Advertisement

Distance Transformation on Two-Dimensional Irregular Isothetic Grids

  • Antoine Vacavant
  • David Coeurjolly
  • Laure Tougne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)

Abstract

In this article, we propose to investigate the extension of the E\(\textsuperscript{2}\)DT (squared Euclidean Distance Transformation) on irregular isothetic grids. We give two algorithms to handle different structurations of grids. We first describe a simple approach based on the complete Voronoi diagram of the background irregular cells. Naturally, this is a fast approach on sparse and chaotic grids. Then, we extend the separable algorithm defined on square regular grids proposed in [22], more convenient for dense grids. Those two methodologies permit to process efficiently E2DT on every irregular isothetic grids.

Keywords

Geographical Information System Voronoi Diagram Voronoi Cell Separable Algorithm Irregular Grid 
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.
    Bentley, J.L.: Multidimensional Binary Search Trees Used for Associative Searching. Communications of the ACM 18(9), 509–517 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Breu, H., Gil, J., Kirkpatrick, D., Werman, M.: Linear Time Euclidean Distance Algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(5), 529–533 (1995)CrossRefGoogle Scholar
  3. 3.
    CGAL, Computational Geometry Algorithms Library, http://www.cgal.org
  4. 4.
    Chehadeh, Y., Coquin, D., Bolon, P.: A Skeletonization Algorithm Using Chamfer Distance Transformation Adapted to Rectangular Grids. In: 13th International Conference on Pattern Recognition (ICPR 1996), vol. 2, pp. 131–135 (1996)Google Scholar
  5. 5.
    Coeurjolly, D.: Supercover Model and Digital Straight Line Recognition on Irregular Isothetic Grids. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 311–322. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Coeurjolly, D., Montanvert, A.: Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(3), 437–448 (2007)CrossRefGoogle Scholar
  7. 7.
    Cuisenaire, O.: Distance Transformations: Fast Algorithms and Applications to Medical Image Processing. PhD Thesis, Université Catholique de Louvain, Louvain-La-Neuve, Belgium (October 1999)Google Scholar
  8. 8.
    Devillers, O.: Improved Incremental Randomized Delaunay Triangulation. In: 14th Annual ACM Symposium on Computational Geometry, 106–115 (1998)Google Scholar
  9. 9.
    Fouard, C., Malandain, G.: 3-D Chamfer Distances and Norms in Anisotropic Grids. Image and Vision Computing 23(2), 143–158 (2005)CrossRefGoogle Scholar
  10. 10.
    Fouard, C., Strand, R., Borgefors, G.: Weighted Distance Transforms Generalized to Modules and their Computation on Point Lattices. Pattern Recognition 40(9), 2453–2474 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Golomb, S.W.: Run-length Encodings. IEEE Transactions on Information Theory 12(3), 399–401 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Guan, W., Ma, S.: A List-Processing Approach to Compute Voronoi Diagrams and the Euclidean Distance Transform. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(7), 757–761 (1998)CrossRefGoogle Scholar
  13. 13.
    Hesselink, W.H., Visser, M., Roerdink, J.B.T.M.: Euclidean Skeletons of 3D Data Sets in Linear Time by the Integer Medial Axis Transform. In: Proceedings of 7th International Symposium on Mathematical Morphology, pp. 259–268 (2005)Google Scholar
  14. 14.
    Jung, D., Gupta, K.K.: Octree-Based Hierarchical Distance Maps for Collision Detection. In: IEEE International Conference on Robotics and Automation, vol. 1, pp. 454–459 (1996)Google Scholar
  15. 15.
    Karavelas, M.I.: Voronoi diagrams in CGAL. In: 22nd European Workshop on Computational Geometry (EWCG 2006), pp. 229–232 (2006)Google Scholar
  16. 16.
    Maurer, C.R., Qi, R., Raghavan, V.: A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(2), 265–270 (2003)CrossRefGoogle Scholar
  17. 17.
    Meijster, A., Roerdink, J.B.T.M., Hesselink, W.H.: A General Algorithm for Computing Distance Transforms in Linear Time. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 331–340 (2000)Google Scholar
  18. 18.
    Paglieroni, D.W.: Distance Transforms: Properties and Machine Vision Applications. In: CVGIP: Graphical Models and Image Processing, vol. 54, pp. 56–74 (1992)Google Scholar
  19. 19.
    Preparata, F.P., Shamos, M.I.: Computational Geometry - An Introduction. Springer, Heidelberg (1985)Google Scholar
  20. 20.
    Rosenfeld, A., Pfaltz, J.L.: Sequential Operations in Digital Picture Processing. Journal of the ACM 13(4), 471–494 (1966)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rosenfeld, A., Pfalz, J.L.: Distance Functions on Digital Pictures. Pattern Recognition 1, 33–61 (1968)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Saito, T., Toriwaki, J.: New Algorithms for n-dimensional Euclidean Distance Transformation. Pattern Recognition 27(11), 1551–1565 (1994)CrossRefGoogle Scholar
  23. 23.
    Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley Longman Publishing Co., Inc, Amsterdam (1990)Google Scholar
  24. 24.
    Schouten, T., Broek, E.: Fast Exact Euclidean Distance (FEED) Transformation. In: 17th International Conference on Pattern Recognition (ICPR 2004), vol. 3, pp. 594–597 (2004)Google Scholar
  25. 25.
    Sintorn, I.M., Borgefors, G.: Weighted Distance Transforms for Volume Images Digitized in Elongated Voxel Grids. Pattern Recognition Letters 25(5), 571–580 (2004)CrossRefGoogle Scholar
  26. 26.
    Vörös, J.: Low-Cost Implementation of Distance Maps for Path Planning Using Matrix Quadtrees and Octrees. Robotics and Computer-Integrated Manufacturing 17(6), 447–459 (2001)CrossRefGoogle Scholar
  27. 27.
    Wang, X., Bertrand, G.: Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(11), 1114–1121 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Antoine Vacavant
    • 1
  • David Coeurjolly
    • 2
  • Laure Tougne
    • 1
  1. 1.LIRIS - UMR 5205Université Lumière Lyon 2Bron cedexFrance
  2. 2.LIRIS - UMR 5205Université Claude Bernard Lyon 1Villeurbanne cedexFrance

Personalised recommendations