Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

TL;DR

This paper connects microscopic nonlocal equations to macroscopic curvature flows in the Heisenberg group, providing a new mathematical framework for image processing and visual cortex modeling, with numerical validation and extensions.

Contribution

It introduces a rigorous upscaling of a nonlocal mean-field equation to Heisenberg mean curvature flow, bridging microscopic models to macroscopic geometric flows.

Findings

Derives curvature flows via asymptotic expansion in the Heisenberg group.

Provides a new approximation and regularization of Heisenberg mean curvature flow.

Numerical algorithms interpolate between nonlocal equations and curvature flow methods.

Abstract

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multiscale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the models of the visual cortex proposed by Petitot and Citti-Sarti, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field equation to curvature flows which are used in image processing. From a pure mathematical point of view, it provides a new approximation and regularization of Heisenberg mean curvature flow. Using the local structure of the rototranslational group, we extend the result to cover the model by Citti and Sarti. Numerically, the parameters in our algorithm interpolate between solving an Ementrout-Cowan type of equation and a Bence-Merriman-Osher algorithm type algorithm for sub-Riemannian mean curvature. We also reproduce some known exact solutions in the Heisenberg case.