# Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in ${\text {SO(3)}}$

Journal of Mathematical Imaging and Vision, Feb 2017

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional $\int \limits _0^l \mathfrak {C}(\gamma (s)) \sqrt{\xi ^2 + k_g^2(s)} \, \mathrm{d}s$ 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 $\mathfrak {C}\ge \delta >0$ 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 $\xi > 0, \mathfrak {C} \ne 1$. For $\mathfrak {C}=1$, 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 $\mathfrak {C} \ne 1$ (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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs10851-017-0705-9.pdf

A. Mashtakov, R. Duits, Yu. Sachkov, E. J. Bekkers, I. Beschastnyi. Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in ${\text {SO(3)}}$, Journal of Mathematical Imaging and Vision, 2017, 239-264, DOI: 10.1007/s10851-017-0705-9