  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 about 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 (https://github.com/matteo-bastico/CoupLap), that allows to
easily generate aligned eigenspaces for multiple registered geometries. We show
that the similarity between those aligned high-dimensional spaces provides a
locally meaningful score to match shapes. We firstly evaluate the performance
of the proposed technique in a point-wise manner, focusing on the task of
object anomaly localization on 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.
