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

Citti, G., Dirr, Nicolas ORCID: https://orcid.org/0000-0003-3634-7367, Dragoni, Federica ORCID: https://orcid.org/0000-0001-6076-9725 and Grande, R. 2024. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group. [Online]. arXiv: Cornell University. Available at: https://arxiv.org/abs/2411.15814

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Abstract

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescal- ing 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 multi-scale approach. The nonlocal equa- tion, 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 mathe- matical 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 cur- vature flow. Using the local structure of the roto-translational 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.

Item Type:	Website Content
Date Type:	Published Online
Status:	Published
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Cornell University
Last Modified:	11 Mar 2025 10:11
URI:	https://orca.cardiff.ac.uk/id/eprint/174420

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