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Resemblance and symmetries of geometric patterns

  • Helmut Alt
  • Johannes Blömer
Geometric Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 594)

Abstract

This article is a survey on research done in recent years at FU Berlin applying methods from Computational Geometry to problems in pattern and shape analysis. In particular, algorithms are described for determining the exact and approximate congruence of finite sets of points in the plane. This problem is generalized to polygonal curves in the plane, where variuos cases are considered depending on the set of rigid motions (translations, rotations) allowed to match the given curves. Furthermore, algorithms are given for approximating convex polygons by ”simpler“ curves. In addition, the problem of finding all symmetries of a finite set of points is considered, the approximate version of which turns out to be NP-complete in general.

Keywords

Symmetry Group Convex Polygon Geometric Object Simple Polygon Polygonal Chain 
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.

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References

  1. [ASS]
    P. K. Agarwal, M. Sharir, P. Shor, Sharp Upper and Lower Bounds on the Length of General Davenport-Schinzel Sequences, J. Comb. Theory, Ser. A 52, 1989, pp. 228–274.Google Scholar
  2. [AHU]
    A. V. Aho, J. E. Hopcraft, J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.Google Scholar
  3. [ABB]
    H. Alt, B. Behrends, J. Blömer, Approximate Matching of Polygonal Shapes, Proceedings of the 7th ACM Symposium on Computational Geometry, 1991, pp. 186–193.Google Scholar
  4. [ABGW]
    H. Alt, J. Blömer, M. Godau, H. Wagener Approximation of Convex Polygons, Proceedings ICALP, International Colloquium on Automata, Languages and Programming, Warwick, England, 1990, pp. 703–716.Google Scholar
  5. [AKMSW]
    E. M. Arkin, K. Kedem, J. S. B. Mitchell, J. Sprinzak, M. Werman Matching Points into Noise Regions: Combinatorial Bounds and Algorithms, in Proceedings SODA, 2nd Symposium on Discrete Algorithms, 1991.Google Scholar
  6. [AMWW]
    H. Alt, K. Mehlhorn, H. Wagener, E. Welzl, Congruence, Similarity and Symmetries of Geometric Objects, Discrete Comp. Geom. 3, 1988, pp. 237–256.Google Scholar
  7. [Atl]
    M. J. Atallah, A Linear Time Algorithm for the Hausdorff-distance between Convex Polygons, Information Processing Letters 17, 1983, pp. 207–209.CrossRefGoogle Scholar
  8. [B]
    B. Behrends, Algorithmen zur Erkennung der ε-Kongruenz von Punktmengen und Polygonen, Diplomarbeit, Freie Universität Berlin, 1990.Google Scholar
  9. [F]
    S. Fortune, A Sweepline — Algorithm for Voronoi-Diagrams, Algorithmica 2, 1987, pp. 153–174.CrossRefGoogle Scholar
  10. [Fr]
    M. Fréchet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Mathematico di Palermo, Vol. 22, 1906, pp. 1–74.Google Scholar
  11. [G1]
    M. Godau, A Natural Metric for Curves-Computing the Distance for Polygonal Chains and Approximation Algorithms, Proceedings Symposium on Theoretical Aspects of Computer Science, STAGS'91, Springer Lecture Notes in Computer Science, Vol. 480, pp. 127–136.Google Scholar
  12. [G2]
    M. Godau, Die Fréchet-Metrik für Polygonzüge — Algorithmen zur Abstandsmessung und Approximation, Diplomarbeit, Fachbereich Mathematik, FU Berlin 1991.Google Scholar
  13. [HS]
    P. J. Heffernan, S. Schirra, Approximate Decision Algorithms for Point Set Congruence, Report MPI-I-91-110, Max-Planck-Institut für Informatik, Saarbrücken, 1991.Google Scholar
  14. [II]
    H. Imai, M. Iri, Polygonal Approximations of a Curve — Formulations and Algorithms. Computational Morphology, G. T. Toussaint (ed)., Elsevier Science Publ., 1988, pp. 71–86.Google Scholar
  15. [ISI]
    K. Imai, S. Sumino, H. Imai, Minimax Geometric Fitting of two Corresponding Sets of Points, in Proceedings of 5th ACM Symp. on Computational Geometry, 1989, pp. 276–282.Google Scholar
  16. [I]
    S. Iwanowski, Approximate Congruence and Symmetry Detection in the Plane, Ph.D. Thesis, Fachbereich Mathematik, FU Berlin, 1990.Google Scholar
  17. [M]
    G. E. Martin, Transformation Geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1982.Google Scholar
  18. [Me]
    N. Megiddo, Applying Parallel Computation Algorithms in the Design of Serial Algorithms, J. of the Assoc. for Comp. Machinery 30, 1983, pp. 852–866.MathSciNetGoogle Scholar
  19. [PS]
    F. P. Preparata, M. I. Shamos, Computational Geometry, Springer-Verlag, New York, 1985.Google Scholar
  20. [S]
    S. Schirra, über die Bitkomplexität der ε-Kongruenz, Diplomarbeit, Fachbereich Informatik, Universität des Saarlandes, 1988.Google Scholar
  21. [Y]
    C. K. Yap, An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments, Discrete Comp. Geom. 2, 1987, pp. 365–393.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Helmut Alt
  • Johannes Blömer
    • 1
  1. 1.Fachbereich MathematikFU BerlinBerlin 33Germany

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