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On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in \({\mathbb Z}^3\)

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Abstract

We show that the construction of a digital sphere by circularly sweeping a digital semi-circle (generatrix) around its diameter results in the appearance of some holes (absentee-voxels) in its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration in the surface of revolution can ensure the required completeness. In this paper, we present a characterization of the absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semi-circular generatrix. Next, we design an algorithm to fill up the absentee-voxels so as to generate a spherical surface of revolution, which is complete and realistic from the viewpoint of visual perception. We also show how the proposed technique for absentee-filling can be used to generate a variety of digital surfaces of revolution by choosing an arbitrary curve as the generatrix. We further show that covering a solid sphere by a set of complete spheres also results to an asymptotically larger count of absentees, which is cubic in the radius of the sphere. A complete characterization of the absentee-voxels that aids the subsequent generation of a solid digital sphere is also presented. Test results have been furnished to substantiate our theoretical findings.

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Notes

  1. Absentee-voxels are also referred to as ‘tunnels’, since they connect the interior and the exterior of an otherwise closed digital surface [16].

References

  1. Andres, E.: Discrete circles, rings and spheres. Comput. Graph. 18(5), 695–706 (1994)

    Article  Google Scholar 

  2. Andres, E., Jacob, M.: The discrete analytical hyperspheres. IEEE Trans. Visual. Comput. Graph. 3(1), 75–86 (1997)

    Article  Google Scholar 

  3. Andres, E., Roussillon, T.: Analytical description of digital circles. In: Proceedings of the Discrete Geometry for Computer Imagery—16th IAPR International Conference, (DGCI’11), LNCS, vol. 6607, pp. 235–246 (2011)

  4. Bastl, B., Kosinka, J., Lávicka, M.: Simple and branched skins of systems of circles and convex shapes. Graph. Models 78, 1–9 (2015)

    Article  Google Scholar 

  5. Bera, S., Bhowmick, P., Bhattacharya, B.B.: A digital-geometric algorithm for generating a complete spherical surface in \({\mathbb{Z}}^{3}\). In: Proceedings of the International Conference on Applied Algorithms (ICAA’14), LNCS, vol. 8321, pp. 49–61 (2014)

  6. Bera, S., Bhowmick, P., Stelldinger, P., Bhattacharya, B.B.: On covering a digital disc with concentric circles in \({{\mathbb{Z}}^2}\). Theor. Comput. Sci. 506, 1–16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhowmick, P., Bhattacharya, B.B.: Number theoretic interpretation and construction of a digital circle. Discret. Appl. Math. 156(12), 2381–2399 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biswas, R., Bhowmick, P.: On finding spherical geodesic paths and circles in \({\mathbb{Z}}^{3}\). In: Proceedings of the Discrete Geometry for Computer Imagery, LNCS, vol. 8668, pp. 396–409. Springer (2014)

  9. Biswas, R., Bhowmick, P.: Layer the sphere—for accurate and additive voxelation by integer operation. Visual Comput. 31(6–8), 787–797 (2015)

    Article  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Biswas, R., Bhowmick, P.: From prima quadraginta octant to lattice sphere through primitive integer operations. Theor. Comput. Sci. doi:10.1016/j.tcs.2015.11.018

  12. Biswas, R., Bhowmick, P., Brimkov, V.E.: On the connectivity and smoothness of discrete spherical circles. In: Combinatorial Image Analysis—17th International Workshop, IWCIA, LNCS, vol. 9448, pp. 86–100 (2015)

  13. Biswas, R., Bhowmick, P., Brimkov, V.E.: On the polyhedra of graceful spheres and circular geodesics. Discret. Appl. Math. doi:10.1016/j.dam.2015.11.017

  14. Brimkov, V.E., Barneva, R.P.: Graceful planes and lines. Theor. Comput. Sci. 283(1), 151–170 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brimkov, V.E., Barneva, R.P.: On the polyhedral complexity of the integer points in a hyperball. Theor. Comput. Sci. 406(1–2), 24–30 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brimkov, V.E., Barneva, R.P., Brimkov, B.: Connected distance-based rasterization of objects in arbitrary dimension. Graph. Models 73, 323–334 (2011)

    Article  Google Scholar 

  17. Brimkov, V.E., Barneva, R.P., Brimkov, B., Vieilleville, F.: Offsetapproach to defining 3d digital lines. In: Proceedings of the 4th International Symposium on Advances in Visual Computing, ISVC ’08, pp. 678–687. Springer, Heidelberg (2008)

  18. Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity—a review. Discret. Appl. Math. 155(4), 468–495 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chamizo, F.: Lattice points in bodies of revolution. Acta Arith. 85(3), 265–277 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Chamizo, F., Cristobal, E.: The sphere problem and the \(L\)-functions. Acta Math. Hung. 135(1–2), 97–115 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chamizo, F., Cristóbal, E., Ubis, A.: Visible lattice points in the sphere. J. Number Theor. 126(2), 200–211 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chamizo, F., Cristóbal, E., Ubis, A.: Lattice points in rational ellipsoids. J. Math. Anal. Appl. 350(1), 283–289 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chan, Y.T., Thomas, S.M.: Cramer-Rao lower bounds for estimation of a circular arc center and its radius. Graph. Models Image Process. 57(6), 527–532 (1995)

    Article  MATH  Google Scholar 

  24. Cheng, H.L., Shi, X.: Quality mesh generation for molecular skin surfaces using restricted union of balls. Comput. Geom. 42(3), 196–206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Christ, T., Pálvölgyi, D., Stojakovic, M.: Consistent digital line segments. Discret. Comput. Geom. 47(4), 691–710 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chun, J., Korman, M., Nöllenburg, M., Tokuyama, T.: Consistent digital rays. Discret. Comput. Geom. 42(3), 359–378 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cochran, J.K.: Ceramic hollow spheres and their applications. Curr. Opin. Solid State Mater. Sci. 3(5), 474–479 (1998)

    Article  Google Scholar 

  28. Davies, E.R.: A hybrid sequential-parallel approach to accurate circle centre location. Pattern Recognit. Lett. 7, 279–290 (1988)

    Article  Google Scholar 

  29. Doros, M.: On some properties of the generation of discrete circular arcs on a square grid. Comput. Vision Graph. Image Process. 28(3), 377–383 (1984)

    Article  Google Scholar 

  30. Draine, B., Flatau, P.: Discrete dipole approximation for scattering calculations. J. Opt. Soc. Am. A 11, 1491–1499 (1994)

    Article  Google Scholar 

  31. Ewell, J.A.: Counting lattice points on spheres. Math. Intell. 22(4), 51–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Feschet, F., Reveillès, J.P.: A generic approach forn-dimensional digital lines. In: Proceedings of the 13th International Conference on Discrete Geometry for Computer Imagery, DGCI’06, pp. 29–40. Springer, Heidelberg (2006)

  33. Fiorio, C., Jamet, D., Toutant, J.L.: Discrete circles: an arithmetical approach with non-constant thickness. In: Longin Jean Latecki, A.Y.W., David M. Mount (eds.) Vision Geometry XIV, Electronic Imaging, SPIE, vol. 6066, p. 60660C. San Jose, USA (2006)

  34. Fiorio, C., Toutant, J.L.: Arithmetic discrete hyperspheres andseparatingness. In: Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery, DGCI’06, pp. 425–436. Springer, Heidelberg (2006)

  35. Foley, J.D., Dam, A.V., Feiner, S.K., Hughes, J.F.: Computer Graphics—Principles and Practice. Addison-Wesley, Reading (1993)

  36. Fomenko, O.: Distribution of lattice points over the four-dimensional sphere. J. Math. Sci. 110(6), 3164–3170 (2002)

    Article  MathSciNet  Google Scholar 

  37. Fukshansky, L., Henshaw, G., Liao, P., Prince, M., Sun, X., Whitehead, S.: On integral well-rounded lattices in the plane. Discrete & Computational Geometry 48(3), 735–748 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ghahramani, M., Garibov, A., Agayev, T.: Production and quality control of radioactive yttrium microspheres for medical applications. Appl. Radiat. Isot. 85, 87–91 (2014)

    Article  Google Scholar 

  39. Guo, L., Dong, X., Cui, X., Cui, F., Shi, J.: Morphology and dispersivity modulation of hollow microporous spheres synthesized by a hard template route. Mater. Lett. 63(1314), 1141–1143 (2009)

    Article  Google Scholar 

  40. Haralick, R.M.: A measure for circularity of digital figures. IEEE Trans. Syst. Man Cybern. 4, 394–396 (1974)

    Article  MATH  Google Scholar 

  41. Heath-Brown, D.R.: Lattice Points in the Sphere. Number Theory in Progress. Walter de Gruyter, Berlin (1999)

    MATH  Google Scholar 

  42. Hiller, J., Lipson, H.: Design and analysis of digital materials for physical 3D voxel printing. Rapid Prototyp. J. 15(2), 137–149 (2009)

    Article  Google Scholar 

  43. Honsberger, R.: Circles, squares, and lattice points. Math. Gems I, 117–127 (1973)

    Google Scholar 

  44. Kawashita, M., Shineha, R., Kim, H.M., Kokubo, T., Inoue, Y., Araki, N., Nagata, Y., Hiraoka, M., Sawada, Y.: Preparation of ceramic microspheres for in situ radiotherapy of deep-seated cancer. Biomaterials 24(17), 2955–2963 (2003)

    Article  Google Scholar 

  45. K\(\ddot{\text{ u }}\)hleitner, M.: On lattice points in rational ellipsoids: an omega estimate for the error term. In: Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg vol. 70(1), pp. 105–111 (2000)

  46. Kenmochi, Y., Buzer, L., Sugimoto, A., Shimizu, I.: Digital planarsurface segmentation using local geometric patterns. In: Proceedingsof the 14th IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI’08, pp. 322–333. Springer, Heidelberg (2008)

  47. Kim, O.: Rapid prototyping of electrically small spherical wire antennas. IEEE Transactions on Antennas and Propagation 62(7), 3839–3842 (2014)

    Article  Google Scholar 

  48. Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Series in Computer Graphics and Geometric Modeling. Morgan Kaufmann, San Francisco (2004)

    MATH  Google Scholar 

  49. Klette, R., Rosenfeld, A.: Digital straightness: a review. Discret. Appl. Math. 139(1–3), 197–230 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kulpa, Z.: On the properties of discrete circles, rings, and disks. Comput. Graph. Image Process. 10(4), 348–365 (1979)

    Article  Google Scholar 

  51. Kulpa, Z., Kruse, B.: Algorithms for circular propagation in discrete images. Comput. Vision Graph. Image Process. 24(3), 305–328 (1983)

    Article  Google Scholar 

  52. Kumar, G., Sharma, N., Bhowmick, P.: Wheel-throwing in digital space using number-theoretic approach. Int. J. Arts Technol. 4(2), 196–215 (2011)

    Article  Google Scholar 

  53. Lipson, H., Pollack, J.B.: Automatic design and manufacture of robotic lifeforms. Nature 406, 974–978 (2000)

    Article  Google Scholar 

  54. Maehara, H.: On a sphere that passes through \(n\) lattice points. Eur. J. Combin. 31(2), 617–621 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Magyar, A.: On the distribution of lattice points on spheres and level surfaces of polynomials. J. Number Theor. 122(1), 69–83 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Mignosi, F.: On the number of factors of Sturmian words. Theor. Comput. Sci. 82(1), 71–84 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  57. Montani, C., Scopigno, R.: Spheres-to-voxels conversion. In: Glassner, A.S. (ed.) Graphics gems, pp. 327–334. Academic Press Professional, Inc., San Diego (1990)

    Chapter  Google Scholar 

  58. Nagy, B.: Characterization of digital circles in triangular grid. Pattern Recognit. Lett. 25(11), 1231–1242 (2004)

    Article  Google Scholar 

  59. Nakamura, A., Aizawa, K.: Digital circles. Comput. Vision Graph. Image Process. 26(2), 242–255 (1984)

    Article  Google Scholar 

  60. Pal, S., Bhowmick, P.: Determining digital circularity using integer intervals. J. Math. Imaging Vis. 42(1), 1–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Roget, B., Sitaraman, J.: Wall distance search algorithm using voxelized marching spheres. J. Comput. Phys. 241, 76–94 (2013)

    Article  Google Scholar 

  62. Rossignac, J., Whited, B., Slabaugh, G., Fang, T., Unal, G.: Pearling: 3D interactive extraction of tubular structures from volumetric images. In: MICCAI Workshop on Interaction in Medical Image Analysis and Visualization (2007)

  63. Sene, F.F., Martinelli, J.R., Okuno, E.: Synthesis and characterization of phosphate glass microspheres for radiotherapy applications. J. Non Cryst. Solids 354(4244), 4887–4893 (2008)

    Article  Google Scholar 

  64. Stelldinger, P.: Image Digitization and its Influence on Shape Properties in Finite Dimensions. IOS Press, Amsterdam (2007)

    MATH  Google Scholar 

  65. Thomas, S.M., Chan, Y.T.: A simple approach for the estimation of circular arc center and its radius. Comput. Vision Graph. Image Process. 45(3), 362–370 (1989)

    Article  Google Scholar 

  66. Toutant, J.L., Andres, E., Roussillon, T.: Digital circles, spheres and hyperspheres: from morphological models to analytical characterizations and topological properties. Discret. Appl. Math. 161, 2662–2677 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. Tsang, K.M.: Counting lattice points in the sphere. Bull. Lond. Math. Soc. 32, 679–688 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wang, C., Liu, Z., Liu, L.: As-rigid-as-possible spherical parametrization. Graph. Models 76(5), 457–467 (2014)

    Article  Google Scholar 

  69. Woo, D.M., Han, S.S., Park, D.C., Nguyen, Q.D.: Extraction of 3d line segment using digital elevation data. In: Proceedings of the2008 Congress on Image and Signal Processing, CISP’08, vol. 2, pp. 734–738. IEEE Computer Society, Washington, DC, USA (2008)

  70. Yuen, P.C., Feng, G.C.: A novel method for parameter estimation of digital arc. Pattern Recognit. Lett. 17(9), 929–938 (1996)

    Article  Google Scholar 

  71. Zheng, M., Cao, J., Chang, X., Wang, J., Liu, J., Ma, X.: Preparation of oxide hollow spheres by colloidal carbon spheres. Mater. Lett. 60(24), 2991–2993 (2006)

    Article  Google Scholar 

  72. Zubko, E., Petrov, D., Grynko, Y., Shkuratov, Y., Okamoto, H., Muinonen, K., Nousiainen, T., Kimura, H., Yamamoto, T., Videen, G.: Validity criteria of the discrete dipole approximation. Appl. Opt. 49(8), 1267–1279 (2010)

    Article  Google Scholar 

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Acknowledgments

The authors are thankful to the anonymous reviewers for their constructive comments and suggestions, which have helped in shaping the paper up to its merit.

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  1. Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India

    Sahadev Bera & Bhargab B. Bhattacharya

  2. Computer Science and Engineering Department, Indian Institute Technology, Kharagpur, Kharagpur, India

    Partha Bhowmick

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  1. Sahadev Bera
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Correspondence to Partha Bhowmick.

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A preliminary version of this work appeared in ICAA’14 [5].

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Bera, S., Bhowmick, P. & Bhattacharya, B.B. On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in \({\mathbb Z}^3\) . J Math Imaging Vis 56, 535–553 (2016). https://doi.org/10.1007/s10851-016-0654-8

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