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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Gian Paolo Leonardi
Universita` degli studi di Modena e Reggio Emilia
Title: Discrete Varifolds: theory and applications

Abstract: The theory of varifolds, initially developed by Almgren, Allard, and Hutchinson in the context of Geometric Measure Theory, provides a theoretical framework for extending unoriented d-dimensional submanifolds of a Riemannian n-manifold, in a way that is suitable for applications to variational problems of least-area type. However, varifolds are mainly known to GMT specialists and, despite the fact that they are substantially available as a mathematical tool since 1972, they have had no significant applications in the fields of discrete geometry and numerical analysis. In this talk I will describe a natural way to adapt/modify the theory of varifolds, in order to allow for very general representations of discrete surfaces (including voxelizations and point clouds) as well as for robust approximations of surface features (mean curvature, second fundamental form) also in presence of singularities. The starting point is a notion of "smoothed first variation", which can be also linked to a notion of "scale" at which a discrete object is "seen". Then, suitable notions of approximate (or discrete) curvatures are introduced, and quantitative error estimates are shown to hold in any dimension and codimension. In support of the theory, I will also show some preliminary tests performed on point clouds. This is a joint work with Blanche Buet and Simon Masnou.

Date: Thursday, April 21, 2016
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Ian Tice