Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Median filter method for mean curvature flow using a random Jacobi algorithm

Anton Ullrich, Tim Laux

TL;DR

This work presents a fast, median-filter-based scheme for approximating the level-set mean curvature flow and proves its convergence to the viscosity solution under mild discretization density. The method extends the MBO thresholding framework to simultaneous, high-dimensional level-set evolution and enables strong convergence results for the discretized heat flow in the optimal sampling regime. The authors establish uniform $L^ty$-convergence and, via TL$^2$-type analyses, derive convergence of the discrete heat flow to the continuous counterpart, complemented by a Gamma-convergence treatment of associated energies under Neumann/Young boundary conditions. The paper also provides implementation details, variants, and extensions to manifolds, densities, and semi-supervised learning contexts, underscoring the method’s practical and theoretical robustness for high-dimensional geometric evolution and clustering tasks.

Abstract

We present an efficient scheme for level set mean curvature flow using a domain discretization and median filters. For this scheme, we show convergence in $L^\infty$-norm under mild assumptions on the number of points in the discretization. In addition, we strengthen the weak convergence result for the MBO thresholding scheme applied to data clustering of Lelmi and one of the authors. This is done through a strong convergence of the discretized heat flow in the optimal regime. Different boundary conditions are also discussed.

Median filter method for mean curvature flow using a random Jacobi algorithm

TL;DR

This work presents a fast, median-filter-based scheme for approximating the level-set mean curvature flow and proves its convergence to the viscosity solution under mild discretization density. The method extends the MBO thresholding framework to simultaneous, high-dimensional level-set evolution and enables strong convergence results for the discretized heat flow in the optimal sampling regime. The authors establish uniform -convergence and, via TL-type analyses, derive convergence of the discrete heat flow to the continuous counterpart, complemented by a Gamma-convergence treatment of associated energies under Neumann/Young boundary conditions. The paper also provides implementation details, variants, and extensions to manifolds, densities, and semi-supervised learning contexts, underscoring the method’s practical and theoretical robustness for high-dimensional geometric evolution and clustering tasks.

Abstract

We present an efficient scheme for level set mean curvature flow using a domain discretization and median filters. For this scheme, we show convergence in -norm under mild assumptions on the number of points in the discretization. In addition, we strengthen the weak convergence result for the MBO thresholding scheme applied to data clustering of Lelmi and one of the authors. This is done through a strong convergence of the discretized heat flow in the optimal regime. Different boundary conditions are also discussed.

Paper Structure

This paper contains 16 sections, 22 theorems, 207 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Setting & Algorithm
  5. Definitions
  6. Uniform convergence - Median Filter
  7. Continuous case
  8. Discrete Case
  9. TL2-convergence - Heat Flow
  10. Implementation and program variants
  11. Program
  12. Neumann conditions
  13. Dirichlet conditions and SSL
  14. Related Work
  15. Outlook
  16. ...and 1 more sections

Key Result

Theorem 1

For an initial datum $g$ there exists a regime of step size and number of samples (step size going to $0$ and sample size to $\infty$) s.t. the solutions of the scheme converge almost surely to the viscosity solution of mean curvature flow locally uniformly in time and space.

Figures (7)

  • Figure 1: The notation of this paper.
  • Figure 2: Visualization of the algorithm.
  • Figure 3: The evolution of a non-symmetric function.
  • Figure 4: The simplifications we employ.
  • Figure 5: Our algorithm applied to three classification problems. From left to right is the evolution over time.
  • ...and 2 more figures

Theorems & Definitions (63)

  • Theorem : Version of Theorem \ref{['Cor:Conv']}
  • Theorem : Version of Theorem \ref{['Cor:HeatFlows']}
  • Definition 1.1: Landau notation
  • Remark
  • Definition 2.1: ${\mathop{\mathrm{TL}}\limits}^2$-space
  • Definition 2.2: ${\mathop{\mathrm{TL}}\limits}^2$-convergence
  • Definition 2.3: Viscosity solution
  • Definition 2.4: Level set Laplacian
  • Remark 1
  • Remark
  • ...and 53 more