Journal of Mathematical Imaging and Vision

, Volume 51, Issue 3, pp 451–464 | Cite as


  • Pedro Duarte
  • Maria Joana TorresEmail author


We provide a characterization of \(r\)-regular sets in terms of the Lipschitz regularity of normal vector fields to the boundary.


\(r\)-regularity Normal vector field \(C^1\)-boundary Lipschitz projection Euclidean distance 

Mathematics Subject Classification

53B21 65D18 



Both authors would like to thank Armando P. Machado and Alessandro Margheri for valuable suggestions on the problems related to the Sturm–Liouville section. The first author was supported by “Fundação para a Ciência e a Tecnologia” through the Program POCI 2010 and the Project “Randomness in Deterministic Dynamical Systems and Applications” (PTDC-MAT-105448-2008). The second author was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Fundação para a Ciência e a Tecnologia”, through the Project PEstOE/MAT/UI0013/2014.


  1. 1.
    Köthe, U., Stelldinger, P.: Shape preservation digitization of ideal and blurred binary images. In: Proceedings of the Discrete Geometry for Computer Imagery, pp. 82–91 (2003).Google Scholar
  2. 2.
    Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8(2), 131–159 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press, Rockville, MD (1982)CrossRefGoogle Scholar
  4. 4.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Waltham (1982)zbMATHGoogle Scholar
  5. 5.
    Stelldinger, P.: Image Digitization and its Influence on Shape Properties in Finite Dimensions. IOS Press, Germany (2007)Google Scholar
  6. 6.
    Stelldinger, P., Köthe, U.: Shape preservation during digitization: tight bounds based on the morphing distance. In: Proceedings of the Symposium Deutsche Arbeitsgemeinschaft für Mustererkennung, pp. 108–115 (2003).Google Scholar
  7. 7.
    Stelldinger, P., Latecki, L.J., Siqueira, M.: Topological equivalence between a 3D Object and the reconstruction of its digital image. IEEE Trans. Pattern Anal. Mach. Intell. 29(1), 126–140 (2007)CrossRefGoogle Scholar
  8. 8.
    Meine, H., Köthe, U., Stelldinger, P.: A topological sampling theorem for robust boundary reconstruction and image segmentation. Discrete Appl. Math. 157(3), 524–541 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Stelldinger, P.: Topological correct surface reconstruction using alpha shapes and relations to ball-pivoting. In: Proceedings of the International Conference on Pattern Recognition (2008).Google Scholar
  10. 10.
    Stelldinger, P., Tcherniavski, L.: Provably correct reconstruction of surfaces from sparse noisy samples. Pattern Recogn. 42(8), 1650–1659 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Duarte, P., Torres, M.J.: Combinatorial stability of non-deterministic systems. Erg. Theory Dyn. Syst. 26, 93–128 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Duarte, P., Torres, M.J.: Smoothness of boundaries of regular sets. J. Math. Imaging Vis. 48(1), 106–113 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Spivak, M.: Calculus On Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Westview Press, Boulder (1971)Google Scholar
  14. 14.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  15. 15.
    Attali, D.: r-Regular shape reconstruction from unorganized points. Comput. Geom.: Theory Appl. 10, 239–247 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Bernardini, F., Bajaj, C.L.: Sampling and reconstructing manifolds using \(\alpha \)-shapes. In: Proceedings of the 9th Canadian Conference on Computational Geometry, pp. 193–198 (1997).Google Scholar
  17. 17.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001).Google Scholar
  18. 18.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel (1990).Google Scholar
  19. 19.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1986)Google Scholar
  20. 20.
    Hinton, D.: Sturm’s 1836 oscillation results evolution of the theory. Sturm-Liouville theory, pp. 1–27. Birkhäuser, Basel (2005).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CMAF, Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisboaPortugal
  2. 2.CMAT, Departamento de Matemática e AplicaçõesUniversidade do MinhoBragaPortugal

Personalised recommendations