Fast-Marching Methods for Curvature Penalized Shortest Paths

Abstract

We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds-Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton-Jacobi-Bellman PDEs, posed on the three dimensional domain R 2 × S 1. Our discretizations involve sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the Fast-Marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.

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

“…The three dimensional improved estimate is established in [50], see Theorem 4.11, using dimension specific techniques.…”

“…For these reasons it is simple to implement, fast to solve numerically independently of the problem instance, and has a wide application scope and generalization potential, see e.g. [50] for a variant devoted to the global optimization of path energies involving curvature.…”

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

“…where "d" denotes the differentiation operator. We refer to [2] for this theory, including the concept of discontinuous solutions to eikonal equations, which is required for some of our problem instances [39] but is out of the scope of this paper. The eikonal PDE (3) involves the dual metric F * : Ω × E * → [0, ∞[, defined by F * p (p) := sup{ p,ṗ ;ṗ ∈ E, F p (ṗ) ≤ 1},…”

“…In the case of isotropic metrics, the first task is essentially a solved problem [51,61], however it becomes challenging when considering (strongly) anisotropic metrics [29,58,8,1,35,36]. For that purpose, we rely on an original Eulerian and causal discretization of the eikonal equation [40,39]. Two robust geodesic backtracking methods are provided for minimal path extraction.…”

Automatic Differentiation of Non-holonomic Fast Marching for Computing Most Threatening Trajectories Under Sensors Surveillance

“…We design weights c ξ (x, y), x, y ∈ X such that for any tangent vectorẋ at x one has Figure 1. Their construction exploits the additive structure of the discretization grid X and relies on techniques from lattice geometry [14], see [6,10,11] for details. The generalized eikonal PDE H ξ (x, ∇ x u ξ (x)) = 1/2 is discretized as…”