Duits R, Meesters S P L, Mirebeau J-M, Portegies J M
1CASA, Eindhoven University of Technology, Eindhoven, The Netherlands.
2University Paris-Sud, CNRS, University Paris-Saclay, 91405 Orsay, France.
J Math Imaging Vis. 2018;60(6):816-848. doi: 10.1007/s10851-018-0795-z. Epub 2018 Feb 20.
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 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 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 and brain connectivity measures from diffusion-weighted MRI data for the case , 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.
我们提出了一种基于偏微分方程的方法来寻找Reeds-Shepp汽车的最优路径。在我们的模型中，我们最小化一个（数据驱动的）泛函，该泛函涉及曲率和长度惩罚，并进行了多种推广。我们的方法涵盖了该模型的二维和三维变体、状态依赖成本，此外，还包括去除车辆倒档的可能性。我们证明了模型的全局和局部可控性结果。通过流形上的程函方程，我们计算关于高度各向异性芬斯勒度量的距离映射，该度量近似于模型所基于的奇异（准）距离。这是通过一种快速行进（FM）方法实现的，该方法基于Mirebeau（《数值数学》126(3):515 - 557，2013；《SIAM数值分析杂志》52(4):1573 - 1599，2014）的工作。FM方法基于特定的离散模板，这些模板适应于芬斯勒度量的优选方向并遵循广义锐度性质。最短路径可以通过在距离映射上的梯度下降方法找到，我们在一个定理中对其进行了形式化。我们通过证明收敛结果来证明使用我们的近似度量的合理性。我们具有数据驱动成本的曲线优化模型允许从医学图像中提取复杂的管状结构，例如交叉点以及由于遮挡或低对比度导致的不完整数据。我们的工作将Sanguinetti等人（《模式识别、图像分析、计算机视觉及应用进展》LNCS 9423，2015）关于数值次黎曼程函方程和Reeds-Shepp汽车的结果扩展到了三维，并与Duits等人（《动力与控制系统杂志》22(4):771 - 805，2016）的精确解进行了比较。数值实验表明我们的方法在两个应用中具有很高的潜力：在视网膜图像中进行血管跟踪（针对情况 ）以及从扩散加权MRI数据中进行脑连接性测量（针对情况 ），扩展了Bekkers等人（《SIAM成像科学杂志》8(4):2740 - 2770，2015）的工作。我们展示了没有倒档的新模型如何更好地处理分叉。
