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

TL;DR

This paper develops a method for detecting salient lines in spherical images by leveraging sub-Riemannian geodesics on SO(3), extending existing models to handle more general cost functions and applying it to vessel tracking in retinal images.

Contribution

The authors extend a contour perception model to general cost functions and spherical geometry, deriving explicit SR geodesics and implementing a fast marching method for data-driven applications.

Findings

Derived explicit SR geodesics for the model.

Evaluated cusp times for different parameters.

Applied the method to retinal vessel tracking.

Abstract

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional $\int \limits_0^l C(\gamma(s)) \sqrt{\xi^2 + k_g^2(s)} \, {\rm d}s$ 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 $C\geq \delta>0$ is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group $SO(3)$ 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 $\xi > 0$, $C \neq 1$. 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 $C \neq 1$ (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.