Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in SO(3)

Abstract

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional $\underset{0}{\overset{l}{\int }}C\left(\gamma \left(s\right)\right)\sqrt{{\xi }^2+k_g^2\left(s\right)}ds$ 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 $C\ge \delta >0$ is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group $\text{SO(3)}$ and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case $\xi >0,C\ne 1$ . For $C=1$ , we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case $C\ne 1$ (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

Keywords: Geometric control; Lie group SO(3); Spherical image; Sub-Riemannian geodesics; Vessel tracking.