Abstract
A varifold approach to surface approximation and curvature estimation on point clouds
Blanche Buet
Université de Paris XI
We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds were introduced by Almgren to study minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete objects as well. While the varifold structure is flexible enough to adapt to both regular and discrete objects, it allows one to define variational notions of mean curvature and second fundamental form based on the divergence theorem.
Thanks to a regularization of these weak formulations, we propose a notion of discrete curvature (actually a family of discrete curvatures associated with a regularization scale) relying only on the varifold structure. We prove nice convergence properties involving a natural growth assumption: the scale of regularization must be large with respect to the accuracy of the discretization. We perform numerical computations of mean curvature and Gaussian curvature on point clouds in \( \mathbb{R}^3 \) to illustrate this approach.
Joint work with Gian Paolo Leonardi (Modena) and Simon Masnou (Lyon).
Thanks to a regularization of these weak formulations, we propose a notion of discrete curvature (actually a family of discrete curvatures associated with a regularization scale) relying only on the varifold structure. We prove nice convergence properties involving a natural growth assumption: the scale of regularization must be large with respect to the accuracy of the discretization. We perform numerical computations of mean curvature and Gaussian curvature on point clouds in \( \mathbb{R}^3 \) to illustrate this approach.
Joint work with Gian Paolo Leonardi (Modena) and Simon Masnou (Lyon).