A Variational Method for Accurate Distance Function Estimation

  • Alexander G. BelyaevEmail author
  • Pierre-Alain Fayolle
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)


Variational problems for accurate approximation of the distance from the boundary of a domain are studied. It is demonstrated that the problems can be efficiently solved by ADMM. Advantages of the proposed distance function estimation methods are demonstrated by numerical experiments.



The Gargoyle mesh model is courtesy of the AIM@SHAPE Shape Repository and the Lucy mesh model is courtesy of the Stanford Graphics Laboratory.


  1. 1.
    Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12, 1–125 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Belyaev, A., Fayolle, P.-A.: On variational and PDE-based distance function approximations. Comput. Graphics Forum 34(8), 104–118 (2015)CrossRefGoogle Scholar
  3. 3.
    Belyaev, A., Fayolle, P.-A., Pasko, A.: Signed L p-distance fields. Comput. Aided Des. 45, 523–528 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bhattacharya, T., DiBenedetto, E., Manfredi, J.: Limits as p → of Δ p u p = f and related extremal problems. Rend. Sem. Mat. Univ. Pol. Torino 47, 15–68 (1989)MathSciNetGoogle Scholar
  5. 5.
    Biswas, A., Shapiro, V., Tsukanov, I.: Heterogeneous material modeling with distance fields. Comput. Aided Geom. Des. 21, 215–242 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Butzer, P., Jongmans, F.: P. L. Chebyshev (1821–1894): a guide to his life and work. J. Approx. Theory 96, 111–138 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cadena, C., Carlone, L., Carrillo, H., Latif, Y., Scaramuzza, D., Neira, J., Reid, I., Leonard, J.J.: Past, present, and future of simultaneous localization and mapping: toward the robust-perception age. IEEE Trans. Robot. 32, 1309–1332 (2016)CrossRefGoogle Scholar
  8. 8.
    Calakli, F., Taubin, G.: SSD: smooth signed distance surface reconstruction. Comput. Graphics Forum 30(7), 1993–2002 (2011)CrossRefGoogle Scholar
  9. 9.
    Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 152:1–152:11 (2013)Google Scholar
  10. 10.
    Fayolle, P.-A., Belyaev, A.: p-Laplace diffusion for distance function estimation, optimal transport approximation, and image enhancement. Comput. Aided Geom. Des. 67, 1–20 (2018)Google Scholar
  11. 11.
    Freytag, M., Shapiro, V., Tsukanov, I.: Finite element analysis in situ. Finite Elem. Anal. Des. 47(9), 957–972 (2011)CrossRefGoogle Scholar
  12. 12.
    Gibou, F., Fedkiw, R., Osher, S.: A review of level-set methods and some recent applications. J. Comput. Phys. 353, 82–109 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. In: Fitzgibbon, W., Kuznetsov, Y.A., Neittaanmäki, P., Pironneau, O. (eds.) Modeling, Simulation and Optimization for Science and Technology, pp. 59–82. Springer, Berlin (2014)Google Scholar
  14. 14.
    Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410(1), 1–22 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lee, B., Darbon, J., Osher, S., Kang, M.: Revisiting the redistancing problem using the Hopf–Lax formula. J. Comput. Phys. 330, 268–281 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Roget, B., Sitaraman, J.: Wall distance search algorithm using voxelized marching spheres. J. Comput. Phys. 241, 76–94 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Royston, M., Pradhana, A., Lee, B., Chow, Y.T., Yin, W., Teran, J., Osher, S.: Parallel redistancing using the Hopf–Lax formula. J. Comput. Phys. 365, 7–17 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tucker, P.G.: Hybrid Hamilton-Jacobi-Poisson wall distance function model. Comput. Fluids 44(1), 130–142 (2011)CrossRefGoogle Scholar
  19. 19.
    Xia, H., Tucker, P. G.: Fast equal and biased distance fields for medial axis transform with meshing in mind. Appl. Math. Model. 35, 5804–5819 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhu, B., Skouras, M., Chen, D., Matusik, W.: Two-scale topology optimization with microstructures. ACM Trans. Graph. 36(5), 36:1–36:14 (2017)Google Scholar
  21. 21.
    Zollhöfer, M., Dai, A., Innmann, M., Wu, C., Stamminger, M., Theobalt, C., Nießner, M.: Shading-based refinement on volumetric signed distance functions. ACM Trans. Graph. 34(4), 96:1–96:14 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Sensors, Signals and Systems, School of Engineering & Physical SciencesHeriot-Watt UniversityEdinburghUK
  2. 2.Computer Graphics LaboratoryUniversity of AizuAizu-WakamatsuJapan

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