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.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022047
Keywords: Point cloud varifolds, Mean curvature motion, Regularization, Singular evolution, Time discretization
@article{M2AN_2022__56_5_1773_0,
author = {Buet, Blanche and Rumpf, Martin},
title = {Mean curvature motion of point cloud varifolds},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1773--1808},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/m2an/2022047},
mrnumber = {4458833},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022047/}
}
TY - JOUR AU - Buet, Blanche AU - Rumpf, Martin TI - Mean curvature motion of point cloud varifolds JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1773 EP - 1808 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022047/ DO - 10.1051/m2an/2022047 LA - en ID - M2AN_2022__56_5_1773_0 ER -
%0 Journal Article %A Buet, Blanche %A Rumpf, Martin %T Mean curvature motion of point cloud varifolds %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1773-1808 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022047/ %R 10.1051/m2an/2022047 %G en %F M2AN_2022__56_5_1773_0
Buet, Blanche; Rumpf, Martin. Mean curvature motion of point cloud varifolds. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1773-1808. doi: 10.1051/m2an/2022047
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