用于三维点云的神经拉普拉斯算子

Bo Pang, Zhongtian Zheng, Yilong Li, Guoping Wang, Peng-Shuai Wang
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引用次数: 0

摘要

离散拉普拉斯算子在三维几何处理中起着至关重要的作用,但在点云上定义离散拉普拉斯算子仍然具有挑战性。以前的工作主要集中在构建每个点周围的局部三角剖分,以接近底层流形来定义拉普拉斯算子,但这可能并不稳健或准确。相比之下,我们只需使用由输入点云构建的 K-最近邻(KNN)图,并利用图神经网络(GNN)在 KNN 图上学习拉普拉斯算子。然而,真实的拉普拉斯算子是在流形网格上定义的,其连接性与 KNN 图不同,因此不能直接用于训练。为了训练 GNN,我们提出了一种新的训练方案,即在一组探测函数上模仿地面实况拉普拉斯算子的行为,使学习到的拉普拉斯算子与地面实况拉普拉斯算子的行为相似。我们在 ShapeNet 的一个子集上训练我们的网络,并通过各种点云对其进行评估。与之前的方法相比,我们的方法将误差降低了一个数量级,并且在处理具有稀疏结构或尖锐特征的稀疏点云方面表现出色。我们的方法还展示了对未知形状的强大泛化能力。利用我们学习的拉普拉斯操作器,我们进一步将一系列基于拉普拉斯的几何处理算法直接应用于点云,并取得了精确的结果,为点云的几何处理带来了许多令人兴奋的可能性。代码和训练模型可在 https://github.com/IntelligentGeometry/NeLo 上获取。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Neural Laplacian Operator for 3D Point Clouds
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.
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