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Discrete & Computational Geometry

, Volume 53, Issue 3, pp 569–586 | Cite as

On the Reconstruction of Convex Sets from Random Normal Measurements

  • Hiba Abdallah
  • Quentin Mérigot
Article

Abstract

We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error \(\eta \), we provide upper bounds on the number of probes that one has to perform in order to obtain an \(\eta \)-approximation of this convex set with high probability. Our result relies on the stability theory related to Minkowski’s theorem.

Keywords

Surface reconstruction Minkowski problem Surface area measure 

Mathematics Subject Classification

52A27 52A39 

Notes

Acknowledgments

The authors would like to acknowledge the support of a grant from Université de Grenoble (MSTIC GEONOR) and a Grant from the French Agence Nationale de la Recherche (Optiform, ANR-12-BS01-0007). The authors would also like to thank the members of the associated team “Géométrie et Capteurs” between Laboratoire Jean Kuntzmann and CEA-Leti, and especially Luc Biard, for bringing up this problem to their attention.

References

  1. 1.
    Alexandrov, A.: On the theory of mixed volumes of convex bodies. Mat. Sb. 3(45), 227–251 (1938)Google Scholar
  2. 2.
    Bronshtein, E.M.: \(\varepsilon \)-Entropy of convex sets and functions. Mat. Sb. 17(3), 508–514 (1976)Google Scholar
  3. 3.
    Carlier, G.: On a theorem of Alexandrov. J. Nonlinear Convex Anal. 5(1), 49–58 (2004)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cheng, S.Y., Yau, S.T.: On the regularity of the solution of the \(n\)-dimensional Minkowski problem. Commun. Pure Appl. Math. 29(5), 495–516 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Devroye, L.: The equivalence of weak, strong and complete convergence in \({\rm L}_{1}\) for kernel density estimates. Ann. Stat. 11, 896–904 (1983)Google Scholar
  6. 6.
    Diskant, V.I.: Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference. Sib. Math. J. 13(4), 529–532 (1972)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Diskant, V.I.: Stability of the solution of the Minkowski equation. Sib. Math. J. 14(3), 466–469 (1973)CrossRefGoogle Scholar
  8. 8.
    Dudley, R.: Real Analysis and Probability, vol. 74. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gardner, R.J., Kiderlen, M., Milanfar, P.: Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Stat. 34, 1331–1374 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gritzmann, P., Hufnagel, A.: On the algorithmic complexity of Minkowski’s reconstruction theorem. J. Lond. Math. Soc. 59(3), 1081–1100 (1999)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Guan, P.: Monge–Ampere equations and related topics (1998). Course notesGoogle Scholar
  12. 12.
    Hug, D., Schneider, R.: Stability results involving surface area measures of convex bodies. Rend. Circ. Mat. Palermo 2(70), 21–51 (2002)MathSciNetGoogle Scholar
  13. 13.
    Lachand-Robert, T., Oudet, É.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16(2), 368–379 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Little, J.: Extended Gaussian images, mixed volumes, shape reconstruction. In: Proceedings of Symposium on Computational Geometry, pp. 15–23. ACM (1985)Google Scholar
  15. 15.
    Minkowski, H.: Volumen und oberfläche. Math. Ann. 57(4), 447–495 (1903)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sprynski, N., Szafran, N., Lacolle, B., Biard, L.: Surface reconstruction via geodesic interpolation. Comput.-Aided Des. 40(4), 480–492 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, LJKGrenobleFrance
  2. 2.CNRS, LJKGrenobleFrance

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