# A Cortical-Inspired Contour Completion Model Based on Contour Orientation and Thickness

## TL;DR

This paper presents a new model for completing visual contours in the brain, incorporating orientation and thickness using advanced mathematical techniques.

## Contribution

The paper introduces a four-dimensional cortical-inspired model for contour completion that includes thickness and orientation.

## Key findings

- The model uses sub-Riemannian geodesics to restore damaged image contours.
- Existence of minimal geodesics between boundary conditions is proven using geometric control theory.
- Explicit solutions and qualitative analysis are provided for the geodesic problem.

## Abstract

An extended four-dimensional version of the traditional Petitot–Citti–Sarti model on contour completion in the visual cortex is examined. The neural configuration space is considered as the group of similarity transformations, denoted as M=SIM(2). The left-invariant subbundle of the tangent bundle models possible directions for establishing neural communication. The sub-Riemannian distance is proportional to the energy expended in interneuron activation between two excited border neurons. According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space M of positions, orientations and thicknesses (scales). We study the geodesic problem in M using geometric control theory techniques. We prove the existence of a minimal geodesic between arbitrary specified boundary conditions. We apply the Pontryagin maximum principle and derive the geodesic equations. In the special cases, we find explicit solutions. In the general case, we provide a qualitative analysis. Finally, we support our model with a simulation of the association field.