到该轴上的对应关系进行区间穿刺来估计沿轴的平移。其次，将二维配准松弛为一维旋转角搜索，并转化为轴对齐矩形的几何重叠二维RMQ问题，该问题通过结合线段树的扫描线算法在多项式时间内确定性求解。在室内3DMatch/3DLoMatch扫描和室外KITTI LiDAR数据集上的实验结果表明，该方法在精度和效率上均优于SOTA方法，同时其时间复杂度为多项式级，空间复杂度随点数线性增长，即使在最坏情况下也是如此。