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Visibility in Discrete Geometry: An Application to Discrete Geodesic Paths

  • David Coeurjolly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2301)

Abstract

In this article, we present a discrete definition of the classical visibility in computational geometry. We present algorithms to compute the set of pixels in a non-convex domain that are visible from a source pixel. Based on these definitions, we define discrete geodesic paths in discrete domain with obstacles. This allows us to introduce a new geodesic metric in discrete geometry.

Keywords

Geodesic Distance Visibility Problem Chain Code Distance Labelling Binary Relationship 
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.

References

  1. 1.
    G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34(3):344–371, June 1986.CrossRefGoogle Scholar
  2. 2.
    J.P. Braquelaire and P. Moreau. Error free construction of generalized euclidean distance maps and generalized discrete voronoï diagrams. Technical report, Université Bordeaux, Laboratoire LaBRI, 1994.Google Scholar
  3. 3.
    D. Coeurjolly, I. Debled-Rennesson, and O. Teytaud. Segmentation and length estimation of 3d discrete curves. In Digital and Image Geometry. to appear, Springer Lecture Notes in Computer Science, 2001.Google Scholar
  4. 4.
    O. Cuisenaire. Distrance Transformations: Fast Algorithms and Applications to Medical Image Processing. PhD thesis, Université Catholique de Louvain, oct 1999.Google Scholar
  5. 5.
    P.E. Danielsson. Euclidean distance mapping. CGIP, 14:227–248, 1980.Google Scholar
  6. 6.
    I. Debled-Rennesson. Etude et reconnaissance des droites et plans discrets. PhD thesis, Thèse. Université Louis Pasteur, Strasbourg, 1995.Google Scholar
  7. 7.
    I. Debled-Rennesson and J.P. Reveillès. A linear algorithm for segmentation of digital curves. In International Journal of Pattern Recognition and Artificial Intelligence, volume 9, pages 635–662, 1995.CrossRefGoogle Scholar
  8. 8.
    G. Sanniti di Baja and S. Svensson. Detecting centres of maximal discs. Discrete Geometry for Computer Imagery, pages 443–452, 2000.Google Scholar
  9. 9.
    L. Dorst and A.W.M. Smeulders. Decomposition of discrete curves into piecewise straight segments in linear time. In Contemporary Mathematics, volume 119, 1991.Google Scholar
  10. 10.
    J. Françon, J.M. Schramm, and M. Tajine. Recognizing arithmetic straight lines and planes. Discrete Geometry for Computer Imagery, 1996.Google Scholar
  11. 11.
    A. Jonas and N. Kiryati. Digital representation schemes for 3d curves. Pattern Recognition, 30(11):1803–1816, 1997.CrossRefGoogle Scholar
  12. 12.
    N. Kiryati and G. Székely. Estimating shortest paths and minimal distances on digitized three-dimension surfaces. Pattern Recognition, 26(11):1623–1637, 1993.CrossRefGoogle Scholar
  13. 13.
    R. Klette and J. Zunic. Convergence of calculated features in image analysis. Technical Report CITR-TR-52, University of Auckland, 1999.Google Scholar
  14. 14.
    V. Kovalevsky and S. Fuchs. Theoritical and experimental analysis of the accuracy of perimeter estimates. In Robust Computer Vision, pages 218–242, 1992.Google Scholar
  15. 15.
    M. Lindenbaum and A. Bruckstein. On recursive, o(n) partitioning of a digitized curve into digital straigth segments. IEEE Transactions on PatternAnalysis and Machine Intelligence, PAMI-15(9):949–953, september 1993.CrossRefGoogle Scholar
  16. 16.
    M. D. McIlroy. A note on discrete representation of lines. Atandt Tech. J., 64(2, Pt. 2):481–490, February 1985.Google Scholar
  17. 17.
    N. Megiddo. Linear programming in linear time when the dimension is fixed. Journal of the ACM, 31(1):114–127, January 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    P. Moreau. Modélisation et génération de dégradés dans le plan discret. PhD thesis, Université Bordeaux I, 1995.Google Scholar
  19. 19.
    J. Piper and E. Granum. Computing distance transformations in convex and nonconvex domains. Pattern Recognition, 20:599–615, 1987.CrossRefGoogle Scholar
  20. 20.
    I. Ragnemalm. Contour processing distance transforms, pages 204–211. World Scientific, 1990.Google Scholar
  21. 21.
    B. J. H. Verwer, P. W. Verbeek, and S.T Dekker. An efficient uniform cost algorithm applied to distance transforms. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-11(4):425–429, April 1989.CrossRefGoogle Scholar
  22. 22.
    B.J.H Verwer. Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters, 12:671–682, november 1991.CrossRefGoogle Scholar
  23. 23.
    J. Vittone and J.M. Chassery. Recognition of digital naive planes and polyhedization. In Discrete Geometry for Computer Imagery, number 1953 in Lecture Notes in Computer Science, pages 296–307. Springer, 2000.CrossRefGoogle Scholar
  24. 24.
    L.D. Wu. On the chain code of a line. IEEE Trans. Pattern Analysis and Machine Intelligence, 4:347–353, 1982.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • David Coeurjolly
    • 1
  1. 1.Laboratoire ERICUniversité Lumière Lyon 2BRON CEDEXFrance

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