Neural Laplacian Operator for 3D Point Clouds

TL;DR

This work addresses the challenge of defining a Laplacian on 3D point clouds by learning a Laplacian operator on the fixed KNN graph derived from the points. It introduces NeLo, a neural Laplacian operator where a graph neural network predicts edge weights $w_{ij}$ and a mass $M_{ii}$ so that the resulting $L$ and $M$ reproduce the ground-truth operator’s behavior on a set of probe functions, i.e., $\Delta f = M^{-1} L f$ approximates $\Delta_{gt} f = M_{gt}^{-1} L_{gt} f$. The method demonstrates an order-of-magnitude improvement over prior triangulation-based approaches on ShapeNet, strong robustness to noise, and good generalization to unseen shapes and real scans, enabling accurate Laplacian-based processing (heat diffusion, geodesic distance, smoothing, spectral filtering, deformation) directly on point clouds. By avoiding reliance on local triangulations and instead learning geometry priors from data, NeLo provides scalable, end-to-end Laplacian computation for point clouds with broad practical impact in geometry processing and 3D analysis.

Abstract

The discrete Laplacian operator holds a crucial role in 3D geometry processing, yet it is still challenging to define it on point clouds. Previous works mainly focused on constructing a local triangulation around each point to approximate the underlying manifold for defining the Laplacian operator, which may not be robust or accurate. In contrast, we simply use the K-nearest neighbors (KNN) graph constructed from the input point cloud and learn the Laplacian operator on the KNN graph with graph neural networks (GNNs). However, the ground-truth Laplacian operator is defined on a manifold mesh with a different connectivity from the KNN graph and thus cannot be directly used for training. To train the GNN, we propose a novel training scheme by imitating the behavior of the ground-truth Laplacian operator on a set of probe functions so that the learned Laplacian operator behaves similarly to the ground-truth Laplacian operator. We train our network on a subset of ShapeNet and evaluate it across a variety of point clouds. Compared with previous methods, our method reduces the error by an order of magnitude and excels in handling sparse point clouds with thin structures or sharp features. Our method also demonstrates a strong generalization ability to unseen shapes. With our learned Laplacian operator, we further apply a series of Laplacian-based geometry processing algorithms directly to point clouds and achieve accurate results, enabling many exciting possibilities for geometry processing on point clouds. The code and trained models are available at https://github.com/IntelligentGeometry/NeLo.

Figures (16)

• Figure 1: Overview. (a): a $K$-nearest neighbors graph (right) is constructed from the input unoriented point cloud (left). The underlying mesh is drawn together with $K$-nearest neighbors graph for better visualization. (b): a GNN is used to learn the vertex-wise features, followed by two MLPs to decode the features to a Laplacian matrix $L$ and a mass matrix $M$. (c): we compute the Laplacians of probe functions with the predicted $L$ and $M$ and the ground-truth $L_{gt}$ and $M_{gt}$ and minimize the difference between the two Laplacians to train the GNN. Although the connectivity of the KNN graph is different from the ground truth mesh, we can still compare the Laplacians of the probe functions to train the GNN.
• Figure 2: Visualization of probe functions. The top row shows spectral probe functions, consisting of the 1st, 16th, 32th, and 64th non-constant eigenvectors of the ground-truth Laplacian matrix. The bottom row shows sample spatial probe functions with different frequencies and phases.
• Figure 3: The network architecture. The U-Net takes a KNN graph as input and outputs a feature vector for each vertex. Then, two MLPs are used to predict the edge weights of the stiffness matrix and elements of the mass matrix, respectively.
• Figure 4: Visual comparisons. From left to right are the probe functions, results of GraphTaubin1995, HeatBelkin2009, NManifoldSharp2020a, our method, and the ground truth Laplacian operator, respectively. Our results are apparently more faithful to the ground truth, especially for the point clouds with sharp features and thin structures.
• Figure 5: The visualization of the results on real-world data. The point cloud is collected in real scenes by a Microsoft Kinect v2 Wang2016a. From left to right are the probe functions, results of GraphTaubin1995, NManifoldSharp2020a, our method, and the ground truth Laplacian operator, respectively. As the image suggests, our method is robust for wild scenes as well.