Topological relationships between complex spatial objects

Abstract

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.

Evaluating the Cognitive Adequacy of the DLine-Region Calculus

“…Several research approaches have shown that the granularity of formal calculi is inadequate for modeling human conceptualizations of both static and dynamic spatial relations. To address this issue the majority of approaches (e.g., Clementini, Di Felice, & van Oosterom, 1993;Schneider & Behr, 2006) define formal criteria on how to cluster topological equivalence classes such that the overall number of topological predicates is reduced. Clementini and collaborators (1993) suggest five basic relations that are derived on the basis of the emptiness and non-emptiness of component intersection, inclusion and non-inclusion of one object in another object, and the dimension of the component intersection.…”

On the Requirements for User-Centric Spatial Data Warehousing and SOLAP

“…Similarly, the Open GIS Consortium also provides a Reference Model [45] as a standard for a representing geo-spatial information. Qualitative spatial operations include topological relations [46] such as disjoint, meet, overlap, equal, inside, contains, covers and coveredBy, and cardinal direction relations. Quantitative relations on spatial objects include metric operations based on the size, shape and metric distances between objects or their components.…”

Plateau Regions: An Implementation Concept for Fuzzy Regions in Spatial Databases and GIS

“…Fuzzy set theory [10] has been a popular approach to modeling vague spatial objects and resulted in a concept of fuzzy regions [5,8]. A crisp region object is conceptually modeled as a particular point set of the Euclidean plane [7,9]. Each of its points belongs definitely and completely to it.…”

“…. , r 9 and their associated membership values. All membership values are different, and any pair of crisp regions is either disjoint (e.g., r 1 and r 9 ) or adjacent (e.g., r 2 and r 5 ).…”

“…Condition (vi) states that any two crisp regions associated with a plateau region are topologically either disjoint or adjacent to each other. This is expressed by the well known topological predicates disjoint and meet on complex regions [9]. Condition (vii) requires that all membership values are different and that all pairs of the sequence are ordered by their membership values.…”