Weak and approximate curvatures of a measure: A varifold perspective. (September 2022)

• Record Type: Journal Article
• Authors: Buet, Blanche; Leonardi, Gian Paolo; Masnou, Simon
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: primary 49Q15 -- secondary 68U05 65D18; Varifold -- Regularized variations -- Approximate second fundamental form -- Point cloud; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112983
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: By revisiting the notion of generalized second fundamental form originally introduced by Hutchinson for a special class of integral varifolds, we define a weak curvature tensor that is particularly well-suited for being extended to general varifolds of any dimension and codimension through regularization. The resulting approximate second fundamental forms are defined not only for piecewise-smooth surfaces, but also for datasets of very general type (like, e.g., point clouds). We obtain explicitly computable formulas for both weak and approximate curvature tensors, we exhibit structural properties and prove convergence results, and lastly we provide some numerical tests on point clouds that confirm the generality and effectiveness of our approach.