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From Metric to Topology: Determining Relations in Discrete Space

机译:从度量到拓扑:确定离散空间中的关系

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This paper considers the nineteen planar discrete topological relations that apply to regions bounded by a digital Jordan curve. Rather than modeling the topological relations with purely topological means, metrics are developed that determine the topological relations. Two sets of five such metrics are found to be minimal and sufficient to uniquely identify each of the nineteen topological relations. Key to distinguishing all nineteen relations are regions' margins (i.e., the neighborhood of their boundaries). Deriving topological relations from metric properties in R~2 vs. Z~2 reveals that the eight binary topological relations between two simple regions in R~2 can be distinguished by a minimal set of six metrics, whereas in Z~2, a more fine-grained set of relations (19) can be distinguished by a smaller set of metrics (5). Determining discrete topological relations from metrics enables not only the refinement of the set of known topological relations in the digital plane, but further enables the processing of raster images where the topological relation is not explicitly stored by reverting to mere pixel counts.
机译:本文考虑了适用于以数字约旦曲线为边界的区域的十九种平面离散拓扑关系。与其使用纯粹的拓扑手段对拓扑关系进行建模,不如开发确定拓扑关系的度量。发现五个这样的度量的两组是最小的,并且足以唯一地标识十九个拓扑关系中的每一个。区分所有十九种关系的关键是区域的边界(即边界的邻域)。从R〜2与Z〜2的度量属性推导拓扑关系表明,R〜2中两个简单区域之间的八个二元拓扑关系可以通过六个度量的最小集合来区分,而在Z〜2中,则更精细关系集(19)可以通过较小的一组度量(5)进行区分。从度量确定离散的拓扑关系不仅可以改进数字平面中一组已知的拓扑关系,而且还可以处理其中的拓扑关系未通过还原为单纯的像素计数而明确存储的光栅图像。

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