Comparison among digital fundamental groups and its applications

• Authors: Sang-Eon Han
• Venue: Information Sciences: an International Journal
• Year: 2008

The notion of digital fundamental group was originated by Khalimsky [E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proc. IEEE Int. Conf. Syst. Man Cybernet. (1987) 227-234]. Motivated by this notion, three kinds of digital k-homotopies as well as the relative k-homotopy were established [R. Ayala, E. Dominguez, A.R. Frances, A. Quintero, Homotopy in digital spaces, Discrete Appl. Math. 125 (1) (2003) 3-24; L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis. 10 (1999) 51-62; S.E. Han, Connected sum of digital closed surfaces, Inform. Sci. 176 (3) (2006a) 332-348; T.Y. Kong, A digital fundamental group, Comput. Graphics 13 (1989) 159-166; R. Malgouyres, Homotopy in 2-dimensional digital images, Theor. Comput. Sci. 230 (2000) 221-233]. These four notions contributed to the development of three kinds of k-fundamental groups of a digital image (X,k). One was established by Kong [Kong, 1989] and Malgouyres [Malgouyres, 2000], and we denote by @p"K"M^k(X) this digital fundamental group. Another was developed by Boxer [Boxer, 1999] and extended by Han [Han, 2006a; S.E. Han, Discrete Homotopy of a closed k-surface, LNCS 4040, Springer-Verlag, Berlin, 2006b, pp. 214-225; S.E. Han, Equivalent (k"0,k"1)-covering and generalized digital lifting, Inform. Sci. 178 (2) (2008) 550-561] by using both the k-homotopic thinning [Han, 2006b; S.E. Han, Remarks on digital k-homotopy equivalence, Honam Math. J. 29 (1) (2007) 101-118] and Han's digital covering theory [S.E. Han, Digital coverings and their applications, J. Appl. Math. Comput. 18 (1-2) (2005) 487-495; Han, 2006b], which is denoted as @p"B"H^k(X) in this paper. The other was established by Ayala et al. by using the framework of a multilevel architecture [Ayala, 2003]. Since each of these digital k-fundamental groups has an intrinsic feature of its own and its usages depend on the situation. This study is focused on the first two notions, @p"K"M^k(X) and @p"B"H^k(X), and intended to show the strong merits of @p"B"H^k(X) in relation to the classification of digital images.