二维弱一致数字射线的最优界
Optimal Bounds for Weak Consistent Digital Rays in 2D
摘要：
在数字空间中表示欧几里德对象已经是30多年来的研究焦点。数字线段特别重要，因为其他数字对象取决于它们的定义（例如，数字凸对象或数字星形对象）。可能期望数字线段系统满足它们的欧几里德对应物也满足的一些好的性质。该系统是一个一致的数字线段系统（CDS），如果它满足五个属性，最显着的子段属性（任何两个数字线段的交点应该是连接的）和延长属性（任何数字线段应该能够扩展成一个数字线）。已知任何CDS必须具有到其欧几里德对应物的$Ω(\log n)$ Hausdorff距离，其中$n$是段上的网格点的数目。事实上，该下限甚至适用于一致数字射线（CDR），其中对于固定的$p \in \mathbb{Z}^2$，我们考虑每个$q \in \mathbb{Z}^2$的从$p$到$q$的数字段。在本文中，我们认为家庭的弱一致性数字射线（WCDR），我们保持四个CDR属性，但不包括延长属性。在本文中，我们给出了一个WCDR结构，具有最佳的Hausdorff距离的确切常数。也就是说，我们给出了一个构造的Hausdorff距离为1.5下的$L_\infty$度量，我们表明，对于每个$ε> 0$，它是不可能有一个WCDR的Hausdorff距离最多$1.5 - ε$。
英文摘要：
Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have $Ω(\log n)$ Hausdorff distance to their Euclidean counterparts, where $n$ is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed $p \in \mathbb{Z}^2$, we consider the digital segments from $p$ to $q$ for each $q \in \mathbb{Z}^2$. In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the $L_\infty$ metric, and we show that for every $ε> 0$, it is not possible to have a WCDR with Hausdorff distance at most $1.5 - ε$.
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