Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model

TL;DR

This paper studies minimal surfaces in the roto-translation group with a subRiemannian metric, linking geometric analysis to neurobiological visual processing and digital image disocclusion.

Contribution

It introduces a method to compute minimal surfaces with fixed boundaries in the roto-translation group, connecting geometric theory with neurobiological and image processing applications.

Findings

Characterization of smooth minimal surfaces as ruled surfaces.

Identification of obstructions to existence and uniqueness.

Conditions under which smooth minimal spanning surfaces exist.

Abstract

We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a subRiemannian metric. By work of G. Citti and A. Sarti, such solutions are amodal completions of occluded visual data when using a model of the first layer of the visual cortex. Using a characterization of smooth minimal surfaces as ruled surfaces, we give a method to compute a minimal spanning surface given fixed boundary data presuming such a surface exists. Moreover, we describe a number of obstructions to existence and uniqueness but also show that under suitable conditions, smooth minimal spanning surfaces with good properties exist. Not only does this provide an explicit realization of the disocclusion process for the neurobiological model, but it also has application to contructing disocclusion algorithms in digital image processing.