Discretization and approximation of surfaces using varifolds
type: Item
instance of: scholarly article
title: Discretization and approximation of surfaces using varifolds
zbMATH Open document ID: 1398.49040
author: Blanche Buet
author: Gian Paolo Leonardi
author: Simon Masnou
published in: Geometric Flows
publication date: 2018-06-27T00:00:00Z
full work available at URL: http://hdl.handle.net/11380/1160813
review text: 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.
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 49M25
zbMATH DE Number: 6894927
zbMATH Keywords: varifold
zbMATH Keywords: approximate mean curvature
zbMATH Keywords: approximate second fundamental form
zbMATH Keywords: point cloud
reviewed by: Krzysztof Gdawiec
MaRDI profile type: Publication
OpenAlex ID: W2795393580
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DOI: 10.1515/GEOFL-2018-0004
description: scientific article; zbMATH DE number 6894927