Volume 56, Number 5, September-October 2022
|Page(s)||1773 - 1808|
|Published online||01 August 2022|
Mean curvature motion of point cloud varifolds
Université Paris-Saclay, INRIA, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France
2 Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
* Corresponding author: email@example.com
Accepted: 6 May 2022
This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate tangent plane for the points in the cloud. The core ingredient of the mean curvature motion model is the regularization of the first variation of the varifold via convolution with kernels with small stencil. Consistency with the evolution velocity for a smooth surface is proven provided that a sufficiently small stencil and a regular sampling are considered. Furthermore, an implicit and a semi-implicit time discretization are derived. The implicit scheme comes with discrete barrier properties known for the smooth, continuous evolution, whereas the semi-implicit still ensures in all our numerical experiments very good approximation properties while being easy to implement. It is shown that the proposed method is robust with respect to noise and recovers the evolution of smooth curves as well as the formation of singularities such as triple points in 2D or minimal cones in 3D.
Mathematics Subject Classification: 49Q20 / 35K55 / 53A70 / 53E10
Key words: Point cloud varifolds / Mean curvature motion / Regularization / Singular evolution / Time discretization
© The authors. Published by EDP Sciences, SMAI 2022
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