From Metric to Topology: Determining Relations in Discrete Space

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 \( {\mathbb{R}}^{2} \) vs. \( {\mathbb{Z}}^{2} \) reveals that the eight binary topological relations between two simple regions in \( {\mathbb{R}}^{2} \) can be distinguished by a minimal set of six metrics, whereas in \( {\mathbb{Z}}^{2} \), 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.