Horizontal Mean Curvature Flow as a Scaling Limit of a Mean Field Equation in the Heisenberg Group

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.