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

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

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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)=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
Original languageEnglish
Title of host publicationScale Space and Variational Methods in Computer Vision (5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31-June 4, 2015, Proceedings)
EditorsJ.-F. Aujol, M. Nikolova, N. Papadakis
PublisherSpringer
Pages613-625
ISBN (Print)978-3-319-18460-9
DOIs
Publication statusPublished - 2015

Publication series

NameLecture Notes in Computer Science
Volume9087
ISSN (Print)0302-9743

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Bekkers, E. J., Duits, R., Mashtakov, A., & Sanguinetti, G. R. (2015). Data-driven sub-Riemannian geodesics in SE(2). In J-F. Aujol, M. Nikolova, & N. Papadakis (Eds.), Scale Space and Variational Methods in Computer Vision (5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31-June 4, 2015, Proceedings) (pp. 613-625). (Lecture Notes in Computer Science; Vol. 9087). Springer. https://doi.org/10.1007/978-3-319-18461-6_49