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

@article{Duits2018OptimalPF, title={Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis}, author={Remco Duits and Stephan P. L. Meesters and Jean-Marie Mirebeau and Jorg M. Portegies}, journal={Journal of Mathematical Imaging and Vision}, year={2018}, volume={60}, pages={816 - 848} }

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…

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## References

SHOWING 1-10 OF 77 REFERENCES

Fast-Marching Methods for Curvature Penalized Shortest Paths

- Computer ScienceJournal of Mathematical Imaging and Vision
- 2017

This work design monotone and causal discretizations of the associated Hamilton–Jacobi–Bellman PDEs, posed on the three-dimensional domain R2×S1, using sparse, adaptive and anisotropic stencils on a cartesian grid built using techniques from lattice geometry.

Efficient fast marching with Finsler metrics

- Computer ScienceNumerische Mathematik
- 2014

A new algorithm, fast marching using anisotropic stencil refinement (FM-ASR), which addresses the discretization of the escape time problem on a two dimensional domain discretized on a cartesian grid, using local stencils produced by arithmetic means.

Existence of planar curves minimizing length and curvature

- Mathematics
- 2009

AbstractWe consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $$
\smallint \sqrt {1 + K_\gamma ^2 ds}
$$
, depending both on the length…

Curve cuspless reconstruction via sub-Riemannian geometry

- Mathematics
- 2012

We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the…

Sub-Riemannian Fast Marching in SE(2)

- Computer ScienceCIARP
- 2015

A Fast Marching based implementation for computing sub-Riemanninan (SR) geodesics in the roto-translation group SE(2), with a metric depending on a cost induced by the image data, using a Riemannian approximation of the SR-metric.

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in SE(2)

- MathematicsSIAM J. Imaging Sci.
- 2015

A new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group with a metric tensor depending on a smooth external cost computed from image data is presented.

Global Minimum for a Finsler Elastica Minimal Path Approach

- Computer ScienceInternational Journal of Computer Vision
- 2016

This paper proposes a novel curvature penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve that embeds a curvature penalty and introduces two anisotropic image data-driven speed functions that are computed by steerable filters.

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

- Mathematics
- 2013

We consider the problem Pcurve of minimizing ∫0Lξ2+κ2(s)ds$\int \limits _{0}^{L} \sqrt {\xi ^{2} + \kappa ^{2}(s)} \, \mathrm {d}s$ for a curve x in ℝ3$\mathbb {R}^{3}$ with fixed boundary points and…

Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle

- MathematicsGSI
- 2017

Experiments show that the projective line bundle structure greatly reduces the presence of cusps in spatial projections of geodesics, and extends previous cortical models for contour perception on \(\mathbb {R}^{2} \times P^{1}\) to the data-driven case.

Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2)

- MathematicsJournal of Mathematical Imaging and Vision
- 2013

S-parametrization simplifies the exponential map, the curvature formulas, the cusp-surface, and the boundary value problem, and shows that sub-Riemannian geodesics solve Petitot’s circle bundle model.