Automatic Differentiation of Non-holonomic Fast Marching for Computing Most Threatening Trajectories Under Sensors Surveillance

  • Jean-Marie MirebeauEmail author
  • Johann Dreo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10589)


We consider a two player game, where a first player has to install a surveillance system within an admissible region. The second player needs to enter the monitored area, visit a target region, and then leave the area, while minimizing his overall probability of detection. Both players know the target region, and the second player knows the surveillance installation details. Optimal trajectories for the second player are computed using a recently developed variant of the fast marching algorithm, which takes into account curvature constraints modeling the second player vehicle maneuverability. The surveillance system optimization leverages a reverse-mode semi-automatic differentiation procedure, estimating the gradient of the value function related to the sensor location in time \({\mathcal O}(N \ln N)\).


Anisotropic fast-marching Motion planning Sensors placement Game theory Optimization 


  1. 1.
    Barbaresco, F., Monnier, B.: Minimal geodesics bundles by active contours: radar application for computation of most threathening trajectories areas & corridors. In: 10th European Signal Processing Conference, Tampere, pp. 1–4 (2000)Google Scholar
  2. 2.
    Barbaresco, F.: Computation of most threatening radar trajectories areas and corridors based on fast-marching & Level Sets. In: IEEE Symposium on Computational Intelligence for Security and Defense Applications (CISDA), Paris, pp. 51–58 (2011)Google Scholar
  3. 3.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Bikhauser, Bosto (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bekkers, E.J., Duits, R., Mashtakov, A., Sanguinetti, G.R.: Data-driven sub-riemannian geodesics in SE(2). In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 613–625. Springer, Cham (2015). doi: 10.1007/978-3-319-18461-6_49 Google Scholar
  5. 5.
    Benmansour, F., Carlier, G., Peyré, G., Santambrogio, F.: Derivatives with respect to metrics and applications: subgradient marching algorithm. Numer. Math. 116(3), 357–381 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Benmansour, F., Cohen, L.: Tubular structure segmentation based on minimal path method and anisotropic enhancement. Int. J. Comput. Vis. 92(2), 192–210 (2011)CrossRefGoogle Scholar
  7. 7.
    Chen, D., Mirebeau, J.-M., Cohen, L.D.: A new finsler minimal path model with curvature penalization for image segmentation and closed contour detection. In: Proceedings of CVPR 2016, Las Vegas, USA (2016)Google Scholar
  8. 8.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Amer. J. Math. 79, 497–516 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Duits, R., Meesters, S.P.L., Mirebeau, J.-M., Portegies, J.M.: Optimal Paths for Variants of the 2D and 3D Reeds-Shepp. Car with Applications in Image Analysis (Preprint available on arXiv)Google Scholar
  10. 10.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Society for Industrial and Applied Mathematics, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mirebeau, J.-M.: Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction. SIAM J. Numer. Anal. 52(4), 1573–1599 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Mirebeau, J.-M.: Anisotropic fast-marching on cartesian grids using Voronois first reduction of quadratic forms (preprint available on HAL)Google Scholar
  13. 13.
    Mirebeau, J.-M.: Fast Marching methods for Curvature Penalized Shortest Paths (preprint available on HAL)Google Scholar
  14. 14.
    Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29(3), 867–884 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Sethian, J., Vladimirsky, A.: Ordered upwind methods for static Hamilton-Jacobi equations. Proc. Natl. Acad. Sci. 98(20), 11069–11074 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Schürmann, A.: Computational geometry of positive definite quadratic forms, University Lecture Series (2009)Google Scholar
  17. 17.
    Strode, C.; Optimising multistatic sensor locations using path planning and game theory. In: IEEE Symposium on Computational Intelligence for Security and Defense Applications (CISDA), Paris, pp. 9–16 (2011)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University Paris-Sud, CNRS, University Paris-SaclayOrsayFrance
  2. 2.THALES Research & TechnologyPalaiseauFrance

Personalised recommendations