Sub-Riemannian Fast Marching in SE(2)

Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

“…We introduce a numerical approach to the computation of sub-Riemannian distances and geodesics, based on solving the eikonal equations associated with a sequence of increasingly anisotropic approximate Riemannian metrics. This approach is related to [66], which however uses a different numerical scheme for the Riemannian problems, and does not establish a convergence rate. More precisely our results apply to the slightly more general class of pre-Riemannian models.…”

“…This property is referred to as non-holonomy, and models for instance a robotic system with fewer controls than degrees of freedom. A fundamental instance is the Reeds-Shepp car model, posed on the configuration space R 2 × S 1 , which can move forward and backward, rotate, but not translate sideways, see [66,28] for a numerical study with applications to image segmentation and motion planning. A variant, presented in this paper §3.2, is related to the penalization of path torsion.…”

Brain-inspired algorithms for retinal image analysis

“…This multi-orientation framework [14,15,21,31] also allows us to generically deal with crossings, as we will show with the application to vessel tracking [6,8,9,52] and segmentation [1,33,65]. Moreover, due to the neat organization of image data on the Lie-group SE(2), we are able to design effective detection algorithms [6,7], geometric feature analysis techniques such as bifurcation detection/analysis [55] and [40].…”

“…In the orientation score, vessels can also be tracked as geodesic curves [8,9,52]. Geodesics (optimal curves minimizing some curve-length functional) are found by making proper use of the curved geometry of the domain.…”

“…The algorithms for sub-Riemannian geodesic extraction are implemented via an anisotropic fast marching scheme [49,52]. As a result, the curve extraction procedure is both fast and robust.…”