Curvature estimation for point cloud 2-manifolds based on the heat kernel

Kai Wang; Xiheng Wang; Xiaoping Wang; Kai Wang; Xiheng Wang; Xiaoping Wang

doi:10.3934/math.20241557

AIMS Mathematics

2024, Volume 9, Issue 11: 32491-32513. doi: 10.3934/math.20241557

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Research article

Curvature estimation for point cloud 2-manifolds based on the heat kernel

1.
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
2.
Wuhan Research Institute of Posts and Telecommunications, Wuhan 430074, China

Received: 30 July 2024 Revised: 24 September 2024 Accepted: 16 October 2024 Published: 15 November 2024
MSC : 53A45, 53A70

The geometry processing of a point cloud 2-manifold (or point cloud surface) heavily depends on the discretization of differential geometry properties such as Gaussian curvature, mean curvature, principal curvature, and principal directions. Most of the existing algorithms indirectly compute these differential geometry properties by seeking a local approximation surface or fitting point clouds with certain polynomial functions and then applying the curvature formulas in classical differential geometry. This paper initially proposed a new discretized Laplace-Beltrami operator by applying an inherent distance parameter, which acts as the foundation for precisely estimating the mean curvature. Subsequently, the estimated mean curvature was taken as a strong constraint condition for estimating the Gaussian curvatures, principal curvatures, and principal directions by determining an optimal ellipse. The proposed methods are mainly based on the heat kernel function and do not require local surface reconstruction, thus belonging to truly mesh-free methods. We demonstrated the correctness of the estimated curvatures in both analytic and non-analytic models. Various experiments indicated that the proposed methods have high accuracy. As an exemplary application, we utilized the mean curvature for detecting features of point clouds.
- automatic fiber placement,
- variable stiffness trajectory,
- point representation,
- meshless method,
- first-order shear deformation theory
Citation: Kai Wang, Xiheng Wang, Xiaoping Wang. Curvature estimation for point cloud 2-manifolds based on the heat kernel[J]. AIMS Mathematics, 2024, 9(11): 32491-32513. doi: 10.3934/math.20241557

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Abstract

The geometry processing of a point cloud 2-manifold (or point cloud surface) heavily depends on the discretization of differential geometry properties such as Gaussian curvature, mean curvature, principal curvature, and principal directions. Most of the existing algorithms indirectly compute these differential geometry properties by seeking a local approximation surface or fitting point clouds with certain polynomial functions and then applying the curvature formulas in classical differential geometry. This paper initially proposed a new discretized Laplace-Beltrami operator by applying an inherent distance parameter, which acts as the foundation for precisely estimating the mean curvature. Subsequently, the estimated mean curvature was taken as a strong constraint condition for estimating the Gaussian curvatures, principal curvatures, and principal directions by determining an optimal ellipse. The proposed methods are mainly based on the heat kernel function and do not require local surface reconstruction, thus belonging to truly mesh-free methods. We demonstrated the correctness of the estimated curvatures in both analytic and non-analytic models. Various experiments indicated that the proposed methods have high accuracy. As an exemplary application, we utilized the mean curvature for detecting features of point clouds.

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