Stability and Statistical Inferences in the Space of Topological Spatial Relationships

Modeling topological properties of the spatial relationship between objects, known as the topological relationship, represent a fundamental research problem in many domains including artificial intelligence and geographical information science. Real-world data are generally finite and exhibit uncertainty. Therefore, when attempting to model topological relationships from such data, it is useful to do so in a manner which is both stable and facilitates statistical inferences. Current models of the topological relationships do not exhibit either of these properties. We propose a novel model of topological relationships between objects in the Euclidean plane, which encodes topological information regarding connected components and holes. Specifically, a representation of the persistent homology, known as a persistence scale space, is used. This representation forms a Banach space that is stable and, as a consequence of the fact that it obeys the strong law of large numbers and the central limit theorem, facilitates statistical inferences. The utility of this model is demonstrated through a number of experiments.

Abstract licence: CC BY-NC-ND

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