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A Heuristic Homotopic Path Simplification Algorithm

  • Shervin Daneshpajouh
  • Mohammad Ghodsi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)

Abstract

We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set of extra points in the plane. We present a heuristic method for homotopy-preserving simplification under any desired measure for general paths. Our algorithm for finding homotopic shortcuts runs in O( mlog(n + m) + nlogn log(nm) + k) time, where k is the number of homotopic shortcuts. Using this method, we obtain an O(n 2 + mlog(n + m) + nlogn log(nm)) time algorithm for simplification under the Hausdorff measure.

Keywords

Computational Geometry Simplification Homotopy Path Line Curve Heuristic 

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References

  1. 1.
    de Berg, M., van Kreveld, M., Schirra, S.: A New Approach to Subdivision Simplification. In: Twelfth International Symposium on Computer Assisted Cartography, vol. 04, pp. 79–88 (1995)Google Scholar
  2. 2.
    de Berg, M., van Kreveld, M., Schirra, S.: Topologically correct subdivision simplification using the bandwidth criterion. Cartography and GIS 25, 243–257 (1998)Google Scholar
  3. 3.
    Buzer, L.: Optimal Simplification of Polygonal Chain for Rendering. In: 23rd ACM Symposium on Computational Geometry (SoCG), pp. 168–174 (2007)Google Scholar
  4. 4.
    Goodrich, M.T.: Efficient piecewise-linear function approximation using the uniform metric. Discrete Computational Geometry 14, 445–462 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Agarwal, P.K., Harpeled, S., Mustafa, N.H., Wang, Y.: Near-linear time approximation algorithms for curve simplification. Algorithmica 42, 203–219 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Aronov, B., Asano, T., Katoh, N., Mehlhorn, K., Tokuyama, T.: Polyline fitting of planar points under min-sum criteria. International Journal of Computational Geometry and Applications 16, 97–116 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Abam, M.A., de Berg, M., Hachenberger, P., Zarei, A.: Streaming algorithms for line simplifications. In: Proc. ACM Symposium on Computational Geometry (SoCG), pp. 175–183 (2007)Google Scholar
  8. 8.
    Douglas, D.H., Peucker, T.K.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Canadian Cartographer 10(2), 112–122 (1973)CrossRefGoogle Scholar
  9. 9.
    Hershberger, J., Snoeyink, J.: Speeding up the Douglas-Peucker line simplification algorithm. In: Proceeding of 5th International Symposium on Spatial Data Handling, pp. 134–143 (1992)Google Scholar
  10. 10.
    Li, Z., Openshaw, S.: Algorithms for automated line generalization based on a natural principle of objective generalization. International Journal of Geographic Information Systems 6, 373–389 (1992)CrossRefGoogle Scholar
  11. 11.
    Kurozumi, Y., Davis, W.A.: Polygonal approximation by the minimax method. Comput. Graph. Image Process 19, 248–264 (1982)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hobby, J.D.: Polygonal approximations that minimize the number of inflections. In: Proceeding of the 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 93–102 (1993)Google Scholar
  13. 13.
    Asano, T., Katoh, N.: Number theory helps line detection in digital images. In: In Proceeding of 4th Annual International Symposium on Algorithms and Computing, vol. 762, pp. 313–322 (1993)Google Scholar
  14. 14.
    Guibas, L.J., Hershberger, J.E., Mitchell, J.S.B., Snoeyink, J.S.: Approximating polygons and subdivisions with minimum link paths. International Journal of Computational Geometry and Applications 3(4), 383–415 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Estkowski, R., Mitchell, J.S.: Simplifying a polygonal subdivision while keeping it simple. In: Proceedings of the 17th Annual Symposium on Computational Geometry, pp. 40–49 (2001)Google Scholar
  16. 16.
    Daneshpajouh, S., Abam, M.A., Deleuran, L., Ghodsi, M.: Computing Strongly Homotopic Line Simplification in the Plane. In: European Workshop on Computational Geometry (2011)Google Scholar
  17. 17.
    Preparata, F.P., Shamos, M.I.: Computational Geometry - an introduction. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  18. 18.
    Toussaint, G.T.: An optimal algorithm for computing the convex hull of a set of points in a polygon. In: Proceeding of Signal Processing III: Theories and Applications, EURASIP 1986, Part 2, pp. 853–856 (1986)Google Scholar
  19. 19.
    Ben-Moshe, B., Hall-Holt, O., Katz, M., Mitchell, J.: Computing the visibility graph of points within a polygon. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry, pp. 27–35 (2004)Google Scholar
  20. 20.
    Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. Journal of Comput. Syst. Sci. 39, 126–152 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Chan, W.S., Chin, F.: Approximation of polygonal curves with minimum number of line segments. In: Proceeding of 3rd Annual International Symposium on Algorithms and Computing, vol. 650, pp. 378–387 (1992)Google Scholar
  22. 22.
    Imai, H., Iri, M.: Polygonal approximations of a curve-formulations and algorithms. In: Toussaint, G.T. (ed.) Computational Morphology, pp. 71–86 (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Shervin Daneshpajouh
    • 1
  • Mohammad Ghodsi
    • 1
    • 2
  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran
  2. 2.School of Computer ScienceInstitute for Research in Fundamental Sciences (IPM)TehranIran

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