Journal of Mathematical Imaging and Vision 10, 51–62 (1999) c  1999 Kluwer Academic Publishers. Manufactured in The Netherlands. A Classical Construction for the Digital Fundamental Group LAURENCE BOXER Department of Computer and Information Sciences, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, State University of New York at Buffalo boxer@niagara.edu 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 ⊂ Z k , k ≤ 3, has a continuous analog C ( X ) ⊂ R k such that X has digital fundamental group isomorphic to  1 (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). Keywords: digital image, digitally continuous, retraction, homotopy, fundamental group, digital topology 1. Introduction The beautiful results of [7] and [8] concerning the digi- tal fundamental group are based on equivalence classes of digital loops. This seems natural, as it is also the approach of algebraic topology. However, the equiv- alence classes of [7] and [8] rely more on loops as point sets rather than as continuous images of simple closed curves or intervals. Thus, it is desirable to refor- mulate the digital fundamental group in a fashion that more closely parallels the classical approach of alge- braic topology, in terms of continuity and homotopy. In this paper, we show the results of [7] and [8] can be obtained via such an approach, which generalizes to digital images of arbitrary dimension. We use the no- tions of digitally continuous function introduced in [10] (slightly generalized to handle various types of digital connectedness) and digital homotopy introduced in [3] as the basis of our construction. A referee pointed out the similarity of our approach to that of [6]. The latter paper uses a different digital topology than we do; however, many of the results of [6] carry over easily into the digital topologies with which we work. 2. Preliminaries 2.1. General Properties Let Z denote the set of integers. Then Z k is the set of lattice points in Euclidean k -dimensional space. A (binary) digital image is a finite subset of Z k . A variety of adjacency relations are used in the study of digital images. The following terminology is used in [7]. Two points p and q in Z 2 are 8-adjacent if they are distinct and differ by at most 1 in each coordinate; p and q in Z 2 are 4-adjacent if they are 8-adjacent and differ in exactly one coordinate. Two points p and q in Z 3 are 26-adjacent if they are distinct and differ by at most 1 in each coordinate; they are 18-adjacent if they are 26-adjacent and differ in at most two coordinates; they are 6-adjacent if they are 18-adjacent and differ in exactly one coordinate. For n ∈{4, 8, 6, 18, 26}, an n-neighbor of a lattice point p is a poin that is n-adjacent to p. We generalize 4-adjacency in Z 2 and 6-adjacency in Z 3 by saying p, q ∈ Z k are 2k -adjacent if p = q and p and q differ by 1 in one coordinate and by 0 in all other coordinates.