Extended Reeb Graphs for Surface Understanding and Description

  • Silvia Biasotti
  • Bianca Falcidieno
  • Michela Spagnuolo
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)

Abstract

The aim of this paper is to describe a conceptual model for surface representation based on topological coding, which de.nes a sketch of a surface usable for classi.cation or compression purposes. Theoretical approaches based on di.erential topology and geometry have been used for surface coding, for example Morse theory and Reeb graphs. To use these approaches in discrete geometry, it is necessary to adapt concepts developed for smooth manifolds to discrete surface models, as for example piecewise linear approximations. A typical problem is represented by degenerate critical points, that is non-isolated critical points such as plateaux and .at areas of the surface. Methods proposed in literature either do not consider the problem or propose local adjustments of the surface, which solve the theoretical problem but may lead to a wrong interpretation of the shape by introducing artefacts, which do not correspond to any shape feature. In this paper, an Extended Reeb Graph representation (ERG) is proposed, which can handle degenerate critical points, and an algorithm is presented for its construction.

Keywords

Critical Area Shape Description Digital Terrain Modelling Height Function Morse Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Download to read the full conference paper text

References

  1. 1.
    B. Falcidieno, M. Spagnuolo. Shape Abstraction Paradigm for Modelling Geometry and Semantics. In Proceedings of Computer Graphics International, pp. 646–656, Hannover, June 1998. 185, 186Google Scholar
  2. 2.
    W. D’Arcy Thompson. On growth and form. MA: University Press, Cambridge, second edition, 1942 185MATHGoogle Scholar
  3. 3.
    P. Pentland. Perceptual organization and representation of natural form. Artificial Intelligence, Vol.28, pp. 293–331, 1986. 185Google Scholar
  4. 4.
    L. R. Nackman. Two-dimensional Critical Point Configuration Graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-6, No. 4, p. 442–450, 1984. 186MATHCrossRefGoogle Scholar
  5. 5.
    Y. Shinagawa, T. L. Kunii, Y. L. Kergosien. Surface Coding Based on Morse Theory. IEEE Computer Graphics & Applications, pp 66–78, September 1991. 186, 187, 189Google Scholar
  6. 6.
    J. L. Pfaltz. Surface Networks. Geographical Analysis, Vol. 8, pp. 77–93, 1990. 186CrossRefGoogle Scholar
  7. 7.
    C. Bajaj, D. R. Schikore. Topology preserving data simplification with error bounds. Computer & Graphics, 22(1), pp. 3–12, 1998. 186, 189CrossRefGoogle Scholar
  8. 8.
    Y. Shinagawa, T. L. Kunii. Constructing a Reeb graph automatically from cross sections. IEEE Computer Graphics and Applications, 11(6), pp 44–51, 1991. 186, 189CrossRefGoogle Scholar
  9. 9.
    S. Takahashi, T. Ikeda, Y. Shinagawa, T. L. Kunii, M. Ueda. Algorithms for Extracting Correct Critical Points and Construction Topological Graphs from Discrete geographical Elevation Data. Eurographics’ 95, Vol. 14, Number 3, 1995. 186, 189Google Scholar
  10. 10.
    A. Fomenko. Visual Geometry and Topology. Springer-Verlag, 1994. 186Google Scholar
  11. 11.
    V. Guillemin, A. Pollack. Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974. 186Google Scholar
  12. 12.
    J. Milnor. Morse Theory. Princeton University Press, New Jersey, 1963. 186MATHGoogle Scholar
  13. 13.
    G. Reeb. Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numérique. Comptes Rendus Acad. Sciences, Paris, 222:847–849, 1946. 187MATHMathSciNetGoogle Scholar
  14. 14.
    S. Biasotti. Rappresentazione di superfici naturali mediante grafi di Reeb. Thesis for the Laurea Degree, Department of Mathematics, University of Genova, September 1998. 189, 191Google Scholar
  15. 15.
    G. Aumann, H. Ebner, L. Tang. Automatic derivation of skeleton lines from digitized contours. ISPRS Journal of Photogrammetry and Remote Sensing, 46, pp. 259–268, 1991. 190, 192CrossRefGoogle Scholar
  16. 16.
    M. De Martino, M. Ferrino. An example of automated shape analysis to solve human perception problems in anthropology. International Journal of Shape Modeling, Vol. 2 No. 1, pp 69–84, 1996. 190, 192MATHCrossRefGoogle Scholar
  17. 17.
    C. Pizzi, M. Spagnuolo. Individuazione di elementi morfologici da curve di livello. Technical Report IMA, No. 1/98, 1998. 190, 191, 192Google Scholar
  18. 18.
    S. Biasotti, M. Mortara, M. Spagnuolo. Surface Compression and Reconstruction using Reeb graphs and Shape Analysis. Spring Conference on Computer Graphics, Bratislava, May 2000. 194, 196Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Silvia Biasotti
    • 1
  • Bianca Falcidieno
    • 1
  • Michela Spagnuolo
    • 1
  1. 1.Istituto per la Matematica ApplicataConsiglio Nazionale delle RicercheItaly

Personalised recommendations