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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].

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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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Authors and Affiliations

  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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