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We study a data-driven sub-Riemannian (SR) curve optimization model forconnecting local orientations in orientation lifts of images. Our model liveson the projective line bundle $\mathbb{R}^{2} \times P^{1}$, with$P^{1}=S^{1}/_{\sim}$ with identification of antipodal points. It extendsprevious cortical models for contour perception on $\mathbb{R}^{2} \timesP^{1}$ to the data-driven case. We provide a complete (mainly numerical)analysis of the dynamics of the 1st Maxwell-set with growing radii ofSR-spheres, revealing the cut-locus. Furthermore, a comparison of thecusp-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 strongreduction of cusps in spatial projections of geodesics. Numerical solutions ofthe model are obtained by a single wavefront propagation method relying on asimple extension of existing anisotropic fast-marching or iterativemorphological scale space methods. Experiments show that the projective linebundle structure greatly reduces the presence of cusps. Another advantage ofincluding $\mathbb{R}^2 \times P^{1}$ instead of $\mathbb{R}^{2} \times S^{1}$in the wavefront propagation is reduction of computational time.

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