Topological information are the most important kind of qualitative spatial information. Current formalisms for the topological aspect of space focus on relations between regions or properties of regions. This work provides a qualitative model for representing the topological internal structure of complex regions, which could be of multiple pieces and or have holes and islands to any finite level. We propose a layered graph model for representing the internal structure of complex plane regions, where each node represents the closure of a connected component of the interior or the exterior of a complex region. The model provides a complete representation in the sense that the (global) nine-intersections between the interiors, the boundaries, and the exteriors of two complex regions can be determined by the (local) RCC8 topological relations between the associated simple regions. Moreover, this graph model has an inherent hierarchy which could be exploited for map generalization.
The notion of simple region with holes and its generalized region are first introduced in @cite_3 . Our definition of simple region with holes is slightly different from that given in @cite_3 , but is consistent with the one in @cite_15 .
@cite_15 @cite_3
1997862802 5053679
For a long time topological relationships between spatial objects have been a focus of research in a number of disciplines like artificial intelligence, cognitive science, linguistics, robotics, and spatial reasoning. Especially as predicates they support the design of suitable query languages for spatial data retrieval and analysis in spatial databases and geographical information systems (GIS). Unfortunately, they have so far only been defined for and applicable to simplified abstractions of spatial objects like single points, continuous lines, and simple regions. With the introduction of complex spatial data types an issue arises regarding the design, definition, and number of topological relationships operating on these complex types. This article closes this gap and first introduces definitions of general and versatile spatial data types for complex points, complex lines, and complex regions. Based on the well known 9-intersection model, it then determines the complete sets of mutually exclusive topological relationships for all type combinations. Completeness and mutual exclusion are shown by a proof technique called proof-by-constraint-and-drawing. Due to the resulting large numbers of predicates and the difficulty of handling them, the user is provided with the concepts of topological cluster predicates and topological predicate groups, which permit one to reduce the number of predicates to be dealt with in a user-defined and or application-specific manner. The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region—the union of the object and its holes — and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A ’s generalized region and B’s generalized region, (2) A ’s generalized region and each of B’s holes, (3) B’s generalized region with each of A ’s holes, and (4) each of A ’s holes with each of B’s holes. As a side product, the same formalism applies to the description of topological relations between 1-spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multi-level treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes.