# A. P. Mashtakov, Remco Duits, Yu. L. Sachkov, Erik Bekkers, I. Yu. Beschastnyi. Sub-Riemannian Geodesics in SO(3) with Application to Vessel Tracking in Spherical Images of Retina.

Title: Sub-Riemannian Geodesics in SO(3) with Application to Vessel Tracking in Spherical Images of Retina
Authors: A. P. Mashtakov, Remco Duits, Yu. L. Sachkov, Erik Bekkers, I. Yu. Beschastnyi
In order to detect vessel locations in spherical images of retina we consider the problem of minimizing the functional $\int \limits_0^l \gothic{C}(\gamma(s)) \sqrt{\xi^2 + k_g^2(s)} \, {\rm d}s$ for a curve $\gamma$ on a sphere with fixed boundary points and directions. The total length $l$ is free, $s$ denotes the spherical arclength, and $k_g$ denotes the geodesic curvature of~$\gamma$. Here the smooth external cost $\gothic{C}\geq \delta>0$ is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group $\SO$ and propose numerical solution to this problem with consequent comparison to exact solution in the case $\gothic{C} =1$. An experiment of vessel tracking in a spherical image of the retina shows a benefit of using $\SO$ geodesics.