Construction of Persistent Voronoi Diagram on 3D Digital Plane

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10256)

Abstract

Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept.

Keywords

Digital Voronoi diagram 3D digital plane Distance metric Digital convexity Digital geometry 

References

  1. 1.
    Andres, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. Graph. Models Image Process. 59(5), 302–309 (1997)CrossRefGoogle Scholar
  2. 2.
    Aurenhammer, F.: Voronoi diagrams–a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)CrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F., Klein, R., Lee, D.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore (2013)CrossRefMATHGoogle Scholar
  4. 4.
    Biswas, R., Bhowmick, P.: On different topological classes of spherical geodesic paths and circles in \({\mathbb{{Z}}}^3\). Theor. Comput. Sci. 605, 146–163 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Biswas, R., Bhowmick, P.: From prima quadraginta octant to lattice sphere through primitive integer operations. Theor. Comput. Sci. 624, 56–72 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brimkov, V.E., Barneva, R.P.: Graceful planes and lines. Theor. Comput. Sci. 283(1), 151–170 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brimkov, V.E., Barneva, R.P.: Connectivity of discrete planes. Theor. Comput. Sci. 319(1–3), 203–227 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brimkov, V.E., Barneva, R.P.: Plane digitization and related combinatorial problems. Discrete Appl. Math. 147(2–3), 169–186 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity–a review. Discrete Appl. Math. 155(4), 468–495 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cao, T.T., Edelsbrunner, H., Tan, T.S.: Triangulations from topologically correct digital Voronoi diagrams. Comput. Geom. 48(7), 507–519 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Coeurjolly, D., Sivignon, I., Dupont, F., Feschet, F., Chassery, J.M.: On digital plane preimage structure. Discrete Appl. Math. 151(1–3), 78–92 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Graph. Models Image Process. 57(6), 453–461 (1995)CrossRefGoogle Scholar
  13. 13.
    Kim, C.E., Rosenfeld, A.: Digital straight lines and convexity of digital regions. IEEE Trans. Pattern Anal. Mach. Intell. 4(2), 149–153 (1982)CrossRefMATHGoogle Scholar
  14. 14.
    Klette, R., Stojmenović, I., Žunić, J.: A parametrization of digital planes by least square fits and generalizations. Graph. Models Image Process. 58, 295–300 (1996)CrossRefGoogle Scholar
  15. 15.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)MATHGoogle Scholar
  16. 16.
    Rong, G., Tan, T.S.: Jump flooding in GPU with applications to Voronoi diagram and distance transform. In: Proceedings of the 2006 Symposium on Interactive 3D Graphics and Games, pp. 109–116 (2006)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyRoorkeeIndia
  2. 2.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia

Personalised recommendations