Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology
Computing the Laplace–Beltrami operator on point clouds is essential for tasks such as smoothing and shape analysis. Unlike meshes, determining the Laplace–Beltrami operator on point clouds requires establishing neighbors for each point. However, traditional k-nearest neighbors (k-NN) methods for es...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-06-01
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| Series: | Graphical Models |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1524070325000086 |
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| author | Ao Zhang Qing Fang Peng Zhou Xiao-Ming Fu |
| author_facet | Ao Zhang Qing Fang Peng Zhou Xiao-Ming Fu |
| author_sort | Ao Zhang |
| collection | DOAJ |
| description | Computing the Laplace–Beltrami operator on point clouds is essential for tasks such as smoothing and shape analysis. Unlike meshes, determining the Laplace–Beltrami operator on point clouds requires establishing neighbors for each point. However, traditional k-nearest neighbors (k-NN) methods for estimating local neighborhoods often introduce spurious connectivities that distort the manifold topology. We propose a novel approach that leverages persistent homology to refine the neighborhood graph by identifying and removing erroneous edges. Starting with an initial k-NN graph, we assign weights based on local tangent plane estimations and construct a Vietoris–Rips complex. Persistent homology is then employed to detect and eliminate spurious edges through a topological optimization process. This iterative refinement results in a more accurate neighborhood graph that better represents the underlying manifold, enabling precise discretization of the Laplace–Beltrami operator. Experimental results on various point cloud datasets demonstrate that our method outperforms traditional k-NN approaches by more accurately capturing the manifold topology and enhancing downstream computations such as spectral analysis. |
| format | Article |
| id | doaj-art-f96b2f695d304e6883a9bf61552be71d |
| institution | OA Journals |
| issn | 1524-0703 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Graphical Models |
| spelling | doaj-art-f96b2f695d304e6883a9bf61552be71d2025-08-20T02:07:31ZengElsevierGraphical Models1524-07032025-06-0113910126110.1016/j.gmod.2025.101261Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homologyAo Zhang0Qing Fang1Peng Zhou2Xiao-Ming Fu3School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR ChinaCorresponding author.; School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR ChinaSchool of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR ChinaSchool of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR ChinaComputing the Laplace–Beltrami operator on point clouds is essential for tasks such as smoothing and shape analysis. Unlike meshes, determining the Laplace–Beltrami operator on point clouds requires establishing neighbors for each point. However, traditional k-nearest neighbors (k-NN) methods for estimating local neighborhoods often introduce spurious connectivities that distort the manifold topology. We propose a novel approach that leverages persistent homology to refine the neighborhood graph by identifying and removing erroneous edges. Starting with an initial k-NN graph, we assign weights based on local tangent plane estimations and construct a Vietoris–Rips complex. Persistent homology is then employed to detect and eliminate spurious edges through a topological optimization process. This iterative refinement results in a more accurate neighborhood graph that better represents the underlying manifold, enabling precise discretization of the Laplace–Beltrami operator. Experimental results on various point cloud datasets demonstrate that our method outperforms traditional k-NN approaches by more accurately capturing the manifold topology and enhancing downstream computations such as spectral analysis.http://www.sciencedirect.com/science/article/pii/S1524070325000086Laplace–Beltrami operatorPoint cloudsPersistent homologyTopology |
| spellingShingle | Ao Zhang Qing Fang Peng Zhou Xiao-Ming Fu Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology Graphical Models Laplace–Beltrami operator Point clouds Persistent homology Topology |
| title | Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology |
| title_full | Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology |
| title_fullStr | Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology |
| title_full_unstemmed | Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology |
| title_short | Topology-controlled Laplace–Beltrami operator on point clouds based on persistent homology |
| title_sort | topology controlled laplace beltrami operator on point clouds based on persistent homology |
| topic | Laplace–Beltrami operator Point clouds Persistent homology Topology |
| url | http://www.sciencedirect.com/science/article/pii/S1524070325000086 |
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