数字凸+幺模映射=8-连通（除一个点外的所有点4-连通）
Digital Convex + Unimodular Mapping =8-Connected (All Points but One 4-Connected)
摘要：
在二维数字几何中，两个格点是4-连通的（分别为8-连通），如果它们的欧氏距离至多为1（分别为$\sqrt{2}$）。一个集合$S \subset Z^2$是4连接的（分别为8-连接），如果对于$S$中的所有点对$p_1, p_2$，存在将$p_1$连接到$p_2$的路径，使得每条边由4-连接（分别为8-连接）对点。数字凸性的原始定义指出，一个集合$S \subset Z^d$是数字凸的，如果$\conv(S) \cap Z^d= S$，其中$\conv(S)$表示$S$的凸包不保证连通性。然而，多个算法假设连通性。本文证明了在二维空间中，任何由$n$点构成的数字凸集$S$单模等价于一个8-连通数字凸集$C$。事实上，所得到的数字凸集$C$是4-连通的，除了至多一个点是8-连通到集合的其余部分。$SL_2(Z)$的矩阵定义了两个单模等价的格多面体$S$和$C$之间的$Z^2$的仿射同构，可以在大约$O(n)$时间内计算。我们还表明，没有类似的结果是可能的，在更高的维度。
英文摘要：
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 $S \subset Z^2$ 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 $S \subset Z^d$ is digital convex if $\conv(S) \cap Z^d= S$, where $\conv(S)$ 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 $SL_2(Z)$ defining the affine isomorphism of $Z^2$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
