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Data-Driven Sub-Riemannian Geodesics in SE(2)

  • E. J. Bekkers
  • R. DuitsEmail author
  • A. Mashtakov
  • G. R. SanguinettiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

Abstract

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 \(\mathcal {C}:SE(2) \rightarrow [\delta ,1]\), \(\delta >0\), 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 \(\mathcal {C}=1\) 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 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Biomedical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

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