Comparison among digital fundamental groups and its applications

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. Domínguez, A.R. Francés, 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 image. One was established by Kong [Kong, 1989] and Malgouyres [Malgouyres, 2000], and we denote by image 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 image-surface, LNCS 4040, Springer-Verlag, Berlin, 2006b, pp. 214–225; S.E. Han, Equivalent image-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 image-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 image 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, image and image, and intended to show the strong merits of image in relation to the classification of digital images.