Sub-Riemannian Fast Marching in SE(2)

TL;DR

This paper introduces a fast, accurate method for computing sub-Riemannian geodesics in SE(2) using a Riemannian approximation and Fast Marching, with applications in retinal vessel segmentation.

Contribution

It presents a novel Fast Marching implementation for SR-geodesics in SE(2) that handles extreme anisotropies efficiently and improves computational speed over previous PDE-based methods.

Findings

High accuracy in SR-sphere computation validated against known formulas.

Significant reduction in computational time compared to prior approaches.

Demonstrated potential for automated retinal vessel segmentation.

Abstract

We propose a Fast Marching based implementation for computing sub-Riemanninan (SR) geodesics in the roto-translation group SE(2), with a metric depending on a cost induced by the image data. The key ingredient is a Riemannian approximation of the SR-metric. Then, a state of the art Fast Marching solver that is able to deal with extreme anisotropies is used to compute a SR-distance map as the solution of a corresponding eikonal equation. Subsequent backtracking on the distance map gives the geodesics. To validate the method, we consider the uniform cost case in which exact formulas for SR-geodesics are known and we show remarkable accuracy of the numerically computed SR-spheres. We also show a dramatic decrease in computational time with respect to a previous PDE-based iterative approach. Regarding image analysis applications, we show the potential of considering these data adaptive geodesics for a fully automated retinal vessel tree segmentation.