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

Abstract

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.