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Elementary discrete diffusion/redistancing schemes for the mean curvature flow

Antonin Chambolle, Daniele De Gennaro, Massimiliano Morini

Abstract

We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the standard CFL condition $h\sim\e^2$, where $h$ is the time discretization step and $\e$ the space step. We discuss extensions to more general convolution/redistancing schemes.

Elementary discrete diffusion/redistancing schemes for the mean curvature flow

Abstract

We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the standard CFL condition , where is the time discretization step and the space step. We discuss extensions to more general convolution/redistancing schemes.

Paper Structure

This paper contains 24 sections, 15 theorems, 161 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A simple algorithm
  3. Laplacian operator and redistancing on a regular discrete grid.
  4. Algorithm
  5. Convergence of the algorithm
  6. Notions of generalized mean curvature flow
  7. Evolution of a ball
  8. Consistency of the algorithm
  9. Examples
  10. Extension: general convolution kernels
  11. Convolution kernels
  12. Kernel properties
  13. Control of the balls
  14. Consistency
  15. Nonlinear redistancing: applications to deep learning based approximation algorithms
  16. ...and 9 more sections

Key Result

Lemma 2.1

Assume still that $(u_i)_{i\in\varepsilon\mathbb{Z}^N}$ is $1$-Lipschitz. Then the distance function $\,\mathrm{sd}^+$ may also be defined as follows: We also deduce that $\,\mathrm{sd}_i^+\ge 0\Leftrightarrow u_i\ge 0$ and $\,\mathrm{sd}_i^+< 0 \Leftrightarrow u_i<0$.

Figures (5)

  • Figure 1: Evolution of a disk (initial radius 50) (left) and decay of the radius (right)
  • Figure 2: Evolution of a shape: left at times $t=0,20,\dots,200$, right at times $t=0,25,50,\dots,250$ and then $t=375,500,\dots,1250$.
  • Figure 3: Evolution of the radii for an initial disk of radius $50$, $\varepsilon=1$, and $h=5, 10, 20$ (the explicit scheme needs $h=.25$). The precision remains quite good for small values of $h$. Compare with Figure \ref{['fig:Explicit']}, right.
  • Figure 4: Evolutions of a disk of radius $50$ with $h=.25$ (explicit scheme as in Fig. \ref{['fig:Explicit']}) and $h=5,10,20$ (implicit scheme \ref{['eq:impl_scheme']}).
  • Figure 5: Evolutions of the mask pattern, comparison between the explicit and implicit schemes. Left two plots: explicit vs implicit ($h=5$) evolution, times $0,5,\dots,100$; Right two plots: same for times $0,100,200,\dots$

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Proposition 2.2: Comparison
  • proof
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • ...and 19 more