关于HV-凸切换分支类的若干几何问题
On some geometric aspects of the class of hv-convex switching components
摘要：
在离散层析成像的通常目的中，考虑借助于沿着离散方向的集合$U$收集的投影数据来重建未知离散集合。当且仅当发生切换分量时，即当且仅当存在沿$U$中的所有方向具有空投影的非空图像时，可能出现模糊重建。为了降低允许重建的数量，人们试图将可能的额外的几何约束的层析成像问题。特别是，水平和垂直凸连通集的类$\mathbb{P}$（简称为$hv$ -凸多角形）已被广泛考虑。本文引入了一类$hv$ -凸切换分支，并证明了关于其几何结构的一些初步结果。该类包括当在$\mathbb{P}$中考虑层析成像问题时产生的所有切换组件，这极大地激发了对此类配置的调查。事实证明，所考虑的类可以划分为两个不相交的子类的封闭模式，称为窗口和卷曲，分别。由此可见，所有的窗口有一个独特的表示，而卷曲组成的交错序列的子模式，称为$Z$ -路径，这导致的问题，理解这种序列的组合结构。我们提供了明确的结构与一些特殊的序列的卷曲的家庭，并通过一些说明性的例子，进一步允许或禁止的配置也给额外的细节。
英文摘要：
In the usual aim of discrete tomography, the reconstruction of an unknown discrete set is considered, by means of projection data collected along a set $U$ of discrete directions. Possible ambiguous reconstructions can arise if and only if switching components occur, namely, if and only if non-empty images exist having null projections along all the directions in $U$. In order to lower the number of allowed reconstructions, one tries to incorporate possible extra geometric constraints in the tomographic problem. In particular, the class $\mathbb{P}$ of horizontally and vertically convex connected sets (briefly, $hv$-convex polyominoes) has been largely considered. In this paper we introduce the class of $hv$-convex switching components, and prove some preliminary results on their geometric structure. The class includes all switching components arising when the tomographic problem is considered in $\mathbb{P}$, which highly motivates the investigation of such configurations. It turns out that the considered class can be partitioned in two disjointed subclasses of closed patterns, called windows and curls, respectively. It follows that all windows have a unique representation, while curls consist of interlaced sequences of sub-patterns, called $Z$-paths, which leads to the problem of understanding the combinatorial structure of such sequences. We provide explicit constructions of families of curls associated to some special sequences, and also give additional details on further allowed or forbidden configurations by means of a number of illustrative examples.
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。
