Inference of Curvature Using Tubular Neighborhoods

  • Frédéric Chazal
  • David Cohen-Steiner
  • André Lieutier
  • Quentin Mérigot
  • Boris Thibert
Part of the Lecture Notes in Mathematics book series (LNM, volume 2184)


Geometric inference deals with the problem of recovering the geometry and topology of a compact subset K of \(\mathbb{R}^{d}\) from an approximation by a finite set P. This problem has seen several important developments in the previous decade. Many of the proposed constructions share a common feature: they estimate the geometry of the underlying compact set K using offsets of P, that is r-sublevel set of the distance function to P. These offset correspond to what is called tubular neighborhoods in differential geometry. First and second-order geometric quantities are encoded in the tube K r around a manifold. For instance, the classical tube formula asserts that it is possible to estimate the curvature of a compact smooth submanifold K from the volume of its offsets. One can hope that if the finite set P is close to K in the Hausdorff sense, some of this geometric information remains in the offsets of P. In this chapter, we will see how this idea can be used to infer generalized notions of curvature such as Federer’s curvature measures.


  1. 1.
    Alliez, P., Cohen-Steiner, D., Tong, Y., Desbrun, M.: Voronoi-based variational reconstruction of unoriented point sets. In: Proceedings of the Eurographics Symposium on Geometry Processing, vol. 7, pp. 39–48 (2007)Google Scholar
  2. 2.
    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discret. Comput. Geom. 22(4), 481–504 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton-Jacobi equations, and optimal control. In: Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser Boston Inc., Boston (2004)Google Scholar
  4. 4.
    Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Shape smoothing using double offsets. In: Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, pp. 183–192. ACM (2007)Google Scholar
  5. 5.
    Chazal, F., Cohen-Steiner, D., Lieutier, A.: Normal cone approximation and offset shape isotopy. Comput. Geom. 42(6), 566–581 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in Euclidean space. Discret. Comput. Geom. 41(3), 461–479 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Stability of curvature measures. Comput. Graphics Forum 28, 1485–1496 (2009)CrossRefGoogle Scholar
  8. 8.
    Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Boundary measures for geometric inference. Found. Comput. Math. 10(2), 221–240 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cohen-Steiner, D., Morvan, J.M.: Second fundamental measure of geometric sets and local approximation of curvatures. J. Differ. Geom. 74(3), 363–394 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cuel, L., Lachaud, J.O., Mérigot, Q., Thibert, B.: Robust geometry estimation using the generalized Voronoi covariance measure. SIAM J. Imag. Sci. 8(2), 1293–1314 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Edelsbrunner, H.: The union of balls and its dual shape. In: Proceedings of the Ninth Annual Symposium on Computational Geometry, pp. 218–231. ACM (1993)Google Scholar
  12. 12.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)MATHGoogle Scholar
  14. 14.
    Fu, J.: Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52(4), 1025–1046 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hug, D., Schneider, R.: Local tensor valuations. Geom. Funct. Anal. 24(5), 1516–1564 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hug, D., Kiderlen, M., Svane, A.M.: Voronoi-based estimation of Minkowski tensors from finite point samples. Discret. Comput. Geom. 57(3), 545–570 (2017).
  17. 17.
    Lieutier, A.: Any open bounded subset of \(\mathbb{R}^{n}\) has the same homotopy type as its medial axis. Comput. Aided Geom. Des. 36(11), 1029–1046 (2004)CrossRefGoogle Scholar
  18. 18.
    Mérigot, Q.: Geometric structure detection in point clouds. Theses, Université Nice Sophia Antipolis (2009). Google Scholar
  19. 19.
    Mérigot, Q., Ovsjanikov, M., Guibas, L.: Voronoi-based curvature and feature estimation from point clouds. IEEE Trans. Vis. Comput. Graph. 17(6), 743–756 (2011)CrossRefGoogle Scholar
  20. 20.
    Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles convexes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 21, 562–567 (1935)MATHGoogle Scholar
  21. 21.
    Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. In: Surveys in Differential Geometry, vol. XI, pp. 137–201. Int. Press, Somerville (2007)Google Scholar
  22. 22.
    Rataj, J., Zähle, M.: Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differ. Geom. Appl. 19(1), 113–126 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rataj, J., Zähle, M.: General normal cycles and Lipschitz manifolds of bounded curvature. Ann. Global Anal. Geom. 27(2), 135–156 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Frédéric Chazal
    • 1
  • David Cohen-Steiner
    • 2
  • André Lieutier
    • 3
  • Quentin Mérigot
    • 4
  • Boris Thibert
    • 5
  1. 1.Inria SaclayPalaiseauFrance
  2. 2.Inria Sophia-AntipolisSophia-AntipolisFrance
  3. 3.Dassault SystèmesAix-en-ProvenceFrance
  4. 4.Laboratoire de Mathématiques d’OrsayUniversity of Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance
  5. 5.Laboratoire Jean KuntzmannUniversité Grenoble-Alpes, CNRSSaint-Martin-d’HèresFrance

Personalised recommendations