Skip to main content
SpringerLink
Log in

Topological Analysis and Characterization of Discrete Scalar Fields

  • Conference paper
  • First Online:
Geometry, Morphology, and Computational Imaging

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2616))

Abstract

In this paper, we address the problem of analyzing the topology of discrete scalar fields defined on triangulated domains. To this aim, we introduce the notions of discrete gradient vector field and of Smalelike decomposition for the domain of a d-dimensional scalar field. We use such notions to extract the most relevant features representing the topology of the field. We describe a decomposition algorithm, which is independent of the dimension of the scalar field, and, based on it, methods for extracting the critical net of a scalar field. A complete classification of the critical points of a 2-dimensional field that corresponds to a piecewise differentiable field is also presented.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. K. Agoston. Algebraic Topology, A First Course. Pure and Applied Mathematics, Marcel Dekker, 1976.

    Google Scholar 

  2. C. L. Bajaj, V. Pascucci, and D. R. Shikore. Visualization of scalar topology for structural enhacement. In Proceedings of the IEEE Conference on Visualization’ 98 1998, pages 51–58, 1998.

    Google Scholar 

  3. C. L. Bajaj and D. R. Shikore. Topology preserving data simplification with error bounds. Journal on Computers and Graphics, 22(1):3–12, 1998.

    Article  Google Scholar 

  4. T. F. Banchoff. Critical Points and Curvature for Embedded Polyhedral Surfaces. Amer. Math. Monthly, 77(1):475–485, 1977.

    MathSciNet  Google Scholar 

  5. S. Biasotti, B. Falcidieno, and M. Spagnuolo. Extended reeb graphs for surface understanding and description. In Proc. 9th DGCI’2000, Upsala, LNCS 1953, Springer-Verlag, pages 185–197, 2000.

    Google Scholar 

  6. L. DeFloriani, M. M. Mesmoudi, F. Morando, and Enrico Puppo. Non-manifold decomposition in arbitrary dimension. In Proc. DGCI’2002, LNCS 2301, pages 96–80, 2002.

    Google Scholar 

  7. H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse complexes for piecewise linear 2-manifolds. In Proc 17th Sympos. Comput. Geom., pages 70–79, 2001.

    Google Scholar 

  8. R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90–145, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Gerstner and R. Pajarola. Topology preserving and controlled topology simplifying multiresolution isosurface extraction. In Proceedings IEEE Visualization 2000, pages 259–266. IEEE Computer Society, 2000.

    Google Scholar 

  10. J. C Hart. Morse theory for implicit surface modeling. In H. C. Hege and K. Poltihier (Eds), Mathematical Visualization, Springer-Verlag, pages 256–268, 1998.

    Google Scholar 

  11. J. C Hart. Using the CW-complex to represent topological structure of implicit surfaces and solids. In Proc. Implicit Surfaces 1999, Eurographics/SIGGRAPH, pages 107–112, 1999. 386, 394

    Google Scholar 

  12. C Johnson, M. Burnett, and W. Dumbar. Crystallographic topology and its applications. In Crystallographic Computing 7: Macromolecular Crystallography data, P. E. Bourne, K. D. Watenpaugh, eds., IUCr Crystallographic Symposia, Oxford University Press,, 2001.

    Google Scholar 

  13. J. Toriwaki and T. Fukumura. Extraction of structural information from grey pictures. Computer Graphics and Image Processing, 7:30–51, 1975.

    Article  MathSciNet  Google Scholar 

  14. F. Meyer. Topographic distance and watershed lines. Signal Processing, 38(1):113–125, 1994.

    Article  MATH  Google Scholar 

  15. J. Milnor. Morse Theory. Princeton University Press, 1963.

    Google Scholar 

  16. L. R. Nackman. Two-dimensional critical point conflguration graph. IEEE Transactions on Pattern Analysisand Machine Intelligence, PAMI-6(4):442–450, 1984.

    Article  Google Scholar 

  17. T. K. Peucker and E. G. Douglas. Detection of surface-specific points by local paprallel processing of discrete terrain elevation data. Graphics Image Processing, 4:475–387, 1975.

    Google Scholar 

  18. S. Smale. Morse inequalities for a dynamical system. Bulletin of American Mathematical Society, 66:43–49, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Thom. Sur une partition en cellule associées a une fonction sur une variété. C.R.A.S., 228:973–975, 1949.

    MATH  MathSciNet  Google Scholar 

  20. L. T. Watson, T. J. Laffey, and R. M. Haralick. Topographic classification of digital image intensity surfaces using generalised splines and the discrete cosine transformation. Computer Vision, Graphics and Image Processing, 29:143–167, 1985.

    Article  Google Scholar 

  21. G. H. Weber, G Scheuermann, H. Hagen, and B. Hamann. Exploring scalar fields usign criticla isovalues. In Proceedings IEEE Visualization 2002, pages 171–178. IEEE Computer Society Press, 2002.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Information Science (DISI), University of Genova, Via Dodecaneso, 35 -16146, Genova, Italy

    Emanuele Danovaro, Leila De Floriani & Mohammed Mostefa Mesmoudi

Authors
  1. Emanuele Danovaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leila De Floriani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammed Mostefa Mesmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. JAIST, School of Information Science, 1-1 Asahidai, Tatsunokuchi, 923-1202, Ishikawa, Japan

    Tetsuo Asano

  2. Computer Science Dept. and CITR, University of Auckland, Tamaki Campus, Glen Innes, 1005, Auckland, New Zealand

    Reinhard Klette

  3. LSIIT UMR 7005 CNRS-ULP, Parc d’Innovation, Boulevard Sebastien Brant, BP 10413, 67412, Illkirch, France

    Chrisitan Ronse

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Danovaro, E., De Floriani, L., Mesmoudi, M.M. (2003). Topological Analysis and Characterization of Discrete Scalar Fields. In: Asano, T., Klette, R., Ronse, C. (eds) Geometry, Morphology, and Computational Imaging. Lecture Notes in Computer Science, vol 2616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36586-9_25

Download citation

  • DOI: https://doi.org/10.1007/3-540-36586-9_25

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00916-0

  • Online ISBN: 978-3-540-36586-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation