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BY 4.0 license Open Access Published by De Gruyter Open Access December 9, 2020

Digital Objects in Rhombic Dodecahedron Grid

Ranita Biswas, Gaëlle Largeteau-Skapin, Rita Zrour and Eric Andres

Abstract

Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system.

MSC 2010: 52C

References

[1] Eric Andres. Modélisation Analytique Discrète d’Objets Géométrique. Habilitation à diriger des recherches, UFR Sciences Fondamentale et Appliquées, Université de Poitiers (France), Décembre 2000.Search in Google Scholar

[2] Eric Andres, Raj Acharya, and Claudio Sibata. Discrete Analytical Hyperplanes. Graphical Models and Image Processing, 59(5):302–309, 1997.10.1006/gmip.1997.0427Search in Google Scholar

[3] Eric Andres and Marie-Andrée Jacob. The Discrete Analytical Hyperspheres. IEEE Transactions on Visualization and Computer Graphics, 3:75–86, 1997.10.1109/2945.582354Search in Google Scholar

[4] Eric Andres, Philippe Nehlig, and Jean Françon. Supercover of straight lines, planes and triangles. In Discrete Geometry for Computer Imagery, pages 243–254, Berlin, Heidelberg, 1997.10.1007/BFb0024845Search in Google Scholar

[5] Ranita Biswas and Partha Bhowmick. From prima quadraginta octant to lattice sphere through primitive integer operations. Theoretical Computer Science, 624:56–72, 2016.10.1016/j.tcs.2015.11.018Search in Google Scholar

[6] Ranita Biswas and Partha Bhowmick. On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere. Journal of Mathematical Imaging and Vision, 59(1):69–83, 2017.10.1007/s10851-017-0718-4Search in Google Scholar

[7] Ranita Biswas, Gaëlle Largeteau-Skapin, Rita Zrour, and Eric Andres. Rhombic Dodecahedron Grid – Coordinate System and 3D Digital Object Definitions. In DGCI’2019, volume 11414, pages 27–37, Paris, France, 2019.10.1007/978-3-030-14085-4_3Search in Google Scholar

[8] Valentin E. Brimkov and Reneta P. Barneva. Analytical honeycomb geometry for raster and volume graphics. The Computer Journal, 48(2):180–199, 2005.10.1093/comjnl/bxh075Search in Google Scholar

[9] Lidija Comic and Paola Magillo. Repairing 3D Binary Images Using the FCC Grid. Journal of Mathematical Imaging and Vision, 61:1301–1321, 2019.10.1007/s10851-019-00904-0Search in Google Scholar

[10] Lidija Comic and Benedek Nagy. A combinatorial 3-coordinate system for the face centered cubic grid. 2015 9th International Symposium on Image and Signal Processing and Analysis (ISPA), pages 298–303, 2015.Search in Google Scholar

[11] Lidija Comic and Benedek Nagy. A combinatorial coordinate system for the body-centered cubic grid. Graphical Models, 87:11–22, 2016.10.1016/j.gmod.2016.08.001Search in Google Scholar

[12] Lidija Comic and Benedek Nagy. A topological 4-coordinate system for the face centered cubic grid. Pattern Recognition Letters, 83:67–74, 2016.10.1016/j.patrec.2016.03.012Search in Google Scholar

[13] Isabelle Debled-Rennesson. Reconnaissance des droites et plans discrets. PhD thesis, Université Louis Pasteur, Strasbourg, France, 1995.Search in Google Scholar

[14] Bernhard Finkbeiner, Usman R. Alim, Dimitri Van De Ville, and Torsten Moller. High-Quality Volumetric Reconstruction on Optimal Lattices for Computed Tomography. Computer Graphics Forum, 28(3):1023–1030, 2009.10.1111/j.1467-8659.2009.01445.xSearch in Google Scholar

[15] Artyom M. Grigoryan and Sos S. Agaian. 2D hexagonal quaternion Fourier transform in color image processing. In Mobile Multimedia/Image Processing, Security, and Applications 2016, volume 9869, page 98690N. International Society for Optics and Photonics, 2016.10.1117/12.2223115Search in Google Scholar

[16] Li-Jun He, Yong-Kui Liu, and Shi-Chang Sun. A Line Generation Algorithm on 3D Face-Centered Cubic Grid. Chinese Journal of Computers, 33(12):2407–2416, 2010.10.3724/SP.J.1016.2010.02407Search in Google Scholar

[17] Xiangjian He and Wenjing Jia. Hexagonal Structure for Intelligent Vision. In Information and Communication Technologies, 2005. ICICT 2005. First International Conference on, pages 52–64. IEEE, 2005.10.1109/ICICT.2005.1598543Search in Google Scholar

[18] Innchyn Her. Description of the F.C.C. Lattice Geometry Through a Four-Dimensional Hypercube. Acta Cryst., A51:659–662, 1995.10.1107/S0108767395001620Search in Google Scholar

[19] Luis Ibáñez, Chafiaâ Hamitouche, and Christian Roux. Ray-Tracing and 3-D Objects Representation in the BCC and FCC Grids. In Discrete Geometry for Computer Imagery, pages 235–241, Berlin, Heidelberg, 1997.10.1007/BFb0024844Search in Google Scholar

[20] Luis Ibáñez, Chafiaâ Hamitouche, and Christian Roux. A Vectorial Algorithm for Tracing Discrete Straight Lines in N-Dimensional Generalized Grids. IEEE Transactions on Visualization and Computer Graphics, 7(2):97–108, 2001.10.1109/2945.928163Search in Google Scholar

[21] Charles Kittel. Introduction to Solid State Physics. Wiley, 8 edition, 2004.Search in Google Scholar

[22] Girish Koshti, Ranita Biswas, Gaëlle Largeteau-Skapin, Rita Zrour, Eric Andres, and Partha Bhowmick. Sphere Construction on the FCC Grid Interpreted as Layered Hexagonal Grids in 3D. In Combinatorial Image Analysis, pages 82–96. Springer International Publishing, 2018.10.1007/978-3-030-05288-1_7Search in Google Scholar

[23] Vladimir A. Kovalevsky. Geometry of Locally Finite Spaces: Computer Agreeable Topology and Algorithms for Computer Imagery. Berlin: Editing House Dr. Barbel Kovalevski, 2008.Search in Google Scholar

[24] Marie-Ange Lebre, Antoine Vacavant, Manuel Grand-Brochier, Hugo Rositi, Robin Strand, Hubert Rosier, Armand Abergel, Pascal Chabrot, and Benoît Magnin. A robust multi-variability model based liver segmentation algorithm for CT-scan and MRI modalities. Computerized medical imaging and graphics : the oflcial journal of the Computerized Medical Imaging Society, 76:101635, September 2019.10.1016/j.compmedimag.2019.05.003Search in Google Scholar PubMed

[25] Troung Kieu Linh, Atsushi Imiya, Robin Strand, and Gunilla Borgefors. Supercover of Non-square and Non-cubic Grids. In Combinatorial Image Analysis, pages 88–97, Berlin, Heidelberg, 2004. IEEE, Springer.10.1007/978-3-540-30503-3_7Search in Google Scholar

[26] S. Matej and R. M. Lewitt. Eflcient 3D grids for image reconstruction using spherically-symmetric volume elements. IEEE Transactions on Nuclear Science, 42(4):1361–1370, 1995.10.1109/23.467854Search in Google Scholar

[27] Kazi Mostafa and Innchyn Her. An Edge Detection Method for Hexagonal Images. International Journal of Image Processing (IJIP), 10(4):161–173, 2016.Search in Google Scholar

[28] Benedek Nagy. Cellular Topology on the Triangular Grid. In Combinatorial Image Analaysis. IWCIA 2012, volume 7655, pages 143–153. Lecture Notes in Computer Science, 2012.10.1007/978-3-642-34732-0_11Search in Google Scholar

[29] Benedek Nagy. Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids. Annals of Mathematics and Artificial Intelligence, 75:117–134, October 2015.10.1007/s10472-014-9404-zSearch in Google Scholar

[30] Benedek Nagy and Robin Strand. A connection between ℤn and generalized triangular grids. Advances in Visual Computing, pages 1157–1166, 2008.10.1007/978-3-540-89646-3_115Search in Google Scholar

[31] Benedek Nagy and Robin Strand. Non-traditional grids embedded in ℤn. International Journal of Shape Modeling, 14:209–228, 2008.10.1142/S0218654308001154Search in Google Scholar

[32] Benedek Nagy and Robin Strand. Neighborhood sequences in the diamond grid: Algorithms with two and three neighbors. International Journal of Imaging Systems and Technology, 19(2):146–157, 2009.Search in Google Scholar

[33] Gergely Ferenc Racz and Balazs Csébfalvi. Cosine-weighted b-spline interpolation on the face-centered cubic lattice. Computer Graphics Forum, 37(3):503–511, 2018.10.1111/cgf.13437Search in Google Scholar

[34] Jean-Pierre Reveillès. Calcul en Nombres Entiers et Algorithmique. PhD thesis, Université Louis Pasteur, Strasbourg, France, 1991.Search in Google Scholar

[35] Ivan Stojmenović. Honeycomb network. In Proc. Math. Foundations of Computer Science - MFCS ‘95, Lecture Notes in Computer Science, volume 969, pages 267–276, 1995.10.1007/3-540-60246-1_133Search in Google Scholar

[36] Robin Strand and Gunilla Borgefors. Resolution Pyramids on the FCC and BCC Grids. In Discrete Geometry for Computer Imagery, pages 68–78, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.10.1007/978-3-540-31965-8_7Search in Google Scholar

[37] Robin Strand, Benedek Nagy, and Gunilla Borgefors. Digital distance functions on three-dimensional grids. Theoretical Computer Science, 412(15):1350–1363, 2011.10.1016/j.tcs.2010.10.027Search in Google Scholar

Received: 2020-02-04
Accepted: 2020-11-17
Published Online: 2020-12-09

© 2020 Ranita Biswas et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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