Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, Cilt: 49 Sayı: 1, 236 - 253, 06.02.2020
https://doi.org/10.15672/hujms.546983

Öz

Kaynakça

  • [1] P. Alexandroff, Diskrete Rume, Mat. Sb. 2, 501–518, 1937.
  • [2] V.E. Brimkov and R.P. Barneva, Plane digitization and related combinatorial problems, Discrete Appl. Math. 147, 169–186, 2005.
  • [3] V.A. Chatyrko, S.-E. Han, and Y. Hattori, Some remarks concerning semi-$T_{\frac{1}{2}}$ spaces, Filomat 28 (1), 21–25, 2014.
  • [4] U. Eckhardta and L.J. Latecki, Topologies for the digital spaces $Z^2$ and $Z^3$, Comput. Vis. Image Underst. 90 (3), 295–312, 2003.
  • [5] A. Gross and L.J. Latecki, A realistic digitization model of straight lines, Comput. Vis. Image Underst. 67 (2), 131–142, 1997.
  • [6] S.-E. Han, On the classification of the digital images up to a digital homotopy equivalence, J. Comput. Commun. Res. 10, 194–207, 2000.
  • [7] S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, J. Math. Imaging Vis. 31 (1), 1–16, 2008.
  • [8] S.-E. Han, KD-$(k_0, k_1)$-homotopy equivalence and its applications, J. Korean Math. Soc. 47, 1031–1054, 2010.
  • [9] S.-E. Han, Homotopy equivalence which is suitable for studying Khalimsky ndimensional spaces, Topol. Appl. 159, 1705–1714, 2012.
  • [10] S.-E. Han, Existence of the category $DTC_2(k)$ which is equivalent to the given category $KAC_2$, Ukranian Math. J. 76 (8), 1264–1276, 2016.
  • [11] S.-E. Han, A digitization method of the Euclidean nD space associated with the Khalimsky adjacency structure, Comput. Appl. Math. 36, 127–144, 2017.
  • [12] S.-E. Han, U(k)- and L(k)-homotopic properties of digitizations of nD Hausdorff spaces, Hacet. J. Math. Stat. 46 (1), 124–144, 2017.
  • [13] S.-E. Han, Homotopic properties of an MA-digitization of 2D Euclidean spaces, J. Comput. Sys. Sci. 95 (3), 165–175, 2018.
  • [14] S.-E. Han and S. Lee, Some properties of lattice-based K- and M-maps, Honam Math. J. 38 (3), 625–642, 2016.
  • [15] S.-E. Han and A. Sostak, A compression of digital images derived from a Khalimsky topological structure, Comput. Appl. Math. 32, 521–536, (2013).
  • [16] S.-E. Han and W. Yao, Homotopy based on Marcus-Wyse topology and its applications, Topol. Appl. 201, 358–371, 2016.
  • [17] S.-E. Han and W. Yao, An MA-digitization of Haussdorff spaces by using a connectedness graph of the Marcus-Wyse topology, Discrete Appl. Math., 216, 335–347, 2017.
  • [18] G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55, 381-396, 1993.
  • [19] J.-M. Kang, S.-E. Han, and K.-C. Min, Digitizations associated with several types of digital topological approaches, Comput. Appl. Math. 36, 571–597, 2017.
  • [20] E. Khalimsky, Pattern analysis of n-dimensional digital images, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics, 1599–1562, 1986.
  • [21] E. Khalimsky, Topological structures in computer sciences, J. Appl. Math. Simulat. 1 (1), 25–40, 1987.
  • [22] E. Khalimsky, R. Kopperman, and P.R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topol. Appl. 36 (1), 1–17, 1990.
  • [23] C.O. Kiselman, Digital geometry and mathematical morphology, Lecture Notes, Uppsala University, Department of Mathematics, available at www.math.uu.se/ kiselman, 2002.
  • [24] R. Klette and A. Rosenfeld, Digital straightness, Discrete Appl. Math. 139, 197–230, 2004.
  • [25] V. Kovalevsky, Axiomatic digital topology, J. Math. Imaging Vis. 26, 41–58, 2006.
  • [26] G. Largeteau-Skapin and E. Andres, Discrete-Euclidean operations, Discrete Appl. Math. 157, 510–523, 2009.
  • [27] E. Melin, Digital surfaces and boundaries in Khalimsky spaces, J. Math. Imaging Vis. 28, 169–177, 2007.
  • [28] E. Melin, Continuous digitization in Khalimsky spaces, J. Approx. Theory 150, 96– 116, 2008.
  • [29] C. Ronse and M. Tajinea, Discretization in Hausdorff space, J. Math. Imaging Vis. 12, 219–242, 2000.
  • [30] A. Rosenfeld, Digital straight line segments, IEEE Trans. Comput. 23(12), 1264–1269, 1974.
  • [31] A. Rosenfeld, Digital topology, Amer. Math. Monthly 86, 76-87, 1979.
  • [32] P. Stelldinger and U. Köthe, Connectivity preserving digitization of blurred binary images in 2D and 3D , Comput. Graph. 30, 70–76, 2006.
  • [33] E.H. Spanier, Algebraic topology, McGraw-Hill Inc., New York, 1966.

Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

Yıl 2020, Cilt: 49 Sayı: 1, 236 - 253, 06.02.2020
https://doi.org/10.15672/hujms.546983

Öz

For $X (\subset R^n)$, assume the subspace $(X, E_X^n)$ induced by the $n$-dimensional Euclidean topological space $(R^n, E^n)$. Let $Z$ be the set of integers. Khalimsky topology on $Z$, denoted by  $(Z, \kappa)$, is generated by the set $\{\{2m-1, 2m, 2m+1\}\,\vert\, m \in {Z}\}$ as a subbase. Besides, Khalimsky topology on  $Z^n, n \in N$, denoted by $(Z^n, \kappa^n)$, is a product topology induced by $({Z}, \kappa)$. Proceeding with a digitization of $(X, E_X^n)$ in terms of the Khalimsky ($K$-, for short) topology, we obtain a $K$-digitized space in ${Z}^n$, denoted by $D_K(X) (\subset {Z}^n$), which is a $K$-topological space. Considering further $D_K(X)$ with $K$-adjacency, we obtain a topological graph related to the $K$-topology (a $KA$-space for short) denoted by $D_{KA}(X)$ (see an algorithm in Section 3). Motivated by an $A$-homotopy between $A$-maps for $KA$-spaces,  the present paper establishes a new homotopy, called an $LA$-homotopy, which is suitable for studying homotopic properties of both $(X, E_X^n)$ and $D_{KA}(X)$ because a homotopy for Euclidean topological spaces has some limitations of digitizing $(X, E_X^n)$. The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces $(X, E_X^n)$, an $LA$-homotopy equivalence for spaces $(X, E_X^n)$, and an $A$-homotopy equivalence for $KA$-spaces $D_{KA}(X)$. Finally, we classify  $KA$-spaces (resp. $(X, E_X^n))$ via an $A$-homotopy equivalence (resp. an $LA$-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.

Kaynakça

  • [1] P. Alexandroff, Diskrete Rume, Mat. Sb. 2, 501–518, 1937.
  • [2] V.E. Brimkov and R.P. Barneva, Plane digitization and related combinatorial problems, Discrete Appl. Math. 147, 169–186, 2005.
  • [3] V.A. Chatyrko, S.-E. Han, and Y. Hattori, Some remarks concerning semi-$T_{\frac{1}{2}}$ spaces, Filomat 28 (1), 21–25, 2014.
  • [4] U. Eckhardta and L.J. Latecki, Topologies for the digital spaces $Z^2$ and $Z^3$, Comput. Vis. Image Underst. 90 (3), 295–312, 2003.
  • [5] A. Gross and L.J. Latecki, A realistic digitization model of straight lines, Comput. Vis. Image Underst. 67 (2), 131–142, 1997.
  • [6] S.-E. Han, On the classification of the digital images up to a digital homotopy equivalence, J. Comput. Commun. Res. 10, 194–207, 2000.
  • [7] S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, J. Math. Imaging Vis. 31 (1), 1–16, 2008.
  • [8] S.-E. Han, KD-$(k_0, k_1)$-homotopy equivalence and its applications, J. Korean Math. Soc. 47, 1031–1054, 2010.
  • [9] S.-E. Han, Homotopy equivalence which is suitable for studying Khalimsky ndimensional spaces, Topol. Appl. 159, 1705–1714, 2012.
  • [10] S.-E. Han, Existence of the category $DTC_2(k)$ which is equivalent to the given category $KAC_2$, Ukranian Math. J. 76 (8), 1264–1276, 2016.
  • [11] S.-E. Han, A digitization method of the Euclidean nD space associated with the Khalimsky adjacency structure, Comput. Appl. Math. 36, 127–144, 2017.
  • [12] S.-E. Han, U(k)- and L(k)-homotopic properties of digitizations of nD Hausdorff spaces, Hacet. J. Math. Stat. 46 (1), 124–144, 2017.
  • [13] S.-E. Han, Homotopic properties of an MA-digitization of 2D Euclidean spaces, J. Comput. Sys. Sci. 95 (3), 165–175, 2018.
  • [14] S.-E. Han and S. Lee, Some properties of lattice-based K- and M-maps, Honam Math. J. 38 (3), 625–642, 2016.
  • [15] S.-E. Han and A. Sostak, A compression of digital images derived from a Khalimsky topological structure, Comput. Appl. Math. 32, 521–536, (2013).
  • [16] S.-E. Han and W. Yao, Homotopy based on Marcus-Wyse topology and its applications, Topol. Appl. 201, 358–371, 2016.
  • [17] S.-E. Han and W. Yao, An MA-digitization of Haussdorff spaces by using a connectedness graph of the Marcus-Wyse topology, Discrete Appl. Math., 216, 335–347, 2017.
  • [18] G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55, 381-396, 1993.
  • [19] J.-M. Kang, S.-E. Han, and K.-C. Min, Digitizations associated with several types of digital topological approaches, Comput. Appl. Math. 36, 571–597, 2017.
  • [20] E. Khalimsky, Pattern analysis of n-dimensional digital images, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics, 1599–1562, 1986.
  • [21] E. Khalimsky, Topological structures in computer sciences, J. Appl. Math. Simulat. 1 (1), 25–40, 1987.
  • [22] E. Khalimsky, R. Kopperman, and P.R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topol. Appl. 36 (1), 1–17, 1990.
  • [23] C.O. Kiselman, Digital geometry and mathematical morphology, Lecture Notes, Uppsala University, Department of Mathematics, available at www.math.uu.se/ kiselman, 2002.
  • [24] R. Klette and A. Rosenfeld, Digital straightness, Discrete Appl. Math. 139, 197–230, 2004.
  • [25] V. Kovalevsky, Axiomatic digital topology, J. Math. Imaging Vis. 26, 41–58, 2006.
  • [26] G. Largeteau-Skapin and E. Andres, Discrete-Euclidean operations, Discrete Appl. Math. 157, 510–523, 2009.
  • [27] E. Melin, Digital surfaces and boundaries in Khalimsky spaces, J. Math. Imaging Vis. 28, 169–177, 2007.
  • [28] E. Melin, Continuous digitization in Khalimsky spaces, J. Approx. Theory 150, 96– 116, 2008.
  • [29] C. Ronse and M. Tajinea, Discretization in Hausdorff space, J. Math. Imaging Vis. 12, 219–242, 2000.
  • [30] A. Rosenfeld, Digital straight line segments, IEEE Trans. Comput. 23(12), 1264–1269, 1974.
  • [31] A. Rosenfeld, Digital topology, Amer. Math. Monthly 86, 76-87, 1979.
  • [32] P. Stelldinger and U. Köthe, Connectivity preserving digitization of blurred binary images in 2D and 3D , Comput. Graph. 30, 70–76, 2006.
  • [33] E.H. Spanier, Algebraic topology, McGraw-Hill Inc., New York, 1966.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Sang-eon Han 0000-0002-8030-8253

Yayımlanma Tarihi 6 Şubat 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 49 Sayı: 1

Kaynak Göster

APA Han, S.-e. (2020). Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics, 49(1), 236-253. https://doi.org/10.15672/hujms.546983
AMA Han Se. Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics. Şubat 2020;49(1):236-253. doi:10.15672/hujms.546983
Chicago Han, Sang-eon. “Homotopic Properties of $KA$-Digitizations of $n$-Dimensional Euclidean Spaces”. Hacettepe Journal of Mathematics and Statistics 49, sy. 1 (Şubat 2020): 236-53. https://doi.org/10.15672/hujms.546983.
EndNote Han S-e (01 Şubat 2020) Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics 49 1 236–253.
IEEE S.-e. Han, “Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces”, Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 1, ss. 236–253, 2020, doi: 10.15672/hujms.546983.
ISNAD Han, Sang-eon. “Homotopic Properties of $KA$-Digitizations of $n$-Dimensional Euclidean Spaces”. Hacettepe Journal of Mathematics and Statistics 49/1 (Şubat 2020), 236-253. https://doi.org/10.15672/hujms.546983.
JAMA Han S-e. Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics. 2020;49:236–253.
MLA Han, Sang-eon. “Homotopic Properties of $KA$-Digitizations of $n$-Dimensional Euclidean Spaces”. Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 1, 2020, ss. 236-53, doi:10.15672/hujms.546983.
Vancouver Han S-e. Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):236-53.