MORPHOLOGY-BASED REPRESENTATIONS OF DISCRETE SCALAR FIELDS

Mohammed Mostefa Mesmoudi, Leila De Floriani

Abstract

Forman introduced in (Forman, 1998) a theory for cell complexes that is a discrete version of the well known Morse theory. Forman theory finds several applications in digital geometry and image processing where the data to be processed are discrete, see for instance (Lewiner et al., 2002a), (Lewiner et al., 2002b). In (DeFloriani et al., 2002b), we have introduced a Smale-like decomposition of a scalar field f defined on a triangulated domain M based on a discrete gradient field that simulates well the behavior of the gradient field in the differentiable case. Here, we extend our discrete gradient vector field so that the extended form coincides with a Forman gradient field. The extended gradient field does not change the Smale-like decomposition components and, thus, inherits properties of both smooth Morse and discrete Forman functions.

References

  1. Agoston, M. (1976). Algebraic Topology, A First Course,. Pure and Applied Mathematics, Marcel Dekker.
  2. Bajaj, C. L., Pascucci, V., and Shikore, D. R. (1998). Visualization of scalar topology for structural enhacement. In Proceedings of the IEEE Conference on Visualization 7898 1998, pages 51-58.
  3. Bajaj, C. L. and Shikore, D. R. (1998). Topology preserving data simplification with error bounds. Journal on Computers and Graphics, 22(1):3-12.
  4. Danovaro, E., De Floriani, L., and Mesmoudi, M. M. (2003). Topological analysis and characterization of discrete scalar fields. In Asano, T., Klette, R., and Ronse, C., editors, Theoretical Foundations of Computer Vision, Geometry, Morphology, and Computational Imaging, volume 2616 of Lecture Notes on Computer Science, pages 386-402. Springer Verlag.
  5. DeFloriani, L., Mesmoudi, M. M., and Danovaro, E. (2002a). Extraction of Critical Points and Nets Based on Discrete Scalar Fields. In To appear In Proceed. of Eurographics conference.
  6. DeFloriani, L., Mesmoudi, M. M., and Danovaro, E. (2002b). Smale-like Decomposition for Discrete Scalar Fields. In To appear In Proceed. of Inter. Conf. on Pattern Recognition.
  7. Edelsbrunner, H., Harer, J., and Zomorodian, A. (2001). Hierarchical morse complexes for piecewise linear 2- manifolds. In Proc 17th Sympos. Comput. Geom., pages 70-79.
  8. Forman, R. (1998). Morse Theory for Cell Complexes. Advances in Mathematics, 134:90-145.
  9. J.Toriwaki and Fukumura, T. (1975). Extraction of structural information from grey pictures. Computer Graphics and Image Processing, 7:30-51.
  10. Lewiner, T., Lopes, H., and Tavares, G. (2002a). Optimal discrete morse functions for 2-manifolds. Technical report, Pontificia Universidade Catolica do Rio de Janero.
  11. Lewiner, T., Lopes, H., and Tavares, G. (2002b). Visualizing forman's discrete vector field. In H.-C. Hege, K. P. E., editor, to appear in Proceed. of Mathematical Visualization III, Springer.
  12. Milnor, J. (1963). Morse Theory. Princeton University Press.
  13. Nackman, L. R. (1984). Two-dimensional critical point configuration graph. IEEE Transactions on Pattern Analysisand Machine Intelligence, PAMI-6(4):442- 450.
  14. Peucker, T. K. and Douglas, E. G. (1975). Detection of surface-specific points by local paprallel processing of discrete terrain elevation data. Graphics Image Processing, 4:475-387.
  15. Smale, S. (1960). Morse inequalities for a dynamical system. Bulletin of American Mathematical Society, 66:43-49.
  16. Thom, R. (1949). Sur une partition en cellule associées a une fonction sur une variété. C.R.A.S., 228:973-975.
  17. Watson, L. T., Laffey, T. J., and Haralick, R. (1985). Topographic classification of digital image intensity surfaces using generalised splines and the discrete cosine transformation. Computer Vision, Graphics and Image Processing, 29:143-167.
Download


Paper Citation


in Harvard Style

Mostefa Mesmoudi M. and De Floriani L. (2007). MORPHOLOGY-BASED REPRESENTATIONS OF DISCRETE SCALAR FIELDS . In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, ISBN 978-972-8865-71-9, pages 137-144. DOI: 10.5220/0002080501370144


in Bibtex Style

@conference{grapp07,
author={Mohammed Mostefa Mesmoudi and Leila De Floriani},
title={MORPHOLOGY-BASED REPRESENTATIONS OF DISCRETE SCALAR FIELDS},
booktitle={Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},
year={2007},
pages={137-144},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002080501370144},
isbn={978-972-8865-71-9},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,
TI - MORPHOLOGY-BASED REPRESENTATIONS OF DISCRETE SCALAR FIELDS
SN - 978-972-8865-71-9
AU - Mostefa Mesmoudi M.
AU - De Floriani L.
PY - 2007
SP - 137
EP - 144
DO - 10.5220/0002080501370144