Generating Distance Maps with Neighbourhood Sequences

  • Robin Strand
  • Benedek Nagy
  • Céline Fouard
  • Gunilla Borgefors
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


A sequential algorithm for computing the distance map using distances based on neighbourhood sequences (of any length) in the 2D square grid; and 3D cubic, face-centered cubic, and body-centered cubic grids is presented. Conditions for the algorithm to produce correct results are derived using a path-based approach. Previous sequential algorithms for this task have been based on algorithms that compute the digital Euclidean distance transform. It is shown that the latter approach is not well-suited for distances based on neighbourhood sequences.


Short Path Grid Point Weighted Distance Image Domain Sequential Algorithm 
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.
    Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. Journal of the ACM 13(4), 471–494 (1966)MATHCrossRefGoogle Scholar
  2. 2.
    Rosenfeld, A., Pfaltz, J.L.: Distance functions in digital pictures. Pattern Recognition 1(1), 33–61 (1968)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Borgefors, G.: Distance transformations in arbitrary dimensions. Computer Vision, Graphics, and Image Processing 27, 321–345 (1984)CrossRefGoogle Scholar
  4. 4.
    Fouard, C., Strand, R., Borgefors, G.: Weighted distance transforms generalized to modules and their computation on point lattices. Technical report, Centre for Image Analysis, Uppsala University, Uppsala, Sweden (Internal report 38) (2006)Google Scholar
  5. 5.
    Yamashita, M., Honda, N.: Distance functions defined by variable neighbourhood sequences. Pattern Recognition 17(5), 509–513 (1984)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Das, P.P., Chakrabarti, P.P., Chatterji, B.N.: Distance functions in digital geometry. Information Sciences 42, 113–136 (1987)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kumar, M.A., Chatterji, B.N., Mukherjee, J., Das, P.P.: Representation of 2D and 3D binary images using medial circles and spheres. International Journal of Pattern Recognition and Artificial Intelligence 10(4), 365–387 (1996)CrossRefGoogle Scholar
  8. 8.
    Mukherjee, J., Kumar, M.A., Chatterji, B.N., Das, P.P.: Discrete shading of three-dimensional objects from medial axis transform. Pattern Recognition Letters 20(14), 1533–1544 (1999)CrossRefGoogle Scholar
  9. 9.
    Danielsson, P.E.: Euclidean distance mapping. Computer Graphics and Image Processing 14, 227–248 (1980)CrossRefGoogle Scholar
  10. 10.
    Danielsson, P.E.: Minimal error octagonal metric in two and three dimensions. Internal report LiTH-ISY-1-1382, Linköping University, Linköping, Sweden (1992)Google Scholar
  11. 11.
    Ragnemalm, I.: The Euclidean distance transform in arbitrary dimensions. Pattern Recognition Letters 14(11), 883–888 (1993)MATHCrossRefGoogle Scholar
  12. 12.
    Matej, S., Lewitt, R.M.: Efficient 3D grids for image reconstruction using spherically-symmetric volume elements. IEEE Transactions on Nuclear Science 42(4), 1361–1370 (1995)CrossRefGoogle Scholar
  13. 13.
    Garduno, E., Herman, G.T.: Optimization of basis functions for both reconstruction and visualization. Electronic Notes in Theoretical Computer Science 46, 1–17 (2001)CrossRefGoogle Scholar
  14. 14.
    Strand, R., Borgefors, G.: Distance transforms for three-dimensional grids with non-cubic voxels. Computer Vision and Image Understanding 100(3), 294–311 (2005)CrossRefGoogle Scholar
  15. 15.
    Strand, R.: The euclidean distance transform applied to the FCC and BCC grids. In: Marques, J.S., Pérez de la Blanca, N., Pina, P. (eds.) IbPRIA 2005. LNCS, vol. 3522, pp. 243–250. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Carvalho, B.M., Garduño, E., Herman, G.T.: Multiseeded fuzzy segmentation on the face centered cubic grid. In: Singh, S., Murshed, N., Kropatsch, W.G. (eds.) ICAPR 2001. LNCS, vol. 2013, p. 339. Springer, Heidelberg (2001)Google Scholar
  17. 17.
    Theussl, T., Möller, T., Gröller, M.E.: Optimal regular volume sampling. In: VIS 2001: Proceedings of the conference on Visualization 2001, pp. 91–98. IEEE Computer Society, Washington (2001)CrossRefGoogle Scholar
  18. 18.
    Nagy, B.: Distance functions based on neighbourhood sequences. Publicationes Mathematicae Debrecen 63(3), 483–493 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Strand, R., Nagy, B.: Some properties for distances based on neighbourhood sequences in the face-centered cubic grid and the body-centered cubic grid. Technical report, Centre for Image Analysis, Uppsala University, Uppsala, Sweden (Internal Report 39) (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Robin Strand
    • 1
  • Benedek Nagy
    • 3
    • 4
  • Céline Fouard
    • 1
  • Gunilla Borgefors
    • 2
  1. 1.Centre for Image AnalysisUppsala UniversityUppsalaSweden
  2. 2.Centre for Image AnalysisSwedish University of Agricultural SciencesUppsalaSweden
  3. 3.Department of Computer Science, Faculty of InformaticsUniversity of DebrecenDebrecenHungary
  4. 4.Research Group on Mathematical LinguisticsRovira i Virgili UniversityTarragonaSpain

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