  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.
