Data-driven sub-Riemannian geodesics in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group SE(2)=R2¿S1 with a metric tensor depending on a smooth external cost C:SE(2)¿[d,1] , d>0 , computed from image data. The method consists of a first step where geodesically equidistant surfaces are computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin’s Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. We show that our method produces geodesically equidistant surfaces. For C=1 we show that our method produces the global minimizers, and comparison with exact solutions shows a remarkable accuracy of the SR-spheres/geodesics. Finally, trackings in synthetic and retinal images show the potential of including the SR-geometry. Keywords: Roto-translation group; Hamilton-Jacobi equations; Vessel tracking; Sub-riemannian geometry; Morphological scale spaces