发表机构
Department of Applied Mathematics, Tel Aviv University; Department of Statistics, Columbia University; Department of Statistics, University of Chicago（特拉维夫大学应用数学系; 哥伦比亚大学统计系; 芝加哥大学统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对点云具有聚类结构的匹配问题，提出基于拉普拉斯最优传输（LapOT）的聚类感知匹配方法，用二次拉普拉斯项正则化最优传输问题，引入RSC方法，经理论分析和实验证明该方法能实现点云间更一致且有意义的对齐。
AI 中文摘要
在许多匹配应用中，待匹配的点云不仅是无结构的点集，而是具有内在聚类结构的分布样本。在此类情况下，由于在连贯区域内单个点通常可互换，找到稳健的区域到区域对齐比建立精确的点对点对应更可取。为此，我们提出一种基于拉普拉斯最优传输（LapOT）的聚类感知匹配新方法。关键思想是用从点云相似性图构建的二次拉普拉斯项正则化最优传输问题，这促使最优耦合尊重两个点集的聚类结构。我们还引入了精细同步聚类（RSC），利用从LapOT获得的聚类感知耦合在点集间产生一致的划分，可克服独立聚类的局限性并产生更稳定和可解释的结果。我们通过理论分析和实证实验证明了该方法的有效性，表明LapOT确实能产生聚类感知匹配，在点云间实现更一致且有意义的对齐。
英文摘要
In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.
