Paper · Mar 10, 2018

Contour Parametrization via Anisotropic Mean Curvature Flows

P. Suárez-Serrato, E. I. Velázquez Richards

math.DGWider flowsarXiv:1803.03724
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As on arXiv

Abstract

We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as background. We include a stability criteria for our numerical approximation.

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Submitted
Mar 10, 2018
Subjects
math.DG, cs.CG, cs.CV, math.AP, math.NA
Journal
Applied Mathematics and Computation, Volume 441, 2023, 127699
DOI
10.1016/j.amc.2022.127699

Comments: 30 pages, 20 images, source code for our numerical implementation is available in this URL https://github.com/V3du4rd0/AMCF

Mean curvature flowAnisotropic & nonlocal flowsStability & uniquenessNumerics & visualization

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Motion of level sets by inverse anisotropic mean curvature (Francesco Della Pietra, Nunzia Gavitone and Chao Xia, 2018)1Della Pietra 2018 On the self-similar solutions of the crystalline mean curvature flow in three dimensions (Norbert Požár, 2018)4Požár 2018 Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (Yoshiki Jikumaru and Miyuki Koiso, 2019)8Jikumaru 2019 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)3Chambolle 2020 Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions (Annalisa Cesaroni, Heiko Kröner and Matteo Novaga, 2021)2Cesaroni 2021 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)7Bulanyi 2025 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)6Gao 2025 On the anisotropic mean curvature flow in the Gaussian space (Di Zhang, 2026)5Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Motion of level sets by inverse anisotropic mean curvature

    Francesco Della Pietra, Nunzia Gavitone and Chao XiaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Shared terms “approximation”

  2. Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

    Annalisa Cesaroni, Heiko Kröner and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness

  3. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness

  4. On the self-similar solutions of the crystalline mean curvature flow in three dimensions

    Norbert PožárWider flows

    Topics Numerics & visualization, Anisotropic & nonlocal flows · Shared terms “implementation”, “numerical”

  5. On the anisotropic mean curvature flow in the Gaussian space

    Di ZhangWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness

  6. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Anisotropic & nonlocal flows

  7. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “approximation”

  8. Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

    Yoshiki Jikumaru and Miyuki KoisoWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.