Qualitative-geometric ‘surrounds’ relations between disjoint regions

This paper explores a class of qualitative-geometric relations between disjoint regions, embedded in the surface of a sphere or in the plane. These relations combine topological information about the configuration of the regions themselves as well as of the geometric Voronoi regions surrounding them. The method uses maptrees to construct a rigorous and exhaustive framework for n-ary relations between ensembles of regions. The resulting uniquely defined relations are akin to the verbal spatial relation ‘surrounds,’ in one common interpretation. The paper provides an exhaustive formal classification of such extended spatial relations between up to three regions. A further exhaustive computational exploration of up to seven regions in the sphere also provides an algorithm to enumerate all possible configurations. The results demonstrate how the approach can be used to provide fine-grained and salient descriptions of qualitative-geometric relations for complex scenes involving ensembles of multiple regions.