III: Small: A Theory of Topological Relations for Compound Spatial Objects

Spatial data collections with an incomplete coverage yield regions with holes and separations that often cannot be filled by interpolation. Geosensor networks typically generate such configurations, and with the proliferation of sensor colonies, there is now an urgent need to provide users with better information technologies of cognitively plausible methods to search for or compare available spatial data sets that may be incomplete. The objective of the investigations is to advance knowledge about qualitative spatial relations for spatial regions with holes and/or separations. The core activity is the study of the interplay between topological spatial relations with holed regions and topological spatial relations with separated regions to address the potentially complex configurations that feature both holes and separations. Three characteristics of such a set of topological relations are addressed: the formalization of a sound set of relations at a granularity that allows for the distinction of the salient features of holed and separated regions, while offering the opportunity to generalize to coarser relations in a meaningful and consistent way; the relaxation of such relations so that the determination of the most similar relations follows immediately from the applied methodology; and the qualitative inference of new information from the composition of such relations to identify inconsistencies and to drawn information that is not immediately available from individual relations. The hypothesis is that combining the relation formalization with sound similarity and composition reasoning yields critical insights for a sufficiently expressive, common approach to modeling topological relations for holed regions and regions with separations. The resulting theory of topological spatial relations highlights a parallelism between relations with holed regions and regions with separations, which is most apparent when these regions are embedded on the surface of the sphere, while some parts of these regularities are often hidden in the usual planar embedding. Since topological relations are qualitative spatial descriptions, they come close to people\u27s own reasoning, so that a better understanding of the relations for compound spatial objects will have ramifications for qualitative spatial reasoning, without a need for drawing graphical depictions to make inferences. It also lays the foundation for linguistic constructs to communicate in natural language spatial configurations, ultimately leading to talking maps. An immediate impact of this theory of topological relations between holed and separated regions is on the querying and reasoning about dataset that are gathered by geosensor networks. Additional information available online: http://www.spatial.maine.edu/~max/holesAndParts.htm

Spatial refinement as collection order relations

An abstract examination of refinement (and conversely, coarsening) with respect to the involved spatial relations gives rise to formulated order relations between spatial coverings, which are defined as complete-coverage representations composed of regional granules. Coverings, which generalize partitions by allowing granules to overlap, enhance hierarchical geocomputations in several ways. Refinement between spatial coverings has underlying patterns with respect to inclusion—formalized as binary topological relations—between their granules. The patterns are captured by collection relations of inclusion, which are obtained by constraining relevant topological relations with cardinality properties such as uniqueness and totality. Conjoining relevant collection relations of equality and proper inclusion with the overlappedness (non-overlapped or overlapped) of the refining and the refined covering yields collection order relations, which serve as specific types of refinement between spatial coverings. The examination results in 75 collection order relations including seven types of equality and 34 pairs of strict or non-strict types of refinement and coarsening, out of which 19 pairs form partial collection orders