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$U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces

Year 2017, Volume: 46 Issue: 1, 127 - 147, 01.02.2017

Abstract


For $X\subset R^n$ let $(X, E_X^n)$ be the usual topological space induced by the $n$D Euclidean topological space $(R^n, E^n)$. Based on the upper limit ($U$-, for short) topology (resp. the lower limit ($L$-, for brevity) topology), after proceeding with a digitization of $(X, E_X^n)$, we obtain a $U$- (resp. an $L$-) digitized space denoted by $D_U(X)$ (resp. $D_L(X)$) in $Z^n$ [16]. Further considering $D_U(X)$ (resp. $D_L(X)$) with a digital $k$-connectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by $D_{U(k)}(X)$ (resp. $D_{L(k)}(X)$) in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of $(X, E_X^n)$, the present paper establishes the so called $U(k)$-homotopy (resp. $L(k)$-homotopy) which can be used to study homotopic properties of both $(X, E_X^n)$ and $D_{U(k)}(X)$ (resp. both $(X, E_X^n)$ and $D_{L(k)}(X)$). The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a $U(k)$-homotopy equivalence, an $L(k)$-homotopy equivalence and $k$-homotopy equivalence. Finally, we classify $(X, E_X^n)$ in terms of a $U(k)$-homotopy equivalence and an $L(k)$-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry. 

References

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  • L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision 10 (1999) 51-62.
  • V. E. Brimkov and, R. P. Barneva, Plane digitization and related combinatorial problems, Discrete Applied Mathematics 147 (2005) 169-186.
  • U. Eckhardt, L. J. Latecki, Topologies for the digital spaces Z2 and Z3, Computer Vision and Image Understanding 90(3) (2003) 295-312.
  • A. Gross and L. J. Latecki, A Realistic Digitization Model of Straight Lines, Computer Vision and Image Understanding 67(2) (1997) 131-142.
  • S.-E. Han, On the classication of the digital images up to a digital homotopy equivalence, The Jour. of Computer and Communications Research 10 (2000) 194-207.
  • S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005) 73-91.
  • S.-E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27(1) (2005) 115-129.
  • S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, Journal of Math- ematical Imaging and Vision 31 (1)(2008) 1-16.
  • S.-E. Han, KD-($k_0,k_1$)-homotopy equivalence and its applications, J. Korean Math. Soc. 47 (2010) 1031-1054.
  • S.-E. Han, Homotopy equivalence which is suitable for studying Khalimsky nD spaces, Topology Appl. 159 (2012) 1705-1714.
  • S.-E. Han, A digitization method of the Euclidean nD space associated with the Khalimsky adjacency structure, Computational & Applied Mathematics (2015), DOI 10.1007/s40314- 015-0223-6 (in press).
  • S.-E. Han and Sik Lee, Some properties of lattice-based K- and M-maps, Honam Mathe- matical Journal 38(3) (2016) 625-642.
  • S.-E. Han and Wei Yao, An MA-Digitization of Hausdor spaces by using a connectedness graph of the Marcus-Wyse topology, Discrete Applies Mathematics, 201 (2016) 358-371.
  • S.-E. Han and B.G. Park, Digital graph ($k_0,k_1$)-homotopy equivalence and its applications, http://atlas-conferences.com/c/a/k/b/35.htm(2003).
  • J.-M. Kang, S.-E. Han, K.-C. Min, Digitizations associated with several types of digital topological approaches, Computational & Applied Mathematics (2015), DOI10.1007/s40314- 015-0245-0.
  • E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and Its Applications 36(1) (1991) 1-17.
  • E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE In- ternational Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  • C. O. Kiselman, Digital geometry and mathematical morphology, Lecture Notes, Uppsala University, Department of Mathematics, available at www.math.uu.se/ kiselman (2002).
  • R. Klette and A. Rosenfeld, Digital straightness, Discrete Applied Mathematics 139 (2004) 197-230.
  • G. Largeteau-Skapin, E. Andres, Discrete-Euclidean operations, Discrete Applied Mathe- matics 157 (2009) 510-523.
  • E. Melin, Continuous digitization in Khalimsky spaces, Journal of Approximation Theory 150 (2008) 96-116.
  • James R. Munkres, Topology, Prentice Hall, Inc. (2000).
  • C. Ronse, M. Tajinea, Discretization in Hausdor space, Journal of Mathematical Imaging and Vision 12 (2000) 219-242.
  • A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), 76-87.
  • A. Rosenfeld, Digital straight line segments, IEEE Trans. Comput, 23(12) (1974) 1264-1269.
  • E. H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.
  • F. Wyse and D. Marcus et al., Solution to problem 5712, Amer. Math. Monthly 77(1970) 1119.
Year 2017, Volume: 46 Issue: 1, 127 - 147, 01.02.2017

Abstract

References

  • P. Alexandor, Diskrete Räume, Mat. Sb. 2 (1937) 501-518.
  • L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision 10 (1999) 51-62.
  • V. E. Brimkov and, R. P. Barneva, Plane digitization and related combinatorial problems, Discrete Applied Mathematics 147 (2005) 169-186.
  • U. Eckhardt, L. J. Latecki, Topologies for the digital spaces Z2 and Z3, Computer Vision and Image Understanding 90(3) (2003) 295-312.
  • A. Gross and L. J. Latecki, A Realistic Digitization Model of Straight Lines, Computer Vision and Image Understanding 67(2) (1997) 131-142.
  • S.-E. Han, On the classication of the digital images up to a digital homotopy equivalence, The Jour. of Computer and Communications Research 10 (2000) 194-207.
  • S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005) 73-91.
  • S.-E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27(1) (2005) 115-129.
  • S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, Journal of Math- ematical Imaging and Vision 31 (1)(2008) 1-16.
  • S.-E. Han, KD-($k_0,k_1$)-homotopy equivalence and its applications, J. Korean Math. Soc. 47 (2010) 1031-1054.
  • S.-E. Han, Homotopy equivalence which is suitable for studying Khalimsky nD spaces, Topology Appl. 159 (2012) 1705-1714.
  • S.-E. Han, A digitization method of the Euclidean nD space associated with the Khalimsky adjacency structure, Computational & Applied Mathematics (2015), DOI 10.1007/s40314- 015-0223-6 (in press).
  • S.-E. Han and Sik Lee, Some properties of lattice-based K- and M-maps, Honam Mathe- matical Journal 38(3) (2016) 625-642.
  • S.-E. Han and Wei Yao, An MA-Digitization of Hausdor spaces by using a connectedness graph of the Marcus-Wyse topology, Discrete Applies Mathematics, 201 (2016) 358-371.
  • S.-E. Han and B.G. Park, Digital graph ($k_0,k_1$)-homotopy equivalence and its applications, http://atlas-conferences.com/c/a/k/b/35.htm(2003).
  • J.-M. Kang, S.-E. Han, K.-C. Min, Digitizations associated with several types of digital topological approaches, Computational & Applied Mathematics (2015), DOI10.1007/s40314- 015-0245-0.
  • E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and Its Applications 36(1) (1991) 1-17.
  • E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE In- ternational Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  • C. O. Kiselman, Digital geometry and mathematical morphology, Lecture Notes, Uppsala University, Department of Mathematics, available at www.math.uu.se/ kiselman (2002).
  • R. Klette and A. Rosenfeld, Digital straightness, Discrete Applied Mathematics 139 (2004) 197-230.
  • G. Largeteau-Skapin, E. Andres, Discrete-Euclidean operations, Discrete Applied Mathe- matics 157 (2009) 510-523.
  • E. Melin, Continuous digitization in Khalimsky spaces, Journal of Approximation Theory 150 (2008) 96-116.
  • James R. Munkres, Topology, Prentice Hall, Inc. (2000).
  • C. Ronse, M. Tajinea, Discretization in Hausdor space, Journal of Mathematical Imaging and Vision 12 (2000) 219-242.
  • A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), 76-87.
  • A. Rosenfeld, Digital straight line segments, IEEE Trans. Comput, 23(12) (1974) 1264-1269.
  • E. H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.
  • F. Wyse and D. Marcus et al., Solution to problem 5712, Amer. Math. Monthly 77(1970) 1119.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Sang-eon Han

Publication Date February 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 1

Cite

APA Han, S.-e. (2017). $U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces. Hacettepe Journal of Mathematics and Statistics, 46(1), 127-147.
AMA Han Se. $U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces. Hacettepe Journal of Mathematics and Statistics. February 2017;46(1):127-147.
Chicago Han, Sang-eon. “$U(k)$- and $L(k)$-Homotopic Properties of Digitizations of $n$D Hausdorff Spaces”. Hacettepe Journal of Mathematics and Statistics 46, no. 1 (February 2017): 127-47.
EndNote Han S-e (February 1, 2017) $U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces. Hacettepe Journal of Mathematics and Statistics 46 1 127–147.
IEEE S.-e. Han, “$U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, pp. 127–147, 2017.
ISNAD Han, Sang-eon. “$U(k)$- and $L(k)$-Homotopic Properties of Digitizations of $n$D Hausdorff Spaces”. Hacettepe Journal of Mathematics and Statistics 46/1 (February 2017), 127-147.
JAMA Han S-e. $U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46:127–147.
MLA Han, Sang-eon. “$U(k)$- and $L(k)$-Homotopic Properties of Digitizations of $n$D Hausdorff Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, 2017, pp. 127-4.
Vancouver Han S-e. $U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46(1):127-4.