Sub-Riemannian fast marching in SE(2)

Abstract

We propose a Fast Marching based implementation for com-puting sub-Riemanninan (SR) geodesics in the roto-translation groupSE(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 withextreme anisotropies is used to compute a SR-distance map as the so-lution of a corresponding eikonal equation. Subsequent backtracking onthe distance map gives the geodesics. To validate the method, we con-sider the uniform cost case in which exact formulas for SR-geodesics areknown and we show remarkable accuracy of the numerically computedSR-spheres. We also show a dramatic decrease in computational timewith respect to a previous PDE-based iterative approach. Regarding im-age analysis applications, we show the potential of considering these dataadaptive geodesics for a fully automated retinal vessel tree segmentation