Sub-Riemannian geodesics in SO(3) with application to vessel tracking in spherical images of retina

Abstract

In order to detect vessel locations in spherical images of retina we consider the problem of minimizing the functional $$\int\limits_0^l {\mathfrak{C}\left( {\gamma \left( s \right)} \right)\sqrt {{\xi ^2} + k_g^2\left( s \right)} ds}$$ for a curve γ 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 γ. Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and propose numerical solution to this problem with consequent comparison to exact solution in the case C = 1. An experiment of vessel tracking in a spherical image of the retina shows a benefit of using SO(3) geodesics.