Citti, Giovanna, Dirr, Nicolas  ORCID: https://orcid.org/0000-0003-3634-7367, Dragoni, Federica ORCID: https://orcid.org/0000-0001-6076-9725 and Grande, Raffaele
2026.
Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group.
Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal
24
(2)
, pp. 720-741.
10.1137/25m1725954
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Abstract
We derive curvature flows in the Heisenberg group by a 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 by Citti and 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 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 type algorithm for sub-Riemannian mean curvature. We also reproduce some known exact solutions in the Heisenberg case.
| Item Type: | Article |
|---|---|
| Date Type: | Publication |
| Status: | Published |
| Schools: | Schools > Mathematics |
| Publisher: | Society for Industrial and Applied Mathematics |
| ISSN: | 1540-3459 |
| Date of First Compliant Deposit: | 17 June 2026 |
| Date of Acceptance: | 6 January 2026 |
| Last Modified: | 17 Jun 2026 08:45 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/187484 |
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