  We introduce a data-driven version of the plus Cartan connection on the
homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate
a theorem that describes all shortest and straight curves (parallel velocity
and parallel momentum, respectively) with respect to this new data-driven
connection and corresponding Riemannian manifold. Then we use these shortest
curves for geodesic tracking of complex vasculature in multi-orientation image
representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection
characterizes the Hamiltonian flow of all geodesics. It also allows for
improved adaptation to curvature and misalignment of the (lifted) vessel
structure that we track via globally optimal geodesics. We compute these
geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$
that we compute by a new modified anisotropic fast-marching method.
  Our experiments range from tracking single blood vessels with fixed endpoints
to tracking complete vascular trees in retinal images. Single vessel tracking
is performed in a single run in the multi-orientation image representation,
where we project the resulting geodesics back onto the underlying image. The
complete vascular tree tracking requires only two runs and avoids prior
segmentation, placement of extra anchor points, and dynamic switching between
geodesic models.
  Altogether we provide a geodesic tracking method using a single, flexible,
transparent, data-driven geodesic model providing globally optimal curves which
correctly follow highly complex vascular structures in retinal images.
  All experiments in this article can be reproduced via documented Mathematica
notebooks available at GitHub
(https://github.com/NickyvdBerg/DataDrivenTracking).
