Homotopy equivalence of digital pictures in $\mathbb{Z}^2$

Abstract

We investigate the properties of digital homotopy in the context of digital pictures $(X,\kappa,\bar \kappa)$, where $X\subsetneq \Z^n$ is a finite set, $\kappa$ is an adjacency relation on $X$, and $\bar \kappa$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\Z^2$. We define a numerical homotopy-type invariant for digital pictures in $\Z^2$ 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.