The Locality Property in Topological Irregular Graph Hierarchies

  • Helmut Kofler
  • Ernst J. Haunschmid
  • Wilfried N. Gansterer
  • Christoph W. Ueberhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1557)


Graph contraction is applied in many areas of computer science, for instance, as a subprocess in parallel graph partitioning. Parallel graph partitioning is usually implemented as a poly-algorithm intended to speed up the solving of systems of linear equations. Image analysis is another field of application for graph contraction. There regular and irregular image hierarchies are built by coarsening images.

In this paper a general structure of (multilevel) graph contraction is given. The graphs of these coarsening processes are given a topological structure which allows to use concepts like the neighborhood, the interior and the boundary of sets in a well-defined manner. It is shown in this paper that the various coarsenings used in practice are continuous and therefore local processes. This fact enables the efficient parallelization of these algorithms. This paper also demonstrates that the efficient parallel implementations which already exist for multilevel partitioning algorithms can easily be applied to general image hierarchies.


Topological Structure Dual Graph Graph Partitioning Random Match Image Pyramid 
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.


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  1. 1.
    Ahronovitz, E., Aubert, J.-P., Fiorio, Chr.: The Star-Topology: A Topology for Image Analysis. 5th International Workshop, DGCI’95 (1995)Google Scholar
  2. 2.
    Alexandroff, P., Hopf, H.: Topologie, Erster Band. Springer-Verlag, Berlin (1935)Google Scholar
  3. 3.
    Eckhardt, U., Hundt, E.: Topological Approach to Mathematical Morphology. preprint (1997)Google Scholar
  4. 4.
    Gupta, A.: Fast and Effective Algorithms for Graph Partitioning and Sparse-Matrix Ordering. IBM, Journal of Research & Development (1997)Google Scholar
  5. 5.
    Heijmans, H.: Morphological Image Operators. Academic Press (1994)Google Scholar
  6. 6.
    Karypis, G., Kumar, V.: A Coarse-Grain Parallel Formulation of Multilevel k-Way Partitioning Algorithm. Proceedings of the 8th SIAM conf. on Parallel Processing for Scientific Computing (1997)Google Scholar
  7. 7.
    Karypis, G., Kumar, V.: METIS, a Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, version 3.0.3. METIS, Minnesota (1997)Google Scholar
  8. 8.
    Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer Graphics and Connected Topologies on Finite Ordered Sets. Topology Appl. 36 (1980)Google Scholar
  9. 9.
    Kofler, H.: Irregular Graph Hierarchies Equipped with a Topological Structure. 14th Intern. Conference ICPR’98, Brisbane, Australia (1998)Google Scholar
  10. 10.
    Kofler, H.: The Topological Consistence of Path Connectedness in Regular and Irregular Structures. 7th Intern. Workshop, SSPR’89, Sydney (1998)Google Scholar
  11. 11.
    Kovalevsky, V.A.: Finite Topology as Applied to Image Analysis. Computer Vision, Graphics and Image Processing 46 (1989)Google Scholar
  12. 12.
    Kropatsch, W.G.: Building Irregular Pyramids by Dual Graph Contraction. IEE Proceedings Vis. Image Signal Process., Vol. 142 (1995)Google Scholar
  13. 13.
    Latecki, L.: Digitale und Allgemeine Topologie in der Bildhaften Wissensrepräsentation. Ph.D.-Thesis, Hamburg (1992)Google Scholar
  14. 14.
    Preis, R., Diekmann, R.: The Party Partitioning-Library. User Guide-Version 1.1. Univ. Paderborn, Germany (1996)Google Scholar
  15. 15.
    Ptak, P., Kofler, H., Kropatsch, W.: Digital Topologies Revisited. 7th International Workshop, DGCI’97, Montpellier, France, Springer series (1997)Google Scholar
  16. 16.
    Ueberhuber, C.W.: Numerical Computation 1 and 2. Methods, Software, and Analysis. Springer-Verlag, Heidelberg (1997)Google Scholar
  17. 17.
    Wyse, F., Marcus, D. et al.: Solution to Problem 5712. Am. Math. Monthly 77 (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Helmut Kofler
    • 1
  • Ernst J. Haunschmid
    • 1
  • Wilfried N. Gansterer
    • 1
  • Christoph W. Ueberhuber
    • 1
  1. 1.Institute for Applied and Numerical MathematicsTechnical University of ViennaViennaAustria

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