Reasoning about gradual changes of topological relationships

Abstract

Geographic objects and phenomena may gradually change their location, orientation, shape, and size over time. A qualitative change occurs if the deformation of an object affects its topological relationship with respect to another object. The observation of such changes is particularly interesting, because qualitative changes frequently require different decisions or trigger new actions. Investigations of a closed set of mutually exclusive binary topological relationships led to a formal model to determine for each topological relationship the relationships closest to it. Applied to the entire set of binary topological relationships between spatial regions, this model describes a partial order over topological relationships and provides a measure to assess how far two relationships are apart from each other. The changes to the binary topological relationship caused by such deformations as translation, rotation, reduction, and expansion of an object are mapped onto this graph. The graphs show characteristic traverses for each kind of deformation. Using these characteristic traverses as knowledge about deformations, one can infer from multiple observations the kind of deformation that caused the change and predict the next topological relationship. Particularly, it provides answers to three kinds of qualitative space-time inferences: (1) Given a process and a state, what is the next most likely state? (2) Given an ordered pair of states, what process may have occurred? (3) Given an ordered pair of states and a process, in what states must the two objects have been in between?.