Discretization and approximation of surfaces using varifolds
The main aim of the study presented in the paper is to provide a framework that can be used for the representation and the analysis of different discrete surface models and that allows for robust convergence results for the approximate curvatures. The presented method does not require the use of discrete surfaces of some special type. It can be applied with any discrete surface type. The first step in the proposed framework is the definition of the regularized first variation of a varifold, that is obtained by convolving the standard first variation with a regularizing kernel. By a similar regularization of the weight function the authors arrive at the notion of approximate mean curvature vector field. They prove some convergence properties of this curvature. The obtained results rely on the notion of bounded Lipschitz distance. Next, the authors study the second fundamental form obtaining the approximate second fundamental form. To show consistency and robustness of the proposed framework the authors present some numerical examples of computations of the approximate mean curvature of various 2D and 3D point clouds. In particular, they present the dependence on the regularization kernel, the parameter used in the regularization and the sampling resolution.
scientific article; zbMATH DE number 6894927
scientific article