Vessel Tracking via SubRiemannian Geodesics on $\mathbb{R}^2 \times P^{1}$
Abstract
We study a datadriven subRiemannian (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 datadriven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwellset with growing radii of SRspheres, revealing the cutlocus. Furthermore, a comparison of the cuspsurface 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 fastmarching 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.04192
 arXiv:
 arXiv:1704.04192
 Bibcode:
 2017arXiv170404192B
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 8 pages, 4 figures, Submitted to GSI2017