Binary topological relations on the digital sphere

Abstract

Topological relations are the predominant backbone of qualitative spatial reasoning, focusing to date mostly on objects that are embedded in R2. With the advent of ball-shaped screen technologies and the growing importance of globally situated data, the digital sphere, known as T2, becomes a viable embedding. While topological relations have been investigated for a spherical embedding S2 within a continuous framework, the lack of a full account of the relations in T2 prevents more refined query systems on global discrete data. This paper extends the concept of spherical topological relations to the digital spherical embedding, building on the model of a planar digital region in Z2 whose boundary comprises the pixels along the region's margin. With the 9-intersection and its detailed descendant, the 9+-intersection, 29 relations between two digital regions are derived. This set of binary digital spherical relations is jointly exhaustive and pairwise disjoint. The relations' conceptual neighborhood graph, capturing the similarity among the relations, further highlights that 18 of the 29 relations are refinements of those found in S2. These topological relations between digital regions coincide with the binary relations found between broad-boundary regions when the broad boundaries are of uniform thickness, a scenario preventing the inclusion of an entire interior of one region within the boundary of the other. Five different rationales for coarsening are investigated, each leading to a different group of the relations along the conceptual neighborhood graph.