Abstract
Among the different discretization schemes that have been proposed and studiedin the literature, the supercover is a very natural one, and furthermore presents some interesting properties. On the other hand, an important structural property does not hold for the supercover in the classical framework: the supercover of a straight line (resp. a plane) is not a discrete curve (resp. surface) in general.
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References
E. Andrès, Modélisation analytique discrète d’objets géométriques, Thèse de HDR, Université de Poitiers (France), 2000.
E. Andrès, C. Sibata, R. Acharya, “Supercover 3D polygon”, Conf. on Discrete Geom. for Comp. Imag., Vol. 1176, Lect. Notes in Comp. Science, Springer Verlag, pp. 237–242, 1996.
P.S. Alexandro., “Diskrete Räume”, Mat. Sbornik, 2, pp. 501–518, 1937.
P.S. Alexandroff, H. Hopf, Topologie, Springer Verlag, 1937.
G. Bertrand, “New notions for discrete topology”, 8th Conf. on Discrete Geom. for Comp. Imag., Vol. 1568, Lect. Notes in Comp. Science, Springer Verlag, pp. 216–226, 1999.
G. Bertrand, M. Couprie, “A model for digital topology”, 8th Conf. on Discrete Geom. for Comp. Imag., Vol. 1568, Lect. Notes in Comp. Science, Springer Verlag, pp. 229–241, 1999.
J. Bresenham, “Algorithm for computer control of digital plotter”, IBM System Journal, Vol. 4, pp. 25–30, 1965.
V.E. Brimkov, E. Andrès, R.P. Barneva, “Object discretization in higher dimensions”, 9th Conf. on Discrete Geom. for Comp. Imag., Vol. 1953, Lect. Notes in Comp. Science, Springer Verlag, pp. 210–221, 2000.
J.M. Chassery, A. Montanvert, Géométrie discrète en imagerie, Hermès, Paris, France, 1991.
D. Cohen-Or, A. Kaufman, “Fundamentals of surface voxelization”, Graphical models and image processing, 57(6), pp. 453–461, 1995.
A.V. Evako, R. Kopperman, Y.V. Mukhin, “Dimensional Properties of Graphs and Digital Spaces”, Jour. of Math. Imaging and Vision, 6, pp. 109–119, 1996.
E.D. Khalimsky, “On topologies of generalized segments”, Soviet Math. Doklady, 10, pp. 1508–1511, 1969.
E.D. Khalimsky, R. Kopperman, P. R. Meyer, “Computer Graphics and Connected Topologies on Finite Ordered Sets”, Topology and its Applications, 36, pp. 1–17, 1990.
R. Klette, “m-dimensional cellular spaces”, internal report, University of Maryland, CAR-TR-6, MCS-82-18408, CS-TR-1281, 1983.
R. Klette, “The m-dimensional grid point space”, Computer vision, graphics, and image processing, 30, pp. 1–12, 1985.
T.Y. Kong and A. Rosenfeld, “Digital topology: introduction and survey”, Comp. Vision, Graphics and Image Proc., 48, pp. 357–393, 1989.
J. Koplowitz, “On the performance of chain codes for quantization of line drawings”, IEEE Trans. on PAMI, 3, pp. 180–185, 1981.
V.A. Kovalevsky, “Topological foundations of shape analysis”, in Shape in Pictures, NATO ASI Series, Series F, Vol. 126, pp. 21–36, 1994.
L.J. Latecki, Discrete representation of spatial objects in computer vision, Kluwer Academic Publishers, 1998.
J-P. Reveillès, Géométrie discrète, calcul en nombres entiers et algorithmique, Thèse d’état, Université Louis Pasteur, Strasbourg (France), 1991.
C. Ronse, M. Tajine, “Hausdorff discretization of algebraic sets and diophantine sets”, 9th Conf. on Discrete Geom. for Comp. Imag., Vol. 1953, Lect. Notes in Comp. Science, Springer Verlag, pp. 216–226, 2000.
C. Ronse, M. Tajine, “Hausdorff discretization for cellular distances, and its relation to cover and supercover discretizations”, Journal of Visual Communication and Image Representation, Vol. 12, no. 2, pp. 169–200, 2001.
A. Rosenfeld, A.C. Kak: Digital picture processing, Academic Press, 1982.
J. Serra, Image Analysis and Mathematical Morphology, Academic Press, 1982.
J. Webster, “Cell complexes and digital convexity”, Digital and image geometry, Vol. 2243, Lect. Notes in Comp. Science, Springer Verlag, pp. 268–278, 2002.
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Couprie, M., Bertrand, G., Kenmochi, Y. (2002). Discretization in 2D and 3D Orders. In: Braquelaire, A., Lachaud, JO., Vialard, A. (eds) Discrete Geometry for Computer Imagery. DGCI 2002. Lecture Notes in Computer Science, vol 2301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45986-3_27
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