Homotopy equivalence of digital pictures in Z2

We investigate the properties of digital homotopy in the context of digital pictures (X,κ,κ¯), where X⊊Zⁿ is a finite set, κ is an adjacency relation on X , and κ¯ is an adjacency relation on the complement of X . In particular we focus on homotopy equivalence between digital pictures in Z². We define a numerical homotopy-type invariant for digital pictures in Z² called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by “filling in the gaps” of any row or column. We show that a digital picture (X,c_i,c_j) is homotopy equivalent to only finitely many other digital pictures (Y,c_i,c_j). At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture.