Vessel tracking via sub-riemannian geodesics on the projective line bundle

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 R² × P¹, with P¹ = S¹/~ with identification of antipodal points. It extends previous cortical models for contour perception on R² × P¹ 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 R² × P¹ to its counterpart in R² × S¹ 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 R² × P¹ instead of R² × S¹ in the wavefront propagation is reduction of computational time.