DC-Reg: Globally Optimal Point Cloud Registration via Tight Bounding with Difference of Convex Programming

2026-03-26

AI Summary

This work proposes DC-Reg, a novel framework addressing the challenge of achieving globally optimal point cloud registration under partial overlap and large initial misalignment. The method introduces a unified difference-of-convex (DC) decomposition of the coupled objective function involving both transformation and correspondence variables, enabling the construction of a globally concave lower bound that significantly tightens the search space in branch-and-bound (BnB) optimization. By jointly exploiting the structural interdependence between transformation and correspondence variables, DC-Reg overcomes the limitations of conventional per-term relaxation strategies. Integrated with rotation-invariant features and an efficient solver for the linear assignment problem, the approach demonstrates faster convergence than existing global methods on both synthetic data and the 3DMatch benchmark, while exhibiting superior robustness under extreme noise and outlier conditions.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• point cloud registration
• global optimality
• partial overlap
• large misalignment
• non-convex optimization

Innovation

• Difference of Convex Programming
• Globally Optimal Registration
• Holistic Concave Underestimator
• Branch-and-Bound
• Point Cloud Registration