Bo Pang, Zhongtian Zheng, Yilong Li, Guoping Wang, Peng-Shuai Wang
{"title":"用于三维点云的神经拉普拉斯算子","authors":"Bo Pang, Zhongtian Zheng, Yilong Li, Guoping Wang, Peng-Shuai Wang","doi":"arxiv-2409.06506","DOIUrl":null,"url":null,"abstract":"The discrete Laplacian operator holds a crucial role in 3D geometry\nprocessing, yet it is still challenging to define it on point clouds. Previous\nworks mainly focused on constructing a local triangulation around each point to\napproximate the underlying manifold for defining the Laplacian operator, which\nmay not be robust or accurate. In contrast, we simply use the K-nearest\nneighbors (KNN) graph constructed from the input point cloud and learn the\nLaplacian operator on the KNN graph with graph neural networks (GNNs). However,\nthe ground-truth Laplacian operator is defined on a manifold mesh with a\ndifferent connectivity from the KNN graph and thus cannot be directly used for\ntraining. To train the GNN, we propose a novel training scheme by imitating the\nbehavior of the ground-truth Laplacian operator on a set of probe functions so\nthat the learned Laplacian operator behaves similarly to the ground-truth\nLaplacian operator. We train our network on a subset of ShapeNet and evaluate\nit across a variety of point clouds. Compared with previous methods, our method\nreduces the error by an order of magnitude and excels in handling sparse point\nclouds with thin structures or sharp features. Our method also demonstrates a\nstrong generalization ability to unseen shapes. With our learned Laplacian\noperator, we further apply a series of Laplacian-based geometry processing\nalgorithms directly to point clouds and achieve accurate results, enabling many\nexciting possibilities for geometry processing on point clouds. The code and\ntrained models are available at https://github.com/IntelligentGeometry/NeLo.","PeriodicalId":501174,"journal":{"name":"arXiv - CS - Graphics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Neural Laplacian Operator for 3D Point Clouds\",\"authors\":\"Bo Pang, Zhongtian Zheng, Yilong Li, Guoping Wang, Peng-Shuai Wang\",\"doi\":\"arxiv-2409.06506\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The discrete Laplacian operator holds a crucial role in 3D geometry\\nprocessing, yet it is still challenging to define it on point clouds. Previous\\nworks mainly focused on constructing a local triangulation around each point to\\napproximate the underlying manifold for defining the Laplacian operator, which\\nmay not be robust or accurate. In contrast, we simply use the K-nearest\\nneighbors (KNN) graph constructed from the input point cloud and learn the\\nLaplacian operator on the KNN graph with graph neural networks (GNNs). However,\\nthe ground-truth Laplacian operator is defined on a manifold mesh with a\\ndifferent connectivity from the KNN graph and thus cannot be directly used for\\ntraining. To train the GNN, we propose a novel training scheme by imitating the\\nbehavior of the ground-truth Laplacian operator on a set of probe functions so\\nthat the learned Laplacian operator behaves similarly to the ground-truth\\nLaplacian operator. We train our network on a subset of ShapeNet and evaluate\\nit across a variety of point clouds. Compared with previous methods, our method\\nreduces the error by an order of magnitude and excels in handling sparse point\\nclouds with thin structures or sharp features. Our method also demonstrates a\\nstrong generalization ability to unseen shapes. With our learned Laplacian\\noperator, we further apply a series of Laplacian-based geometry processing\\nalgorithms directly to point clouds and achieve accurate results, enabling many\\nexciting possibilities for geometry processing on point clouds. 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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.