The Mereotopology of Discrete Space

Chapter 6: A Theory of Granular Partitions

“…(A similar simplifying assumption is proposed by Galton, 1999.) This is a simplifying assumption, which our present theory of granular partitions will enable us to avoid.…”

Axioms, Algebras and Topology

“…There are many applications, such as describing or reasoning about video images, where one is dealing with a discrete spatial structure. Axiomatic theories which allow atomic regions have been investigated in [74] and [41]; and [28] presents a generalisation of BCAs, in which the extensionality axiom is dropped, and proves a representation theorem in terms of discrete proximity spaces.…”

“…the general BCA theory together with the extensionality axiom. For certain purposes, in particular the modelling of discrete space, one may wish to remove the extensionality condition [41]. The resulting very general BCAs will not be considered further here; however, a representation theorem (in terms of atomic algebras over proximity spaces) is given in [28].…”

Qualitative Spatial Representation and Reasoning

“…There is a tension between the continuous-space models favoured by high-level approaches to handling spatial information and discrete, digital representations used at the lower level. An attempt to bridge this gap by developing a high-level qualitative spatial theory based on a discrete model of space is [91]. For another investigation into discrete vs continuous space, see [139].…”

“…Similarly with a type of the form �2, 3, k�. An example of a theory with a type 3 connection relation interpreted in boundary free models and a separate parthood relation is [128]-influenced by [177] this generalises the RCC system and the discrete mereotopology of Galton [91] to allow for discrete models of RCC (not possible in the standard theory cited above).…”