Abstract
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 Mirebeau (Numer Math 126(3):515–557, 2013; SIAM J Numer Anal 52(4):1573–1599, 2014). 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. (Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications LNCS 9423, 2015) on numerical sub-Riemannian eikonal equations and the Reeds–Shepp car to 3D, with comparisons to exact solutions by Duits et al. (J Dyn Control Syst 22(4):771–805, 2016). 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. (SIAM J Imaging Sci 8(4):2740–2770, 2015). We demonstrate how the new model without reverse gear better handles bifurcations.
Highlights
The precise theoretical statement and proof are found in Theorem 3. – we show in Theorem 4 how the geodesics can be obtained from the distance map, for a general Finsler metric, and in the more specific cases that we use in this paper
We have extended the existing methodology for modeling and solving the problem of finding optimal paths for a Reeds– Shepp car to 3D and to a case without reverse gear
We have shown that the use of the constrained model leads to more meaningful shortest paths in some cases and that the extension to 3D has opened up the possibility for tractography in diffusion-weighted magnetic resonance imaging (dMRI) data
Summary
Reeds and Shepp consider in [50] the same problem, but for a car that does have the possibility for backward motion In both papers, the focus lies on describing and proving the general shape of the optimal paths, without giving explicit solutions for the shortest paths. The focus lies on describing and proving the general shape of the optimal paths, without giving explicit solutions for the shortest paths This can be considered a curve optimization problem in the space R2×(R/2π Z), equipped with the natural Euclidean metric but only among curves γ (t) = (x(t), y(t), θ (t)) subject to the constraint that (x(t), y(t)) is proportional to (cos θ (t), sin θ (t)). We would like to extend the Finsler metric using two data-driven factors that can vary with position and orientation This can be used to compute shortest paths for a car, where for example road conditions and obstacles are taken into account. A global minimizer of (1) is referred to as minimizing geodesic or minimizer
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