Advertisement

Matching convex shapes with respect to the symmetric difference

  • Helmut Alt
  • Ulrich Fuchs
  • Günter Rote
  • Gerald Weber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1136)

Abstract

This paper deals with questions from convex geometry related to shape matching. In particular, we consider the problem of matching convex figures minimizing the area of the symmetric difference. The main theorem of this paper states, that if we just match the two centers of gravity the resulting symmetric difference is within a factor of 11/3 from the optimal one. This leads to efficient approximate matching algorithms for convex figures.

Keywords

Reference Point Convex Body Convex Polygon Rigid Motion Steiner Point 
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. AAR94.
    H. Alt, O. Aichholzer, and G. Rote. Matching shapes with a reference point. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 85–92, 1994. To appear in Int. J. Comput. Geom. Appl.Google Scholar
  2. ABB91.
    H. Alt, B. Behrends, and J. Blömer. Approximate matching of polygonal shapes. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 186–193, 1991.Google Scholar
  3. ABGW90.
    H. Alt, J. Blömer, M. Godau, and H. Wagener. Approximation of convex polygons. In Proc. 17th Internat. Colloq. Automata Lang. Program., volume 443 of Lecture Notes in Computer Science, pages 703–716. Springer-Verlag, 1990.Google Scholar
  4. AST94.
    P. K. Agarwal, M. Sharir, and S. Toledo. Applications of parametric searching in geometric optimization. J. Algorithms, 17:292–318, 1994.CrossRefGoogle Scholar
  5. BW48.
    T. Bonnesen and W.Fenchel. Theorie der konvexen Körper, volume 3 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Chelsea Publ. Comp., 1948.Google Scholar
  6. CGH+93.
    L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. In Proc. 5th Canad. Conf. Comput. Geom., pages 151–156, Waterloo, Canada, 1993.Google Scholar
  7. dBDvK+96.
    M. de Berg, O. Devillers, M. van Kreveld, O. Schwarzkopf, and M. Teillaud. Computing the maximum overlap of two convex polygons under translations. Technical Report, Dept. of Comp. Science, Univ. of Utrecht, 1996.Google Scholar
  8. HKS93.
    D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267–291, 1993.Google Scholar
  9. Sch82.
    A. Schönhage. The fundamental theorem of algebra in terms of computational complexity. Technical Report, University of Tübingen, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Helmut Alt
    • 1
  • Ulrich Fuchs
    • 1
  • Günter Rote
    • 2
  • Gerald Weber
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany
  2. 2.Institut für MathematikTechnische Universität GrazGrazAustria

Personalised recommendations