  We present a PDE-based approach for finding optimal paths for the Reeds-Shepp
car. In our model we minimize a (data-driven) functional involving both
curvature and length penalization, with several generalizations. Our approach
encompasses the two and three dimensional variants of this model, state
dependent costs, and moreover, the possibility of removing the reverse gear of
the vehicle. We prove both global and local controllability results of the
models.
  Via eikonal equations on the manifold $\mathbb{R}^d \times \mathbb{S}^{d-1}$
we compute distance maps w.r.t. highly anisotropic Finsler metrics, which
approximate the singular (quasi)-distances underlying the model. This is
achieved using a Fast-Marching (FM) method, building on work by Mirebeau. The
FM method is based on specific discretization stencils which are adapted to the
preferred directions of the Finsler metric and obey a generalized acuteness
property. The shortest paths can be found with a gradient descent method on the
distance map, which we formalize in a theorem. We justify the use of our
approximating metrics by proving convergence results.
  Our curve optimization model in $\mathbb{R}^{d} \times \mathbb{S}^{d-1}$ with
data-driven cost allows to extract complex tubular structures from medical
images, e.g. crossings, and incomplete data due to occlusions or low contrast.
Our work extends the results of Sanguinetti et al. on numerical sub-Riemannian
eikonal equations and the Reeds-Shepp Car to 3D, with comparisons to exact
solutions by Duits et al.
  Numerical experiments show the high potential of our method in two
applications: vessel tracking in retinal images for the case $d=2$, and brain
connectivity measures from diffusion weighted MRI-data for the case $d=3$,
extending the work of Bekkers et al. We demonstrate how the new model without
reverse gear better handles bifurcations.
