Connectedness of Offset Digitizations in Higher Dimensions

  • Valentin E. Brimkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6026)


In this paper we investigate properties of a digital object obtained by taking the integer points within an offset of a certain radius of the object. Our considerations apply to digitizations of arbitrary path-connected sets in an arbitrary dimension n. Corollaries are derived for the important special case of surfaces, as well as for offsets of disconnected sets.


digital geometry digital curve digital surface digital object connectedness set offset path-connected set edge-connected/ disconnected graph spanning tree 


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  1. 1.
    Andres, E.: Discrete linear objects in dimension n: the standard model. Graphical Models 65(1-3), 92–111 (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    Andres, E.: Discrete circles, rings and spheres. Computers & Graphics 18(5), 695–706 (1994)CrossRefGoogle Scholar
  3. 3.
    Andres, E., Jacob, M.-A.: The discrete analytical hyperspheres. IEEE Trans. Vis. Comput. Graph. 3(1), 75–86 (1997)CrossRefGoogle Scholar
  4. 4.
    Anton, F.: Voronoi Diagrams of Semi-algebraic Sets. Ph.D. thesis, The University of British Columbia, Vancouver, British Columbia, Canada (2004)Google Scholar
  5. 5.
    Anton, F., Emiris, I., Mourrain, B., Teillaud, M.: The offset to an algebraic curve and an application to conics. In: Gervasi, O., Gavrilova, M.L., Kumar, V., Laganá, A., Lee, H.P., Mun, Y., Taniar, D., Tan, C.J.K. (eds.) ICCSA 2005. LNCS, vol. 3480, pp. 683–696. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. J. Pure and Applied Algebra 136(3), 199–209 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brimkov, V.E., Barneva, R.P., Brimkov, B., de Vieilleville, F.: Offset approach to defining 3D digital lines. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Porikli, F., Peters, J., Klosowski, J., Arns, L., Chun, Y.K., Rhyne, T.-M., Monroe, L. (eds.) ISVC 2008, Part I. LNCS, vol. 5358, pp. 678–687. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Brimkov, V.E., Barneva, R.P., Brimkov, B.: Minimal offsets that guarantee maximal or minimal connectivity of digital curves in nD. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 337–349. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Cohen-Or, D., Kaufman, A.: 3D line voxelization and connectivity control. IEEE Computer Graphics & Applications 17(6), 80–87 (1997)CrossRefGoogle Scholar
  10. 10.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, New York (1998)zbMATHGoogle Scholar
  11. 11.
    Debled-Rennesson, I.: Etude et Reconnaissance des Droites et Plans Discrets. Ph.D. Thesis, Université Louis Pasteur, Strasbourg, France (1995)Google Scholar
  12. 12.
    Debled-Rennesson, I., Domenjoud, E., Jamet, D.: Arithmetic discrete parabolas. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A., Meenakshisundaram, G., Pascucci, V., Zara, J., Molineros, J., Theisel, H., Malzbender, T. (eds.) ISVC 2006. LNCS, vol. 4292, pp. 480–489. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Figueiredo, O., Reveillès, J.-P.: New results about 3D digital lines. In: Melter, R.A., Wu, A.Y., Latecki, L. (eds.) Internat. Conference Vision Geometry V. SPIE, vol. 2826, pp. 98–108 (1996)Google Scholar
  14. 14.
    Fiorio, C., Jamet, D., Toutant, J.-L.: Discrete circles: An arithmetical approach based on norms. In: Internat. Conference Vision-Geometry XIV. San Jose, CA. SPIE, vol. 6066, 60660C (2006)Google Scholar
  15. 15.
    Hoffmann, C.M., Vermeer, P.J.: Eliminating extraneous solutions for the sparse resultant and the mixed volume. J. Symbolic Geom. Appl. 1(1), 47–66 (1991)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Jonas, A., Kiryati, N.: Digital representation schemes for 3D curves. Pattern Recognition 30(11), 1803–1816 (1997)CrossRefGoogle Scholar
  17. 17.
    Kim, C.E.: Three dimensional digital line segments. IEEE Transactions on Pattern Analysis and Machine Intelligence 5(2), 231–234 (1983)zbMATHCrossRefGoogle Scholar
  18. 18.
    Klette, R., Rosenfeld, A.: Digital Geometry – Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  19. 19.
    Kong, T.Y.: Digital topology. In: Davis, L.S. (ed.) Foundations of Image Understanding, pp. 33–71. Kluwer, Boston (2001)Google Scholar
  20. 20.
    Menger, K.: Kurventheorie. Teubner (1932)Google Scholar
  21. 21.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957)Google Scholar
  22. 22.
    Rosenfeld, A.: Connectivity in digital pictures. Journal of the ACM 17(3), 146–160 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rosenfeld, A.: Arcs and curves in digital pictures. Journal of the ACM 20(1), 81–87 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Toutant, J.-L.: Characterization of the closest discrete approximation of a line in the 3-dimensional space. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A., Meenakshisundaram, G., Pascucci, V., Zara, J., Molineros, J., Theisel, H., Malzbender, T. (eds.) ISVC 2006. LNCS, vol. 4291, pp. 618–627. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Urysohn, O.: Über die allegemeinen Cantorischen Kurven. In: Proc. Ann. Meeting, Deutsche Mathematiker Vereinigung (1923)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Valentin E. Brimkov
    • 1
  1. 1.Mathematics DepartmentSUNY Buffalo State CollegeBuffaloUSA

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