Tight bounds in the quadtree complexity theorem and the maximal number of pixels crossed by a curve of given length

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• 作者：Yan Gerard ; ^{yan.gerard@udamail.fr" class="auth_mail" title="E-mail the corresponding author} ; Antoine Vacavant ^{antoine.vacavant@udamail.fr" class="auth_mail" title="E-mail the corresponding author} ; Jean-Marie Favreau ^{J-Marie.Favreau@udamail.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quadtree complexity theorem ; Digital cover cardinality ; Rectifiable curve ; Length
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：18 April 2016
• 年：2016
• 卷：624
• 期：Complete
• 页码：41-55
• 全文大小：2597 K

文摘

The main purpose of this work is to determine the exact maximum number of pixels (a bi-dimensional sequence of unit squares tiling a plane) that a rectifiable curve of given length l   can cross. In other words, given l∈R$l\in \mathbb{R}$, we provide the value N(l)$N(l)$ of the maximal cardinality of the digital cover of a rectifiable curve of length l  . The optimal curves are polygonal curves with integer vertices, 0, 1 or 2 vertical or horizontal steps and an arbitrary number of diagonal steps. We also report the properties of the staircase function N(l)$N(l)$, which is affinely periodic in the sense that $N(l+\sqrt{2})=N(l)+3$ and a bound .

Our second aim is to look at the restricted class of closed curves and offer some conjectures on the maximum number N_closed(l)$N_{\mathrm{closed}}(l)$ of pixels that a closed curve of length l can cross.

This work finds its application in the quadtree complexity theorem. This well-known result bounds the number of quads with a shape of perimeter p   by 16q−11+16p$16q-11+16p$. However, this linear bound is not tight. From our new upper bound we derive a new improved multiresolution complexity theorem: $\text{Number(quads)}\le 16q-11+6\sqrt{2}p$. Lastly, we show that this new bound is tight up to a maximal error of 16(q−1)$16(q-1)$.