Advertisement

Algorithmica

, Volume 15, Issue 1, pp 82–102 | Cite as

Using topological sweep to extract the boundaries of regions in maps represented by region quadtrees

  • M. B. Dillencourt
  • H. Samet
Article

Abstract

A variant of the plane-sweep paradigm known as topological sweep is adapted to solve geometric problems involving two-dimensional regions when the underlying representation is a region quadtree. The utility of this technique is illustrated by showing how it can be used to extract the boundaries of a map inO(M) space andO(Mα(M)) time, whereM is the number of quadtree blocks in the map, andα(·) is the (extremely slowly growing) inverse of Ackerman's function. The algorithm works for maps that contain multiple regions as well as holes. The algorithm makes use of active objects (in the form of regions) and an active border. It keeps track of the current position in the active border so that at each step no search is necessary. The algorithm represents a considerable improvement over a previous approach whose worst-case execution time is proportional to the product of the number of blocks in the map and the resolution of the quadtree (i.e., the maximum level of decomposition). The algorithm works for many different quadtree representations including those where the quadtree is stored in external storage.

Key words

Computational geometry Topological sweep Plane sweep Region representation Boundary extraction Active borders Region quadtrees Computer graphics Image Processing Geographic information systems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. J. Abel and J. L. Smith. A data structure and algorithm based on a linear key for a rectangle retrieval problem.Computer Vision, Graphics, and Image Processing, 24(1):1–13, October 1983.Google Scholar
  2. [2]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman.The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA, 1974.Google Scholar
  3. [3]
    H. S. Baird. Fast algorithms for LSI artworks analysis.Journal of Design Automation & Fault-Tolerant Computing, 2:179–209, 1978.Google Scholar
  4. [4]
    M. Ben-Or. Lower bounds for algebraic computation trees.Proceedings of the Fifteenth Annual ACM Symposium on the Theory of Computing, pp. 80–86, Boston, MA, April 1983.Google Scholar
  5. [5]
    J. L. Bentley. Algorithms for Klee's rectangle problems. Unpublished manuscript, Computer Science Department, Carnegie-Mellon University, Pittsburgh, PA, 1977.Google Scholar
  6. [6]
    J. L. Bentley and D. Wood. An optimal worst-case algorithm for reporting intersections of rectangles.IEEE Transactions on Computers, 29(7):571–576, July 1980.Google Scholar
  7. [7]
    M. B. Dillencourt, H. Samet, and M. Tamminen. A general approach to connected-component labeling for arbitrary image representations.Journal of the ACM, 39(3):253–280, April 1992. Also see Corrigenda,Journal of the ACM, 39(4):985–985, October 1992.CrossRefGoogle Scholar
  8. [8]
    C. R. Dyer, A. Rosenfeld, and H. Samet. Region representation: Boundary codes from quadtrees.Communications of the ACM, 23(3):171–179, March 1980.CrossRefGoogle Scholar
  9. [9]
    H. Edelsbrunner. A new approach to rectangle intersections: part I.International Journal of Computer Mathematics, 13(3–4):209–219, 1983.Google Scholar
  10. [10]
    H. Edelsbrunner.Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer-Verlag, Berlin, 1987.Google Scholar
  11. [11]
    H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement.Proceedings of the Eighteenth Annual ACM Symposium on the Theory of Computing, pp. 389–403, Berkeley, CA, May 1986.Google Scholar
  12. [12]
    H. Freeman. Computer processing of line-drawing images.ACM Computing Surveys, 6(1):57–97, March 1974.CrossRefGoogle Scholar
  13. [13]
    I. Gargantini. An effective way to represent quadtrees.Communications of the ACM, 25(12):905–910, December 1982.CrossRefGoogle Scholar
  14. [14]
    E. Kawaguchi, T. Endo, and J. Matsunaga. Depth-first picture expression viewed from digital picture processing.IEEE Transactions on Pattern Analysis andMachine Intelligence, 5(4):373–384, July 1983.Google Scholar
  15. [15]
    D. T. Lee. Maximum clique problem of rectangle graphs. In F. P. Preparata, editor,Computational Geometry, pp. 91–107. Advances in Computing Research, vol. 1. JAI Press, Greenwich, CT, 1983.Google Scholar
  16. [16]
    E. M. McCreight. Priority search trees.S1AM Journal on Computing, 14(2):257–276, May 1985.CrossRefGoogle Scholar
  17. [17]
    J. Nievergelt and F. P. Preparata. Plane-sweep algorithms for intersecting geometric figures.Communications of the ACM, 25(10):739–747, October 1982.CrossRefGoogle Scholar
  18. [18]
    F. P. Preparata and M. I. Shamos.Computational Geometry: An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  19. [19]
    H. Samet. Hierarchical representations of collections of small rectangles.ACM Computing Surveys, 20(2):271–309, December 1988.CrossRefGoogle Scholar
  20. [20]
    H. Samet.The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA, 1990.Google Scholar
  21. [21]
    H. Samet and M. Tamminen. Computing geometric properties of images represented by linear quadtrees.IEEE Transactions on Pattern Analysis and Machine Intelligence, 7(3):229–240, March 1985.Google Scholar
  22. [22]
    H. Samet and M. Tamminen. Efficient component labeling of images of arbitrary dimension represented by linear bintrees.IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(4):579–586, July 1988.CrossRefGoogle Scholar
  23. [23]
    H. Samet and R. E. Webber. Storing a collection of polygons using quadtrees.ACM Transactions on Graphics, 4(3):182–222, July 1985.CrossRefGoogle Scholar
  24. [24]
    M. Tamminen, Encoding pixel trees.Computer Graphics, Vision, and Image Processing, 28(1):44–57, October 1984.Google Scholar
  25. [25]
    R. E. Tarjan, Efficiency of a good but not linear set union algorithm.Journal of the ACM, 22(2):215–225, April 1975.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • M. B. Dillencourt
    • 1
  • H. Samet
    • 2
  1. 1.Department of Information and Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.Center for Automation Research and Computer Science Department, and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA

Personalised recommendations