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Discrete Varifolds, Point Clouds, and Surface Approximation

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If you have a question about this talk, please contact Carola-Bibiane Schoenlieb.

There are many models for the discrete approximations of a surface: point clouds, triangulated surfaces, pixel or voxel approximations, etc. We claim that it is possible to study these various approximations in a common setting using the notion of varifolds. Varifolds are tools from geometric measure theory which were introduced by Almgren in the context of Plateau’s problem. They carry both spatial and tangential informations, and they have nice properties in a variational context : compactness, continuity of mass, multiplicity information, control of regularity, and a generalized notion of mean curvature. The aforementioned approximations can be associated with “discrete varifolds”. The talk will be devoted to approximation properties of such discrete varifolds, to a notion of approximated mean curvature for these objects, and to the convergence properties of this approximated curvature. Numerical evaluations on various 2D and 3D point clouds will illustrate these notions. This is joint work with Blanche Buet and Gian Paolo Leonardi.

This talk is part of the Applied and Computational Analysis series.

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