Digital Convex + Unimodular Mapping = 8-Connected (All Points but One 4-Connected)

Abstract

In two dimensional digital geometry, two lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one (resp. $\sqrt{2}$). A set $S \subset \mathbb {Z}^2$ is 4-connected (resp. 8-connected) if for all pair of points $p_1, p_2$ in S there is a path connecting $p_1$ to $p_2$ such that every edge consists of a 4-connected (resp. 8-connected) pair of points. The original definition of digital convexity which states that a set $S \subset \mathbb {Z}^d$ is digital convex if $\mathrm {conv}(S) \cap \mathbb {Z}^d= S$, where $\mathrm {conv}(S)$ denotes the convex hull of S does not guarantee connectivity. However, multiple algorithms assume connectivity. In this paper, we show that in two dimensional space, any digital convex set S of n points is unimodularly equivalent to a 8-connected digital convex set C. In fact, the resulting digital convex set C is 4-connected except for at most one point which is 8-connected to the rest of the set. The matrix of $SL_2(\mathbb {Z})$ defining the affine isomorphism of $\mathbb {Z}^2$ between the two unimodularly equivalent lattice polytopes S and C can be computed in roughly O(n) time. We also show that no similar result is possible in higher dimension.