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

We present a new flexible wavefront propagation algorithm for the boundary value problem for subRiemannian (SR) geodesics in the roto-translation group SE(2) = R S with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton–Jacobi–Bellman system derived via Pontryagin’s maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For C = 1 we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case C = 1. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.