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

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Scale Space and Variational Methods in Computer Vision (SSVM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,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.

E.J. Bekkers, R. Duits, A. Mashtakov and G.R. Sanguinetti— Joint main Authors. The research leading to the results of this article has received funding from the European Research Council under the ECs 7th Framework Programme (FP7/2007 2014)/ERC grant agreement No. 335555 and from (FP7-PEOPLE-2013-ITN)/EU Marie-Curie ag. no. 607643.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    R. Duits & G. R. Sanguinetti

  2. Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    E. J. Bekkers, R. Duits & A. Mashtakov

Authors
  1. E. J. Bekkers
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  2. R. Duits
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  3. A. Mashtakov
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  4. G. R. Sanguinetti
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Corresponding authors

Correspondence to R. Duits or G. R. Sanguinetti .

Editor information

Editors and Affiliations

  1. University of Bordeaux, Talence, France

    Jean-François Aujol

  2. ENS Cachan, Cachan, France

    Mila Nikolova

  3. University of Bordeaux, Talence, France

    Nicolas Papadakis

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Bekkers, E.J., Duits, R., Mashtakov, A., Sanguinetti, G.R. (2015). Data-Driven Sub-Riemannian Geodesics in SE(2). In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_49

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  • DOI: https://doi.org/10.1007/978-3-319-18461-6_49

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18460-9

  • Online ISBN: 978-3-319-18461-6

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