Advertisement

Approximating polygons and subdivisions with minimum link paths

  • Leonidas J. Guibas
  • John E. Hershberger
  • Joseph S. B. Mitchell
  • Jack Scott Snoeyink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 557)

Abstract

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We also discuss additional topological constraints such as simplicity.

Keywords

Homotopy Class Support Point Simple Polygon Support Line Dynamic Programming Approach 
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]
    P. K. Agarwal, M. Sharir, and P. Shor. Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences. J. Comb. Theory A, 52:228–274, 1989.Google Scholar
  2. [2]
    H. Alt, J. Blömer, M. Godau, and H. Wagener. Approximation of convex polygons. In Seventeenth ICALP, number 443 in LNCS, pages 703–716. Springer-Verlag, 1990.Google Scholar
  3. [3]
    R. Bellman. On the approximation of curves by line segments using dynamic programming. CACM, 4:284, 1961.Google Scholar
  4. [4]
    M. Blakemore. Generalisation and error in spatial data bases. Cartographica, 21:131–139, 1984.Google Scholar
  5. [5]
    B. Buttenfield. Treatment of the cartographic line. Cartographica, 22:1–26, 1985.Google Scholar
  6. [6]
    D. H. Douglas and T. K. Peucker. Algorithms for the reduction of the number of points required to represent a line or its caricature. The Canadian Cartographer, 10(2):112–122, 1973.Google Scholar
  7. [7]
    P. Egyed and R. Wenger. Ordered stabbing of pairwise disjoint convex sets in linear time. Disc. App. Math., to appear.Google Scholar
  8. [8]
    M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Comp. Sci., 1:237–267, 1976.Google Scholar
  9. [9]
    S. K. Ghosh. Computing the visibility polygon from a convex set and related problems. J. Alg., 12:75–95, 1991.Google Scholar
  10. [10]
    R. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Info. Proc. Let., 1:132–133, 1972.Google Scholar
  11. [11]
    L. Guibas, L. Ramshaw, and J. Stolfi. A kinetic framework for computational geometry. In Proc. 24th FOCS, pages 100–211, 1983.Google Scholar
  12. [12]
    S. L. Hakimi and E. F. Schmeichel. Fitting polygonal functions to a set of points in the plane. CVGIP: Graph. Mod. Image Proc., 53(2):132–136, 1991.Google Scholar
  13. [13]
    J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. In WADS '91 Proceedings, 1991.Google Scholar
  14. [14]
    J. Hershberger and J. Snoeyink. An implementation of the Douglas-Peucker line simplification algorithm using at most cn log n operations. In preparation, 1991.Google Scholar
  15. [15]
    H. Imai and M. Iri. Computational-geometric methods for polygonal approximations of a curve. Comp. Vis. Graph. Image Proc., 36:31–41, 1986.Google Scholar
  16. [16]
    H. Imai and M. Iri. An optimal algorithm for approximating a piecewise linear function. J. Info. Proc., 9(3):159–162, 1986.Google Scholar
  17. [17]
    H. Imai and M. Iri. Polygonal approximations of a curve—formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology. North Holland, 1988.Google Scholar
  18. [18]
    K. Kedem, R. Livne, J. Pach, and M. Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Disc. & Comp. Geom., 1:59–71, 1986.Google Scholar
  19. [19]
    R. B. McMaster. A statistical analysis of mathematical measures for linear simplification, Amer. Cartog., 13:103–116, 1986.Google Scholar
  20. [20]
    R. B. McMaster. Automated line generalization. Cartographica, 24(2):74–111, 1987.Google Scholar
  21. [21]
    R. B. McMaster. The integration of simplification and smoothing algorithms in line generalization. Cartographica, 26(1):101–121, 1989.Google Scholar
  22. [22]
    A. Melkman and J. O'Rourke. On polygonal chain approximation. In G. T. Toussaint, editor, Computational Morphology. North Holland, 1988.Google Scholar
  23. [23]
    J. R. Munkres. Topology: A First Course. Prentice-Hall, Englewood Cliffs, N.J., 1975.Google Scholar
  24. [24]
    J. O'Rourke. An on-line algorithm for fitting straight lines between data ranges. CACM, 24(9):574–578, Sept. 1981.Google Scholar
  25. [25]
    J. Perkal. On the length of empirical curves. In Discussion Paper 10, Michigan Inter-University Community of Mathematical Geographers, University of Michigan, Ann Arbor, 1966.Google Scholar
  26. [26]
    U. Ramer. An iterative procedure for the polygonal approximation of plane curves. Comp. Vis. Graph. Image Proc., 1:244–256, 1972.Google Scholar
  27. [27]
    A. Rosenfeld. Axial representation of shape. Comp. Vis. Graph. Image Proc., 33:156–173, 1986.Google Scholar
  28. [28]
    S. Suri. A linear time algorithm for minimum link paths inside a simple polygon. Comp. Vis. Graph. Image Proc., 35:99–110, 1986.Google Scholar
  29. [29]
    G. Toussaint. On the complexity of approximating polygonal curves in the plane. In Proc. IASTED, International Symposium on Robotics and Automation, Lugano, Switzerland, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Leonidas J. Guibas
    • 1
  • John E. Hershberger
    • 2
  • Joseph S. B. Mitchell
    • 3
  • Jack Scott Snoeyink
    • 4
  1. 1.Stanford and DEC SRCUSA
  2. 2.DEC SRCUSA
  3. 3.Cornell UniversityUSA
  4. 4.Utrecht UniversityThe Netherlands

Personalised recommendations