Irregular Pyramids with Combinatorial Maps

  • Luc Brun
  • Walter Kropatsch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1876)

Abstract

This paper presents a new formalism for irregular pyramids based on combinatorial maps. Such pyramid consists of a stack of successively reduced graph. Each smaller graph is deduced from the preceding one by a set of edges which have to be contracted or removed. In order to perform parallel contractions or removals, the set of edges to be contracted or removed has to verify some properties. Such a set of edges is called a Decimation Parameter. A combinatorial map encodes a planar graph thanks to two permutations encoding the edges and their orientation around the vertices. Combining the useful properties of both combinatorial maps and irregular pyramids offers a potential alternative for representing structures at multiple levels of abstraction.

Keywords

Planar Graph Dual Graph Removal Operation Independent Vertex Contraction Operation 
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.

References

  1. [1]
    J. P. Braquelaire and L. Brun. Image segmentation with topological maps and inter-pixel representation. Journal of Visual Communication and Image representation, 9(1), 1998.Google Scholar
  2. [2]
    L. Brun, J.P. Domenger, and J.P. Braquelaire. Discrete maps: a framework for region segmentation algorithms. In Workshop on Graph based representations, Lyon, April 1997. published in Advances in Computing (Springer).Google Scholar
  3. [3]
    L. Brun and Walter Kropatsch. Dual contraction of combinatorial maps. In GbR99. IAPR TC15, 1999.Google Scholar
  4. [4]
    L. Brun and Walter Kropatsch. Dual contractions of combinatorial maps. Technical Report 54, Institute of Computer Aided Design, Vienna University of Technology, lstr. 3/1832,A-1040 Vienna AUSTRIA, January 1999. avalaible through http://www.prip.tuwien.ac.at/.
  5. [5]
    A. Jones Gareth and David Singerman. Theory of maps on orientable surfaces, volume 3, pages 273–307. London Mathematical Society, 1978.Google Scholar
  6. [6]
    S.L. Horowitz and T. Pavlidis. Picture segmentation by a tree traversal algorithm. Journal of The Association for Computing Machinery, 23(2):368–388, April 1976.Google Scholar
  7. [7]
    J.M. Jolion and A. Montanvert. The adaptative pyramid: A framework for 2d image analysis. Computer Vision, Graphics, and Image Processing, 55(3):339–348, May 1992.Google Scholar
  8. [8]
    Walter G. Kropatsch. Building Irregular Pyramids by Dual Graph Contraction. IEE-Proc. Vision, Image and Signal Processing, Vol. 142(No. 6):366–374, December 1995.CrossRefGoogle Scholar
  9. [9]
    Walter G. Kropatsch. Equivalent Contraction Kernels and The Domain of Dual Irregular Pyramids. Technical Report PRIP-TR-42, Institute f. Automation 183/2, Dept. for Pattern Recognition and Image Processing, TU Wien, Austria, 1995.Google Scholar
  10. [10]
    Walter G. Kropatsch. Property Preserving Hierarchical Graph Transformations. In Carlo Arcelli, Luigi P. Cordella, and Gabriella Sanniti di Baja, editors, Advances in Visual Form Analysis, pages 340–349. World Scientific Publishing Company, 1998.Google Scholar
  11. [11]
    Walter G. Kropatsch and Souheil BenYacoub. Universal Segmentation with PIRRamids. In Axel Pinz, editor, Pattern Recognition 1996, Proc. of 20th OAGM Workshop, pages 171–182. OCG-Schriftenreihe, Österr. Arbeitsgemeinschaft für Mustererkennung, R. Oldenburg, 1996. Band 90.Google Scholar
  12. [12]
    Walter G. Kropatsch and Herwig Macho. Finding the structure of connected components using dual irregular pyramids. In Cinquième Colloque DGCI, pages 147–158. LLAIC1, Université d’Auvergne, ISBN 2-87663-040-0, September 1995.Google Scholar
  13. [13]
    M.D. Levine. Vision in man and machine. Mc Graw-Hill Book Compagny, 1985.Google Scholar
  14. [14]
    Lienhardt. Subdivisions of n-dimensional spaces and n-dimensional generalized maps. In Annual ACM Symposium on Computational Geometry, all, volume 5, 1989.Google Scholar
  15. [15]
    Jean-Gerard Pailloncy, Walter G. Kropatsch, and Jean-Michel Jolion. Object Matching on Irregular Pyramid. In Anil K Jain, Svetha Venkatesh, and Brian C Lovell, editors, 14th International Conference on Pattern Recognition, volume II, pages 1721–1723. IEEE Comp.Soc., 1998.Google Scholar
  16. [16]
    W.T. Tutte. Graph Theory, volume 21. Addison-Wesley, encyclopedia of mathematics and its applications edition, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Luc Brun
    • 1
  • Walter Kropatsch
    • 2
  1. 1.LERII.U.T. Léonard de VinciReimsFrance
  2. 2.Institute for Computer-aided Automation Pattern Recognition and Image Processing GroupVienna Univ. of TechnologyViennaAustria

Personalised recommendations