On the digital homology groups of digital images

In this article we study the digital homology groups of digital images which are based on the singular homology groups of topological spaces in algebraic topology. Specifically, we define a digitally standard $n$-simplex, a digitally singular $n$-simplex, and the digital homology groups of digital images with $k$-adjacency relations. We then construct a covariant functor from a category of digital images and digitally continuous functions to the one of abelian groups and group homomorphisms, and investigate some fundamental and interesting properties of digital homology groups of digital images, such as the digital version of the dimension axiom which is one of the Eilenberg-Steenrod axioms.