GMOR: A Lightweight Robust Point Cloud Registration Framework via Geometric Maximum Overlapping

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.

💡 Research Summary

The paper addresses the fundamental problem of 6‑DoF rigid registration of point clouds given a set of noisy correspondences, where the goal is to find the transformation that maximizes the number (or weighted overlap) of inliers within a prescribed noise threshold. Existing global registration approaches fall into two main families: spatial‑compatibility graph methods and deterministic branch‑and‑bound (BnB) searches. Graph‑based techniques achieve strong outlier rejection but require at least quadratic time and memory for graph construction, limiting scalability. Multi‑stage BnB methods reduce dimensionality by decomposing the problem, yet they suffer from local‑optimum traps between stages and often treat all correspondences equally, ignoring confidence weights.

GMOR (Geometric Maximum Overlapping Registration) proposes a novel two‑stage, rotation‑only BnB framework that leverages Chasles’ theorem to decompose any rigid motion into (i) a translation along a rotation axis and (ii) a 2‑D rigid transformation in the plane orthogonal to that axis. This decomposition enables a unified reduction of the original 6‑DoF search to a sequence of low‑dimensional sub‑problems while preserving global optimality.

Stage 1 – Rotation‑axis search.
The unit sphere is restricted to a hemisphere and parametrized by cube‑mapping, yielding a uniform grid of candidate axis cells. A BnB algorithm iteratively tightens upper and lower bounds for each cell. For a given axis candidate, all correspondences are projected onto the axis, producing 1‑D intervals. Instead of counting uniform inliers, GMOR formulates a weighted overlap maximization as a 1‑D range‑maximum‑query (RMQ) problem, solved by an interval‑stabbing algorithm in O(N) time. The optimal translation along the axis is the point of maximal weighted overlap. Only the top‑k axis candidates (by bound value) are forwarded to the next stage, mitigating the risk of early local‑optimum selection.

Stage 2 – Residual rotation‑angle search.
With a fixed axis, the remaining unknown is the rotation angle about that axis. GMOR reduces this to a 1‑D search over the angle domain. For each angle, the induced 2‑D rotation center is approximated by an axis‑aligned bounding rectangle that encloses the transformed correspondences. The problem of finding the angle that yields the maximal number of overlapping rectangles is equivalent to Klee’s rectangle stabbing problem, a classic 2‑D RMQ. The authors solve it deterministically using a sweep‑line algorithm augmented with a segment tree, achieving O(N log N) time per angle evaluation. The angle delivering the highest overlap provides the optimal 2‑D rigid component, which together with the translation from Stage 1 reconstructs the full 6‑DoF transformation via the inverse of Chasles’ decomposition.

Complexity and resources.
Stage 1’s BnB over the hemispherical grid incurs polynomial time proportional to the number of cells; each cell’s interval‑stabbing is linear in the number of correspondences. Stage 2’s sweep‑line RMQ is O(N log N). Overall memory consumption is linear, O(N), because only the correspondence list and a few auxiliary data structures (segment tree, event list) are stored. This contrasts sharply with graph‑based methods that need O(N²) adjacency structures.

Experimental validation.
The authors evaluate GMOR on indoor datasets (3DMatch, 3DLoMatch) and an outdoor LiDAR dataset (KITTI). They vary outlier ratios up to 95 % and add Gaussian noise to simulate realistic sensor errors. Compared against state‑of‑the‑art methods—TEASER++, GO‑ICP, TR‑DE, GORE, and several graph‑clique approaches—GMOR consistently achieves lower rotation and translation errors while requiring less runtime. Notably, on large point clouds where graph‑based methods exceed memory limits, GMOR remains functional. The method also benefits from weighted correspondences, improving discrimination when match confidence is heterogeneous.

Contributions and impact.

1. A unified rotation‑only BnB framework that directly searches the SO(3) space via axis parametrization, avoiding nested translation searches.
2. An adaptive interval‑stabbing RMQ that maximizes weighted overlap rather than raw inlier count, providing robustness to non‑uniform correspondence quality.
3. A deterministic polynomial‑time solution for the residual 2‑D registration by casting it as a Klee rectangle stabbing problem, solved with sweep‑line and segment‑tree techniques.
4. Linear‑space, near‑linear‑time complexity with empirical validation on diverse real‑world datasets, demonstrating scalability to large‑scale LiDAR and RGB‑D scans.

The released code (GitHub) and thorough ablation studies further strengthen reproducibility. GMOR’s blend of classical computational‑geometry tools with modern BnB search offers a compelling alternative for robust global registration in SLAM, autonomous driving, medical imaging, and AR/VR pipelines where high outlier ratios and limited memory are common challenges.