Paper · Mar 3, 2016

A diffusion driven curvature flow

Emre Baspinar, Giovanna Citti

math.APWider flowsarXiv:1603.01156
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As on arXiv

Abstract

Bence-Merriman-Osher algorithm computes numerically mean curvature flow via solutions of heat equation iteratively initialized at the end of each short time interval. Inspired by the convergence proof of Evans of this algorithm where he employed nonlinear semigroup theory and level set approach to mean curvature flow, we prove here that a similar procedure, but now with a different definition of evolved surface at the end of each short time interval, results in a motion by curvature. The interest of this version of the algorithm is its application corresponding to model of the visual cortex.

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Submitted
Mar 3, 2016
Subjects
math.AP
Mean curvature flowOther curvature flowsWeak & singular flowsNumerics & visualization

Its neighbourhood

Connected papers

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 Uniqueness of vanishing viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)2Baspinar 2016 Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures (Emre Baspinar and Giovanna Citti, 2016)3Baspinar 2016 On the self-similar solutions of the crystalline mean curvature flow in three dimensions (Norbert Požár, 2018)5Požár 2018 Theory and numerics for Chen's flow of curves (Matthew Cooper, Glen Wheeler and Valentina-Mira Wheeler, 2020)4Cooper 2020 Sphere Under Advection and Mean Curvature Flow (Bryce A. Besler, Tannis D. Kemp, Nils D. Forkert et al., 2020)6Besler 2020 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)8Bahadursingh 2024 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)1Citti 2024 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)7He 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows

    Topics Numerics & visualization, Other curvature flows · Same author Giovanna Citti · Shared terms “visual cortex”, “algorithm”, “Bence Merriman”

  2. Uniqueness of vanishing viscosity mean curvature flow solution in two sub-Riemannian structures

    Emre Baspinar and Giovanna CittiWider flows

    Topics Weak & singular flows · Same authors Emre Baspinar, Giovanna Citti · Shared terms “visual cortex”

  3. Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures

    Emre Baspinar and Giovanna CittiWider flows

    Topics Weak & singular flows · Same authors Emre Baspinar, Giovanna Citti · Shared terms “visual cortex”

  4. Theory and numerics for Chen's flow of curves

    Matthew Cooper, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Numerics & visualization, Other curvature flows · Shared terms “diffusion”, “algorithm”

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

    Norbert PožárWider flows

    Topics Numerics & visualization, Weak & singular flows · Shared terms “algorithm”, “level set”

  6. Sphere Under Advection and Mean Curvature Flow

    Bryce A. Besler, Tannis D. Kemp, Nils D. Forkert et al.Wider flows

    Topics Numerics & visualization

  7. A curvature flow approach to dorsal closure modelling

    Shuhui He, Ben Whale, Glen Wheeler et al.Wider flows

    Topics Numerics & visualization, Other curvature flows

  8. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows

    Topics Numerics & visualization

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 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.