Vessel Tracking via Sub-Riemannian Geodesics on $\mathbb{R}^2 \times P^{1}$

We study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle $\mathbb{R}^{2} \times P^{1}$, with $P^{1}=S^{1}/_{\sim}$ with identification of antipodal points. It extends previous cortical models for contour perception on $\mathbb{R}^{2} \times P^{1}$ to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cut-locus. Furthermore, a comparison of the cusp-surface in $\mathbb{R}^{2} \times P^{1}$ to its counterpart in $\mathbb{R}^{2} \times S^{1}$ of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including $\mathbb{R}^2 \times P^{1}$ instead of $\mathbb{R}^{2} \times S^{1}$ in the wavefront propagation is reduction of computational time.