Point cloud matching, a crucial technique in computer vision, medical and robotics fields, is primarily concerned with finding correspondences between pairs of point clouds or voxels. In some practical scenarios, emphasizing local differences is crucial for accurately identifying a correct match, thereby enhancing the overall robustness and reliability of the matching process. Commonly used shape descriptors have several limitations and often fail to provide meaningful local insights on the paired geometries. In this work, we propose a new technique, based on graph Laplacian eigenmaps, to match point clouds by taking into account fine local structures. To deal with the order and sign ambiguity of Laplacian eigenmaps, we introduce a new operator, called Coupled Laplacian, that allows to easily generate aligned eigenspaces for multiple rigidly-registered geometries. We show that the similarity between those aligned high-dimensional spaces provides a locally meaningful score to match shapes. We initially evaluate the performance of the proposed technique in a point-wise manner, specifically focusing on the task of object anomaly localization using the MVTec 3D-AD dataset. Additionally, we define a new medical task, called automatic Bone Side Estimation (BSE), which we address through a global similarity score derived from coupled eigenspaces. In order to test it, we propose a benchmark collecting bone surface structures from various public datasets. Our matching technique, based on Coupled Laplacian, outperforms other methods by reaching an impressive accuracy on both tasks. The code to reproduce our experiments is publicly available at https://github.com/matteo-bastico/CoupledLaplacian and in the Supplementary Code.

翻译：点云匹配是计算机视觉、医学和机器人学领域中的关键技术，主要关注于寻找点云对或体素对之间的对应关系。在某些实际场景中，强调局部差异对于准确识别正确匹配至关重要，从而提升匹配过程的整体鲁棒性和可靠性。常用的形状描述符存在若干局限性，且往往无法为成对几何体提供有意义的局部信息。本文提出了一种基于图拉普拉斯特征映射的新技术，通过考虑精细的局部结构来实现点云匹配。为解决拉普拉斯特征映射的排序和符号模糊性问题，我们引入了一种名为耦合拉普拉斯的新算子，该算子能够轻松生成多个刚性配准几何体的对齐特征空间。研究表明，这些对齐的高维空间之间的相似性为形状匹配提供了具有局部意义的评分。我们首先以逐点方式评估了所提技术的性能，具体聚焦于使用MVTec 3D-AD数据集进行目标异常定位的任务。此外，我们定义了一项新的医学任务——自动骨骼侧别估计（BSE），并通过从耦合特征空间导出的全局相似性评分来解决该任务。为验证该方法，我们提出了一个基准测试集，收集了来自多个公开数据集的骨骼表面结构。基于耦合拉普拉斯算子的匹配技术在两项任务上均以令人印象深刻的准确率超越了其他方法。用于复现实验的代码已公开于 https://github.com/matteo-bastico/CoupledLaplacian 及补充代码中。