A New Visibility Partition for Affine Pattern Matching

  • Michiel Hagedoorn
  • Mark Overmars
  • Remco C. Veltkamp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)


Visibility partitions play an important role in computer vision and pattern matching. This paper studies a new type of visibility, reflection-visibility, with applications in affine pattern matching: it is used in the definition of the reflection metric between two patterns consisting of line segments. This metric is affine invariant, and robust against noise, deformation, blurring, and cracks. We present algorithms that compute the reflection visibility partition in O((n+k) log(n)+v) randomised time, where k is the number of visibility edges (at most O(n 2 )), and v is the number of vertices in the partition (at most O(n 2 +k 2 ). We use this partition to compute the reflection metric in O(r(n A + n B )) randomised time, for two line segment unions, with n A and n B line segments, separately, where r is the complexity of the overlay of two reflection-visibility partitions (at most O(nA 4 + nB 4)).


Line Segment Pattern Match Computational Geometry Visibility Graph Visibility Edge 
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.


  1. 1.
    P. K. Agarwal and M. Sharir. On the number of views of polyhedral terrains. Discrete & Computational Geometry, 12:177–182, 1994. 359zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    B. Aronov, L. J. Guibas, M. Teichmann, and L. Zhang. Visibility queries in simple polygons and applications. In ISAAC, pages 357–366, 1998. 359Google Scholar
  3. 3.
    T. Asano, T. Asano, L. Guibas, J. Hershberger, and H. Imai. Visibility of disjoint polygons. Algorithmica, 1:49–63, 1986. 359, 361zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    P. Bose, A. Lubiw, and I. Munro. Efficient visibility queries in simple polygons. In 4th Canadian Conference on Computational Geometry, 1992. To appear in International Journal of Computational Geometry and Applications. 359Google Scholar
  5. 5.
    K. W. Bowyer and C. R. Dyer. Aspect graphs: An introduction and survey of recent results. Int. J. of Imaging Systems and Technology, 2:315–328, 1990. 359CrossRefGoogle Scholar
  6. 6.
    I. Chakravarty and H. Freeman. Characteristic views as a basis for threedimensional object recognition. In Proc. SPIE: Robot Vision, pages 37–45, 1982. 358Google Scholar
  7. 7.
    M. de Berg, D. Halperin, M. H. Overmars, and M. van Kreveld. Sparse arrangements and the number of views of polyhedral scenes. Int. J. Computational Geometry & Applications, 7(3):175–195, 1997. 359CrossRefGoogle Scholar
  8. 8.
    H. Everett. Visibility graph recognition. PhD thesis, Department of Computer Science, University of Toronto, 1990. 359Google Scholar
  9. 9.
    S. K. Ghosh. On recognizing and characterizing visibility graphs of simple polygons. Discrete & Computational Geometry, 17(2):143–162, 1997. 359zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    S. K. Ghosh and D. M. Mount. An output-sensitive algorithm for computing visibility graphs. SIAM Journal on Computing, 20:888–910, 1991. 364zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Z. Gigus, J. Canny, and R. Seidel. Efficiently computing and representing aspect graphs of polyhedral objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13:542–551, 1991. 359CrossRefGoogle Scholar
  12. 12.
    N. Grimal, J. Hallof, and D. van der Plas. Hieroglyphica, sign list., 1993. 360
  13. 13.
    L. J. Guibas, Rajeev Motwani, and Prabhakar Raghavan. The robot localisation problem. SIAM J. Computing, 26(4), 1997. 359Google Scholar
  14. 14.
    M. Hagedoorn. Pattern matching using similarity measures PhD thesis, Department of Computer Science, Utrecht University, ISBN90-393-2460-3, 2000. 359Google Scholar
  15. 15.
    M. Hagedoorn, M. H. Overmars, and R. C. Veltkamp. New visibility partitions with applications in affine pattern matching. Technical Report UU-CS-1999-21, Utrecht University, Padualaan 14, 3584 CH Utrecht, the Netherlands, July 1999. 364, 365, 367
  16. 16.
    M. Hagedoorn and R. C. Veltkamp. Measuring resemblance of complex patterns. In L. Perroton G. Bertrand, M. Couprie, editor, Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science 1568, pages 286–298, 1999. Springer. 359, 367CrossRefGoogle Scholar
  17. 17.
    D. J. Kriegman and J. Ponce. Computing exact aspect graphs of curved objects: Solids of revolution. ACM Symp. Computational Geometry, 5:119–135, 1990. 359Google Scholar
  18. 18.
    D. T. Lee. Proximity and Reachability in the plane. PhD thesis, University of Illinois at Urbana-Champaign, 1978. 358Google Scholar
  19. 19.
    Y. Lin and S. S. Skiena. Complexity aspects of visibility graphs. International Journal of Computational Geometry and Applications, 5:289–312, 1995. 359zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    K. Mulmuley. Computational Geometry: An introduction through randomized algorithms. Prentice Hall, 1994. 361, 364Google Scholar
  21. 21.
    W. H. Plantinga and C. R. Dyer. Visibility, occlusion and the aspect graph. ACM Symp. Computational Geometry, 5(2):137–160, 1990. 359Google Scholar
  22. 22.
    M. Pocchiola and G. Vegter. Topologically sweeping visibility complexes via pseudotriangulations. Discrete & Computational Geometry, 16(4):419–453, 1996. 359, 364zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    E. Welzl. Constructing the visibility graph for n-line segments in O(n2) time. Inform. Process. Lett., 20:167–171, 1985. 359zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Michiel Hagedoorn
    • 1
  • Mark Overmars
    • 1
  • Remco C. Veltkamp
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations