# Coupled Laplacian Eigenmaps for Locally-Aware 3D Rigid Point Cloud Matching

arXiv: https://arxiv.org/abs/2402.17372 · HTML: https://mlsanity.com/paper/2402.17372/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 82 references that are in the map.
- First posted on arXiv 27 Feb 2024.

## Builds on (most cited first, 21 of 21)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [3DMatch: Learning Local Geometric Descriptors from RGB-D Reconstructions](https://mlsanity.com/paper/1603.08182/) (2016)
- [Deep Closest Point: Learning Representations for Point Cloud Registration](https://mlsanity.com/paper/1905.03304/) (2019)
- [LIFT: Learned Invariant Feature Transform](https://mlsanity.com/paper/1603.09114/) (2016)
- [PPFNet: Global Context Aware Local Features for Robust 3D Point Matching](https://mlsanity.com/paper/1802.02669/) (2018)
- [The MVTec 3D-AD Dataset for Unsupervised 3D Anomaly Detection and Localization](https://mlsanity.com/paper/2112.09045/) (2021)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [3DFeat-Net: Weakly Supervised Local 3D Features for Point Cloud Registration](https://mlsanity.com/paper/1807.09413/) (2018)
- [Anomaly Detection in 3D Point Clouds using Deep Geometric Descriptors](https://mlsanity.com/paper/2202.11660/) (2022)
- [FilterReg: Robust and Efficient Probabilistic Point-Set Registration using Gaussian Filter and Twist Parameterization](https://mlsanity.com/paper/1811.10136/) (2018)
- [Rotational Projection Statistics for 3D Local Surface Description and Object Recognition](https://mlsanity.com/paper/1304.3192/) (2013)
- [Robot Object Retrieval with Contextual Natural Language Queries](https://mlsanity.com/paper/2006.13253/) (2020)
- [Learning Local Shape Descriptors from Part Correspondences With Multi-view Convolutional Networks](https://mlsanity.com/paper/1706.04496/) (2017)
- [Real-time multimodal image registration with partial intraoperative point-set data](https://mlsanity.com/paper/2109.05023/) (2021)
- [Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey](https://mlsanity.com/paper/2106.02154/) (2021)
- [Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration](https://mlsanity.com/paper/2012.07340/) (2020)
- [Fast and Accurate Point Cloud Registration using Trees of Gaussian Mixtures](https://mlsanity.com/paper/1807.02587/) (2018)
- [Point Transformer for Shape Classification and Retrieval of 3D and ALS Roof PointClouds](https://mlsanity.com/paper/2011.03921/) (2020)
- [3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching](https://mlsanity.com/paper/2106.11166/) (2021)

## Built on by (most cited first, 3 of 3)

- [Modality-Aware Feature Matching in Visual and Vision-Language Applications: A Comprehensive Survey](https://mlsanity.com/paper/2507.22791/) (2025)
- [A Survey of Medical Point Cloud Shape Learning: Registration, Reconstruction and Variation](https://mlsanity.com/paper/2508.03057/) (2025)
- [LiDAR-based Human Activity Recognition through Laplacian Spectral Analysis](https://mlsanity.com/paper/2509.23255/) (2025)

## Abstract (arXiv)

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.