Advertisement

Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space

  • Shivam Dwivedi
  • Aniket Gupta
  • Siddhant Roy
  • Ranita Biswas
  • Partha Bhowmick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10502)

Abstract

Space filling circles and spheres have various applications in mathematical imaging and physical modeling. In this paper, we first show how the thinnest (i.e., 2-minimal) model of digital sphere can be augmented to a space filling model by fixing certain “simple voxels” and “filler voxels” associated with it. Based on elementary number-theoretic properties of such voxels, we design an efficient incremental algorithm for generation of these space filling spheres with successively increasing radius. The novelty of the proposed technique is established further through circular space filling on 3D digital plane. As evident from a preliminary set of experimental result, this can particularly be useful for parallel computing of 3D Voronoi diagrams in the digital space.

Keywords

Digital circle Digital sphere Space filling curve Space filling surface Spherical propagation Voronoi diagram 

References

  1. 1.
    Andres, E., Jacob, M.-A.: The discrete analytical hyperspheres. IEEE TVCG 3, 75–86 (1997)Google Scholar
  2. 2.
    Aurenhammer, F.: Voronoi diagrams–a survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)CrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F., Klein, R., Lee, D.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bera, S., Bhowmick, P., Bhattacharya, B.B.: A digital-geometric algorithm for generating a complete spherical surface in \({\mathbb{{Z}}}^3\). In: Gupta, P., Zaroliagis, C. (eds.) ICAA 2014. LNCS, vol. 8321, pp. 49–61. Springer, Cham (2014). doi: 10.1007/978-3-319-04126-1_5 CrossRefGoogle Scholar
  5. 5.
    Bera, S., Bhowmick, P., Bhattacharya, B.B.: On the characterization of absentee-voxels in a spherical surface and volume of revolution in \({\mathbb{{Z}}}^3\). JMIV 56, 535–553 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bera, S., Bhowmick, P., Stelldinger, P., Bhattacharya, B.B.: On covering a digital disc with concentric circles in \({\mathbb{{Z}}}^2\). TCS 506, 1–16 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Biswas, R., Bhowmick, P.: Layer the sphere. Vis. Comput. 31, 787–797 (2015)CrossRefGoogle Scholar
  8. 8.
    Biswas, R., Bhowmick, P.: From prima quadraginta octant to lattice sphere through primitive integer operations. TCS 624, 56–72 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Biswas, R., Bhowmick, P.: On the functionality and usefulness of quadraginta octants of naive sphere. JMIV (2017). doi: 10.1007/s10851-017-0718-4 zbMATHGoogle Scholar
  10. 10.
    Biswas, R., Bhowmick, P., Brimkov, V.E.: On the connectivity and smoothness of discrete spherical circles. In: Barneva, R.P., Bhattacharya, B.B., Brimkov, V.E. (eds.) IWCIA 2015. LNCS, vol. 9448, pp. 86–100. Springer, Cham (2015). doi: 10.1007/978-3-319-26145-4_7 CrossRefGoogle Scholar
  11. 11.
    Brimkov, V.E.: Formulas for the number of \((n-2)\)-gaps of binary objects in arbitrary dimension. DAM 157, 452–463 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cao, T., Edelsbrunner, H., Tan, T.: Triangulations from topologically correct digital Voronoi diagrams. Comput. Geom. 48, 507–519 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cohen-Or, D., Kaufman, A.: 3D line voxelization and connectivity control. IEEE Comput. Graph 17, 80–87 (1997)CrossRefGoogle Scholar
  14. 14.
    Gouraud, H.: Continuous shading of curved surfaces. IEEE Trans. Comput. 20, 623–629 (1971)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kaufman, A.: Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes. SIGGRAPH 21, 171–179 (1987)CrossRefGoogle Scholar
  16. 16.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  17. 17.
    Roget, B., Sitaraman, J.: Wall distance search algorithm using voxelized marching spheres. In: ICCFD 2012, pp. 1–23 (2012)Google Scholar
  18. 18.
    Roget, B., Sitaraman, J.: Wall distance search algorithm using voxelized marching spheres. J. Comput. Phys. 241, 76–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rong, G., Tan, T.: Jump flooding in GPU with applications to Voronoi diagram and distance transform. In: Symposium on International 3D Graphics & Games, pp. 109–116 (2006)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Shivam Dwivedi
    • 1
  • Aniket Gupta
    • 1
  • Siddhant Roy
    • 1
  • Ranita Biswas
    • 1
  • Partha Bhowmick
    • 2
  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyRoorkeeIndia
  2. 2.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia

Personalised recommendations