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We present a PDE-based approach for finding optimal paths for the Reeds-Sheppcar. In our model we minimize a (data-driven) functional involving bothcurvature and length penalization, with several generalizations. Our approachencompasses the two and three dimensional variants of this model, statedependent costs, and moreover, the possibility of removing the reverse gear ofthe vehicle. We prove both global and local controllability results of themodels.  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, whichapproximate the singular (quasi)-distances underlying the model. This isachieved using a Fast-Marching (FM) method, building on work by Mirebeau. TheFM method is based on specific discretization stencils which are adapted to thepreferred directions of the Finsler metric and obey a generalized acutenessproperty. The shortest paths can be found with a gradient descent method on thedistance map, which we formalize in a theorem. We justify the use of ourapproximating metrics by proving convergence results.  Our curve optimization model in $\mathbb{R}^{d} \times \mathbb{S}^{d-1}$ withdata-driven cost allows to extract complex tubular structures from medicalimages, e.g. crossings, and incomplete data due to occlusions or low contrast.Our work extends the results of Sanguinetti et al. on numerical sub-Riemannianeikonal equations and the Reeds-Shepp Car to 3D, with comparisons to exactsolutions by Duits et al.  Numerical experiments show the high potential of our method in twoapplications: vessel tracking in retinal images for the case $d=2$, and brainconnectivity measures from diffusion weighted MRI-data for the case $d=3$,extending the work of Bekkers et al. We demonstrate how the new model withoutreverse gear better handles bifurcations.

Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis