Subdivision-based homotopy equivalence of digital circles

A direct translation of the notion of homotopy equivalence from algebraic topology to digital images leads to a much more rigid definition in the context of digital topology. This results in two digital circles of different radii being not homotopic. A more suitable idea of equivalences, called "subdivision-based homotopy equivalence" which is introduced by Lupton et al., involves the concept of subdivision of digital images. We prove that, in this sense, any digital circle is subdivision-based homotopic to the Diamond, the prototypical digital circle of radius 1, which implies, as a consequence, that the fundamental group of any digital circle is isomorphic to Z.