Digital Khalimsky manifolds. (English) Zbl 1523.68125
Summary: We consider different possibilities to define digital manifolds that are locally homeomorphic to Khalimsky \(n\)-space. We prove existence and non-existence of certain types of Khalimsky manifolds. An embedding theorem is proved.
We introduce the join operator and use it to analyze the structure of adjacency neighborhoods and of intersections of neighborhoods in \(\mathbb Z^{n }\).
We introduce the join operator and use it to analyze the structure of adjacency neighborhoods and of intersections of neighborhoods in \(\mathbb Z^{n }\).
MSC:
| 68U03 | Computational aspects of digital topology |
References:
| [1] | Alexandrov, P.: Diskrete Räume. Mat. Sb. 2(44), 501–519 (1937) · Zbl 0018.09105 |
| [2] | Alexandrov, P., Hopf, H.: Topologie I. Springer, Berlin (1935) |
| [3] | Birkhoff, G.: Rings of sets. Duke Math. J. 3(3), 443–454 (1937) · Zbl 0017.19403 · doi:10.1215/S0012-7094-37-00334-X |
| [4] | Ciria, J.C., De Miguel, A., Domínguez, E., Francés, A.R., Quintero, A.: A maximum set of (26,6)-connected digital surfaces. In: Klette, R., Žunić, J. (eds.) Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 3322, pp. 291–306. Springer, Berlin (2004) · Zbl 1113.68573 |
| [5] | Couprie, M., Bertrand, G.: Simplicity surfaces: a new definition of surfaces in \(\mathbb{Z}\)3. In: Melter, R.A., Wu, A.Y., Latecki, L.J. (eds.) SPIE Vision Geometry, VII, vol. 3454, pp. 40–51 (1998) |
| [6] | Daragon, X., Couprie, M., Bertrand, G.: Derived neighborhoods and frontier orders. Discrete Appl. Math. 147(2–3), 227–243 (2005). MR MR2127076 (2006i:06008) · Zbl 1061.68165 · doi:10.1016/j.dam.2004.09.013 |
| [7] | Evako, A.V., Kopperman, R., Mukhin, Y.V.: Dimensional properties of graphs and digital spaces. J. Math. Imaging Vis. 6(2–3), 109–119 (1996). MR 97e:68143 · Zbl 1191.05088 · doi:10.1007/BF00119834 |
| [8] | Herman, G.T.: Geometry of Digital Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Cambridge (1998). MR 2000h:68228 |
| [9] | Herman, G.T., Webster, D.: A topological proof of a surface tracking algorithm. Comput. Vis. Graph. Image Process. 23, 162–177 (1983) · Zbl 0585.68087 · doi:10.1016/0734-189X(83)90110-X |
| [10] | Kenmochi, Y., Imiya, A., Ichikawa, A.: Discrete combinatorial geometry. Pattern Recogn. 30, 1719–1728 (1997) · Zbl 0886.68118 · doi:10.1016/S0031-3203(97)00001-0 |
| [11] | Khalimsky, E.D.: Finite, primitive and Euclidean spaces. J. Appl. Math. Simul. 1(3), 177–196 (1988). MR MR964806 (90g:54026) · Zbl 0679.54032 |
| [12] | Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Its Appl. 36(1), 1–17 (1990). MR 92c:54037 · Zbl 0709.54017 · doi:10.1016/0166-8641(90)90031-V |
| [13] | Klette, R.: Topologies on the planar orthogonal grid. In: 6th International Conference on Pattern Recognition (ICPR’02), pp. 354–357 (2002) |
| [14] | Yung Kong, T., Khalimsky, E.D.: Polyhedral analogs of locally finite topological spaces. In: General Topology and Applications (Middletown, CT, 1988) (New York). Lecture Notes in Pure and Appl. Math., vol. 123, pp. 153–164. Dekker, New York (1990). MR MR1057635 (91e:54106) |
| [15] | Yung Kong, T.: The Khalimsky topologies are precisely those simply connected topologies on \(\mathbb{Z}\) n whose connected sets include all 2n-connected sets but no (3 n )-disconnected sets. Theor. Comput. Sci. 305, 221–235 (2003) · Zbl 1072.68110 · doi:10.1016/S0304-3975(02)00710-7 |
| [16] | Kopperman, R.: The Khalimsky line as a foundation for digital topology. In: Ying-Lie, O., et al. (eds.) Shape in Picture. NATO ASI, vol. 126, pp. 3–20. Springer, Berlin (1994) |
| [17] | Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital spaces. Discrete Comput. Geom. 6(2), 155–161 (1991). MR 92f:68185 · Zbl 0738.68086 · doi:10.1007/BF02574681 |
| [18] | Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process 46, 141–161 (1989) · doi:10.1016/0734-189X(89)90165-5 |
| [19] | Massey, W.S.: A Basic Course in Algebraic Topology. Graduate Texts in Mathematics, vol. 127. Springer, New York (1991) MR MR1095046 (92c:55001) · Zbl 0725.55001 |
| [20] | Melin, E.: Digital surfaces and boundaries in Khalimsky spaces. J. Math. Imaging Vis. 28, 169–177 (2007) · Zbl 1523.68133 · doi:10.1007/s10851-007-0006-9 |
| [21] | Melin, E.: Locally finite spaces and the join operator. In: Banon, G., Barrera, J. (eds.) Mathematical Morphology and its Applications to Signal and Image Processing (Rio de Janeiro, Brazil) ISMM, pp. 63–74 (2007) |
| [22] | Melin, E.: Digital geometry and Khalimsky spaces. Uppsala Dissertations in Mathematics 54, Uppsala University (2008). http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8419 · Zbl 1138.65013 |
| [23] | Melin, E.: Continuous extension in topological digital spaces. Appl. Gen. Topol. 9, 51–61 (2008) |
| [24] | Morgenthaler, D.G., Rosenfeld, A.: Surfaces in three-dimensional digital images. Inf. Control 51(3), 227–247 (1981). MR 84j:68062 · Zbl 0502.68017 · doi:10.1016/S0019-9958(81)90290-4 |
| [25] | Munkres, J.R.: Topology. Prentice Hall, New York (2000) |
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