A PDE approach to data-driven sub-Riemannian geodesics in SE(2)

E.J. Bekkers, R. Duits, A. Mashtakov, G.R. Sanguinetti

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31 Citations (Scopus)
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We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, 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 $\mathcal{C}=1$ 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 $\mathcal{C}=1$. 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.
Original languageEnglish
Pages (from-to)2740-2770
Number of pages31
JournalSIAM Journal on Imaging Sciences
Issue number4
Publication statusPublished - 1 Dec 2015


  • roto-translation group
  • Hamilton-Jacobi equations
  • vessel tracking
  • sub-Riemannian geometry
  • morphological scale spaces


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