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

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.