Abstract
A version of topology's fundamental group is developed for digital images in dimension at most 3 in [7] and [8]. In the latter paper, it is shown that such a digital image X ⊂ \(\mathcal{Z}^k \), k ≤ 3, has a continuous analog C(X) ⊂ Rk such that X has digital fundamental group isomorphic to Π1(C(X)). However, the construction of the digital fundamental group in [7] and [8] does not greatly resemble the classical construction of the fundamental group of a topological space. In the current paper, we show how classical methods of algebraic topology may be used to construct the digital fundamental group. We construct the digital fundamental group based on the notions of digitally continuous functions presented in [10] and digital homotopy [3]. Our methods are very similar to those of [6], which uses different notions of digital topology. We show that the resulting theory of digital fundamental groups is related to that of [7] and [8] in that it yields isomorphic fundamental groups for the digital images considered in the latter papers (for certain connectedness types).
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References
B.J. Ball and J. Ford, “Spaces of ANR's,” Fundamenta Mathematicae, Vol. 77, pp. 33–49, 1972.
K. Borsuk, Theory of Retracts, Polish Scientific Publishers: Warsaw, 1967.
L. Boxer, “Digitally continuous functions,” Pattern Recognition Letters, Vol. 15, pp. 833–839, 1994.
J. Dugundji, Topology, Allyn and Bacon: Boston, 1966.
G.T. Herman, “Oriented surfaces in digital spaces,” CVGIP: Graphical Models and Image Processing, Vol. 55, pp. 381–396, 1993.
E. Khalimsky, “Motion, deformation, and homotopy in finite spaces,” in Proceedings IEEE Intl. Conf. on Systems; Man; and Cybernetics, 1987, pp. 227–234.
T.Y. Kong, “A digital fundamental group,” Computers and Graphics, Vol. 13, pp. 159–166, 1989.
T.Y. Kong, A.W. Roscoe, and A. Rosenfeld, “Concepts of digital topology,” Topology and its Applications, Vol. 46, pp. 219–262, 1992.
W.S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace, and World: New York, 1967.
A. Rosenfeld, “'Continuous' functions on digital pictures,” Pattern Recognition Letters, Vol. 4, pp. 177–184, 1986.
Q.F. Stout, “Topological matching,” in Proc. 15th Annual Symp. on Theory of Computing, 1983, pp. 24–31.
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Boxer, L. A Classical Construction for the Digital Fundamental Group. Journal of Mathematical Imaging and Vision 10, 51–62 (1999). https://doi.org/10.1023/A:1008370600456
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DOI: https://doi.org/10.1023/A:1008370600456