Advertisement

Voronoi Hierarchies

  • Christopher Gold
  • Paul Angel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4197)

Abstract

Voronoi diagrams are widely used to represent geographical distributions of information, but they are not readily stacked in a hierarchical fashion. We propose a simple mechanism whereby each index Voronoi cell contains the generators of several Voronoi cells in the next lower level. This allows various processes of indexing, paging, visualization and generalization to be performed on various types of data. While one-level Voronoi indexes have been used before, and hierarchies of the dual Delaunay triangulation have been used for fast point location, we believe that this is the first time that the advantages of their integration have been demonstrated.

Keywords

Base Point Voronoi Diagram Delaunay Triangulation Index Generator Voronoi Cell 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amenta, N., Choi, S., Rote, G.: Incremental constructions con BRIO. In: 19th ACM Symposium on Computational Geometry, San Diego (2003)Google Scholar
  2. 2.
    Aurenhammer, F.: Voronoï diagrams - a survey of a fundamental geometric data structure. ACM Computing Surveys 23, 345–405 (1991)CrossRefGoogle Scholar
  3. 3.
    Boots, B., Shiode, N.: Recursive Voronoi diagrams. Environment and Planning B 30(1), 113–124 (2003)CrossRefGoogle Scholar
  4. 4.
    Christaller, W.: Central Places in Southern Germany (translation). Prentice Hall, NJ (1966)Google Scholar
  5. 5.
    Dakowicz, M., Gold, C.M.: Extracting Meaningful Slopes from Terrain Contours. International Journal of Computational Geometry Applications 13, 339–357 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dakowicz, M., Gold, C.M.: Structuring Kinetic Maps. In: Proceedings of the Twelfth International Symposium on Spatial Data Handling, Vienna (in press, July 2006)Google Scholar
  7. 7.
    De Floriani, L.: A pyramidal data structure for triangle-based surface description. Computer Graphics and Applications 9(2), 67–78 (1989)CrossRefGoogle Scholar
  8. 8.
    Devillers, O.: On deletion in Delaunay triangulation. In: Proc. 15th ACM Symp. Comp. Geom., pp. 181–188 (1999)Google Scholar
  9. 9.
    Devillers, O.: The Delaunay Hierarchy. Int. J. Foundations of Computer Science 13(2), 163–180 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dobashi, Y., Haga, T., Johan, H., Nishita, T.: A Method for Creating Mosaic Images Using Voronoi Diagrams. In: Proc. EUROGRAPHICS 2002, pp. 341–348 (2002)Google Scholar
  11. 11.
    Gold, C.M.: An Algorithmic Approach to Marine GIS, ch.4. In: Wright, D., Bartlett, D. (eds.) Marine and Coastal Geographical Information Systems, pp. 37–52. Taylor and Francis, London (2000)Google Scholar
  12. 12.
    Gold, C.M., Charters, T.D., Ramsden, J.: Automated contour mapping using triangular element data structures and an interpolant over each triangular domain. In: George, J. (ed.) Proceedings of Sigraph 1977. Computer Graphics, San Francisco, USA, vol. 11, pp. 170–175 (1977)Google Scholar
  13. 13.
    Gold, C.M., Dakowicz, M.: The Crust and Skeleton – Applications in GIS. In: Proceedings, 2nd International Symposium on Voronoi Diagrams in Science and Engineering, Seoul, Korea, pp. 33–42 (2005)Google Scholar
  14. 14.
    Gold, C.M., Maydell, U.M.: Triangulation and spatial ordering in computer cartography. In: Proceedings, Canadian Cartographic Association third annual meeting, Vancouver, BC, Canada, pp. 69–81 (1978)Google Scholar
  15. 15.
    Gold, C.M., Nantel, J., Yang, W.: Outside-in: an alternative approach to forest map digitizing. International Journal of Geographical Information Systems 10, 291–310 (1996)Google Scholar
  16. 16.
    Guibas, L., Mitchell, J.S.B., Roos, T.: Voronoï diagrams of moving points in the plane. In: Schmidt, G., Berghammer, R. (eds.) WG 1991. LNCS, vol. 570, pp. 113–125. Springer, Heidelberg (1992)Google Scholar
  17. 17.
    Guibas, L., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of Voronoï diagrams. Transactions on Graphics 4, 74–123 (1985)MATHCrossRefGoogle Scholar
  18. 18.
    Haggett, P.: Geography – A Modern Synthesis, 3rd edn., p. 627. Harper and Row, NY (1979)Google Scholar
  19. 19.
    Luebke, D., Reddy, M., Cohen, J., Varshney, A., Watson, B., Huebner, R.: Level of Detail for 3D Graphics, p. 390. Morgan Kaufmann, San Francisco (2003)Google Scholar
  20. 20.
    Lukatela, H.: A Seamless Global Terrain Model in the Hipparchus System. In: International Conference on Discrete Global Grids, Santa Barbara (2000), http://www.geodyssey.com/
  21. 21.
    Mucke, E.P., Saias, I., Zhu, B.: Fast randomized point location without preprocessing in two and three dimensional Delaunay triangulations. In: Proc. 12th ACM Symp. Comp. Geom., pp. 274–283 (1996)Google Scholar
  22. 22.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations - Concepts and Applications of Voronoi Diagrams, 2nd edn., p. 671. John Wiley and Sons, Chichester (2000)MATHGoogle Scholar
  23. 23.
    Okabe, A., Sadahiro, Y.: An illusion of spatial hierarchy: spatial hierarchy in a random configuration. Environment and Planning A 28, 1533–1552 (1996)CrossRefGoogle Scholar
  24. 24.
    Pulo, K.J.: Recursive space decomposition in force-directed graph drawing algorithms. In: Proceedings, Australian Symposium on Information Visualization, Sydney, vol. 9, pp. 95–102 (December 2001)Google Scholar
  25. 25.
    Schussman, S., Bertram, M., Hamann, B., Joy, K.I.: Hierarchical data representations based on planar Voronoi diagrams. In: van Liere, R., Hermann, I., Ribarsky, W. (eds.) Proceedings of VisSym 2000 – The Joint Eurographics and IEEE TVCG Conference on Visualization, Amsterdam, The Netherlands (May 2000)Google Scholar
  26. 26.
    Sibson, R.: A brief description of natural neighbour interpolation. In: Barnett, V. (ed.) Interpreting Multivariate Data, pp. 21–36. John Wiley and Sons, New York (1981)Google Scholar
  27. 27.
    Swets, D.L., Weng, J.: Hierarchical discriminant analysis for image retrieval. IEEE Trans. PAMI 21(5), 386–401 (1999)Google Scholar
  28. 28.
    Tse, R., Gold, C.M., Kidner, D.: A New Approach to Urban Modelling Based on LIDAR. In: Proceedings, WSCG 2006, Plzen, Czech Republic (2006), http://wscg.zcu.cz/WSCG2006/contents.htm

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christopher Gold
    • 1
  • Paul Angel
    • 1
  1. 1.School of ComputingUniversity of GlamorganPontypridd, WalesUK

Personalised recommendations