STRONG k-DEFORMATION RETRACT AND ITS APPLICATIONS

Title & Authors
STRONG k-DEFORMATION RETRACT AND ITS APPLICATIONS
Han, Sang-Eon;

Abstract
In this paper, we study a strong k-deformation retract derived from a relative k-homotopy and investigate its properties in relation to both a k-homotopic thinning and the k-fundamental group. Moreover, we show that the k-fundamental group of a wedge product of closed k-curves not k-contractible is a free group by the use of some properties of both a strong k-deformation retract and a digital covering. Finally, we write an algorithm for calculating the k-fundamental group of a dosed k-curve by the use of a k-homotopic thinning.
Keywords
digital image;digital k-graph;$\small{(k_0,k_1)}$-homeomorphism;$\small{(k_0,k_1)}$-isomorphism;strongly local $\small{(k_0,k_1)}$-isomorphism;k-fundamental group;simple k-curve point;simple k-point;k-thinning algorithm;simply k-connected;k-homotopy equivalence;$\small{(k_0,k_1)}$-homotopy equivalence;k-homotopic thinning;strong k-defprmation retract;digital covering;discrete topology;digital topology;
Language
English
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REMARKS ON DIGITAL HOMOTOPY EQUIVALENCE,;

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2.
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6.
REMARK ON GENERALIZED UNIVERSAL COVERING SPACE IN DIGITAL COVERING THEORY,;

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7.
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8.
COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3,;

호남수학학술지, 2010. vol.32. 1, pp.141-155
9.
PROPERTIES OF A GENERALIZED UNIVERSAL COVERING SPACE OVER A DIGITAL WEDGE,;

호남수학학술지, 2010. vol.32. 3, pp.375-387
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KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS,;

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11.
EXTENSION PROBLEM OF SEVERAL CONTINUITIES IN COMPUTER TOPOLOGY,;

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ARRANGEMENT OF ELEMENTS OF LOCALLY FINITE TOPOLOGICAL SPACES UP TO AN ALF-HOMEOMORPHISM,;;

호남수학학술지, 2011. vol.33. 4, pp.617-628
13.
CLASSIFICATION OF SPACES IN TERMS OF BOTH A DIGITIZATION AND A MARCUS WYSE TOPOLOGICAL STRUCTURE,;;

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14.
IDENTIFICATION METHOD FOR DIGITAL SPACES,;

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AN EQUIVALENT PROPERTY OF A NORMAL ADJACENCY OF A DIGITAL PRODUCT,;

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16.
REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY,;

호남수학학술지, 2014. vol.36. 3, pp.519-530
17.
UTILITY OF DIGITAL COVERING THEORY,;;

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18.
COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT,;

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3.
REGULAR COVERING SPACE IN DIGITAL COVERING THEORY AND ITS APPLICATIONS, Honam Mathematical Journal, 2009, 31, 3, 279
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Ultra regular covering space and its automorphism group, International Journal of Applied Mathematics and Computer Science, 2010, 20, 4
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PROPERTIES OF A GENERALIZED UNIVERSAL COVERING SPACE OVER A DIGITAL WEDGE, Honam Mathematical Journal, 2010, 32, 3, 375
8.
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9.
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Cartesian Product of the Universal Covering Property, Acta Applicandae Mathematicae, 2009, 108, 2, 363
11.
COMMUTATIVE MONOID OF THE SET OF k-ISOMORPHISM CLASSES OF SIMPLE CLOSED k-SURFACES IN Z3, Honam Mathematical Journal, 2010, 32, 1, 141
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UTILITY OF DIGITAL COVERING THEORY, Honam Mathematical Journal, 2014, 36, 3, 695
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ARRANGEMENT OF ELEMENTS OF LOCALLY FINITE TOPOLOGICAL SPACES UP TO AN ALF-HOMEOMORPHISM, Honam Mathematical Journal, 2011, 33, 4, 617
15.
CLASSIFICATION OF SPACES IN TERMS OF BOTH A DIGITIZATION AND A MARCUS WYSE TOPOLOGICAL STRUCTURE, Honam Mathematical Journal, 2011, 33, 4, 575
16.
COMPARISON AMONG SEVERAL ADJACENCY PROPERTIES FOR A DIGITAL PRODUCT, Honam Mathematical Journal, 2015, 37, 1, 135
17.
EXTENSION PROBLEM OF SEVERAL CONTINUITIES IN COMPUTER TOPOLOGY, Bulletin of the Korean Mathematical Society, 2010, 47, 5, 915
18.
REMARKS ON DIGITAL HOMOTOPY EQUIVALENCE, Honam Mathematical Journal, 2007, 29, 1, 101
19.
AN EQUIVALENT PROPERTY OF A NORMAL ADJACENCY OF A DIGITAL PRODUCT, Honam Mathematical Journal, 2014, 36, 1, 199
20.
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21.
IDENTIFICATION METHOD FOR DIGITAL SPACES, Honam Mathematical Journal, 2011, 33, 1, 51
22.
Existence Problem of a Generalized Universal Covering Space, Acta Applicandae Mathematicae, 2010, 109, 3, 805
23.
REMARK ON GENERALIZED UNIVERSAL COVERING SPACE IN DIGITAL COVERING THEORY, Honam Mathematical Journal, 2009, 31, 3, 267
24.
Multiplicative Property of the Digital Fundamental Group, Acta Applicandae Mathematicae, 2010, 110, 2, 921
25.
Extension of continuity of maps between axiomatic locally finite spaces, International Journal of Computer Mathematics, 2011, 88, 14, 2889
26.
KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS, Journal of the Korean Mathematical Society, 2010, 47, 5, 1031
27.
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References
1.
G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters 15 (1994), no. 10, 1003-1011

2.
G. Bertrand and R. Malgouyres, ome topological properties of surfaces in $Z^3$, J. Math. Imaging Vision 11 (1999), no. 3, 207-221

3.
L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10 (1999), no. 1, 51-62

4.
A. I. Bykov, L. G. Zerkalov, and M. A. Rodriguez Pineda, Index of point of 3-D digital binary image and algorithm for computing its Euler characteristic, Pattern Recognition 32 (1999), no. 5, 845-50

5.
J. Dontchev and H. Maki, Groups of $\theta$ -generalized homeomorphisms and the digital line, Topology Appl. 95 (1999), no. 2, 113–28

6.
S. E. Han, Algorithm for discriminating digital images w. r. t. a digital $(k_0, k_1)$- homoeomorphism, J. Appl. Math. Comput. 18 (2005), no. 1-2, 505–12

7.
S. E. Han, Comparison between digital continuity and computer continuity, Honam Math. J. 26 (2004), no. 3, 331-339

8.
S. E. Han, Computer topology and its applications, Honam Math. J. 25 (2003), no. 1, 153-162

9.
S. E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (2006), no. 3, 332-348

10.
S. E. Han, Digital coverings and their applications, J. Appl. Math. Comput. 18 (2005), no. 1-2, 487-495

11.
S. E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences 177 (2007), no. 16, 3314-326

12.
S. E. Han, Discrete Homotopy of a Closed k-Surface, LNCS 4040, Springer-Verlag Berlin, pp.214–25 (2006)

13.
S. E. Han, Equivalent $(k_0, k_1$ -covering and generalized digital lifting, Information Sciences, www.sciencedirect.com (Articles in press), 2007

14.
S. E. Han, Erratum to: 'Non-product property of the digital fundamental group' , Inform. Sci. 176 (2006), no. 2, 215-216

15.
S. E. Han, Generalized digital $(k_0, k_1$ -homeomorphism, Note Mat. 22 (2003/04), no. 2, 157-166

16.
S. E. Han, Minimal digital pseudotorus with k-adjacency, $k\;{\in}$ {6, 18, 26}, Honam Math. J. 26 (2004), no. 2, 237-246

17.
S. E. Han, Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Inform. Sci. 176 (2006), no. 2, 120-134

18.
S. E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171 (2005), no. 1-3, 73-91

19.
S. E. Han, On the simplicial complex stemmed from a digital graph, Honam Math. J. 27 (2005), no. 1, 115-129

20.
S. E. Han, Remarks on Digital k-homotopy equivalence, Honam Math. J. 29 (2007), no. 1, 101-118

21.
S. E. Han, The k-fundamental group of a closed k-surface, Inform. Sci. 177 (2007), no. 18, 3731-3748

22.
F. Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969

23.
E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987), 227-234

24.
E. Khalimsky, R. Kopperman, and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl. 36 (1990), no. 1, 1-17

25.
T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), no. 2, 159-166

26.
T. Y. Kong and A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996

27.
V. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphic, and Image processing 46 (1989), 141-161

28.
R. Malgouyres, Homotopy in two-dimensional digital images, Theoret. Comput. Sci. 230 (2000), no. 1-2, 221-233

29.
R. Malgouyres and A. Lenoir, Topology preservation within digital surfaces, Graphical Models 62 (2000), no. 2, 71-84

30.
W. S. Massey, Algebraic Topology, Springer-Verlag, New York, 1977

31.
A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters 4 (1986), no. 3, 177-184