A Classical Construction for the Digital Fundamental Group

• Authors: Laurence Boxer
• Venue: Journal of Mathematical Imaging and Vision
• Year: 1999

A version of topology‘s fundamental group is developed fordigital images in dimension at most 3 in [7] and [8].In the latter paper, it is shown that such a digital imageX ⊂ {\cal Z}k, k ≤ 3, hasa continuous analog C(X) ⊂ Rksuch that X has digital fundamental group isomorphic toΠ1(C(X)). However, the construction of the digital fundamentalgroup in [7] and [8] does not greatly resemblethe classical construction of the fundamental group of a topologicalspace. In the current paper, we show how classical methods ofalgebraic topology may be used to construct the digital fundamentalgroup. We construct the digital fundamental group based onthe notions of digitally continuous functions presented in [10]and digital homotopy [3]. Our methods are very similar tothose of [6], which uses different notions of digitaltopology. We show that the resulting theory ofdigital fundamental groups is related to thatof [7] and [8] in that it yields isomorphic fundamental groups for the digital imagesconsidered in the latter papers (for certain connectedness types).