Skip to main content
Log in

Cite this article


We prove that the boundary of an r-regular set is a codimension one manifold of class C 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. Attali, D.: r-regular shape reconstruction from unorganized points. Comput. Geom. Theory Appl. 10, 239–247 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bernardini, F., Bajaj, C.L.: Sampling and reconstructing manifolds using α-shapes. In: Proc. 9th Canadian Conf. Comput. Geom., pp. 193–198 (1997)

    Google Scholar 

  3. Duarte, P., Torres, M.J.: Combinatorial stability of non-deterministic systems. Ergod. Theory Dyn. Syst. 26, 93–128 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  5. Holmes, R.B.: Smoothness of certain metric projections on Hilbert space. Trans. Am. Math. Soc. 184, 87–100 (1973)

    Article  Google Scholar 

  6. Holmes, R.B.: A Course on Optimization and Best Approximation. Lecture Notes in Mathematics, vol. 257. Springer, Berlin (1972)

    MATH  Google Scholar 

  7. Köthe, U., Stelldinger, P.: Shape preservation digitization of ideal and blurred binary images. In: Proc. Discrete Geometry for Computer Imagery, pp. 82–91 (2003)

    Chapter  Google Scholar 

  8. Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8(2), 131–159 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)

    MATH  MathSciNet  Google Scholar 

  11. Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press, New York (1982)

    Book  Google Scholar 

  12. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982)

    MATH  Google Scholar 

  13. Spivak, M.: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Westview, Boulder (1971)

    Google Scholar 

  14. Stelldinger, P.: Topological correct surface reconstruction using alpha shapes and relations to ball-pivoting. In: International Conference on Pattern Recognition (2008)

    Google Scholar 

  15. Stelldinger, P.: Image Digitization and Its Influence on Shape Properties in Finite Dimensions. IOS Press, Amsterdam (2007)

    Google Scholar 

  16. Stelldinger, P., Köthe, U.: Shape preservation during digitization: tight bounds based on the morphing distance. In: Proc. Symp. Deutsche Arbeitsgemeinschaft für Mustererkennung, pp. 108–115 (2003)

    Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. Stelldinger, P., Tcherniavski, L.: Provably correct reconstruction of surfaces from sparse noisy samples. Pattern Recognit. 42(8), 1650–1659 (2009)

    Article  MATH  Google Scholar 

Download references


PD 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). MJT was partially financed by FEDER Funds through “Programa Operacional Factores de Competitividade—COMPETE” and by Portuguese Funds through FCT—“Fundação para a Ciência e a Tecnologia”, within the Project PEst-C/MAT/UI0013/2011.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Maria Joana Torres.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Duarte, P., Torres, M.J. Smoothness of Boundaries of Regular Sets. J Math Imaging Vis 48, 106–113 (2014).

Download citation

  • Published:

  • Issue Date:

  • DOI: