Spatial reasoning with augmented points: Extending cardinal directions with local distances

Abstract

We present an approach for supplying existing qualitative direction calculi with a distance component to support fully fledged positional reasoning. The general underlying idea of augmenting points with local reference properties has already been applied in the OPRAm calculus. In this existing calculus, point objects are attached with a local reference direction to obtain oriented points and able to express relative direction using binary relations. We show how this approach can be extended to attach a granular distance concept to direction calculi such as the cardinal direction calculus or adjustable granularity calculi such as OPRAm or the Star calculus. We focus on the cardinal direction calculus and extend it to a multi-granular positional calculus called EPRAm. We provide a formal specification of EPRAm including a composition table for EPRA2 automatically determined using real algebraic geometry. We also report on an experimental performance analysis of EPRA2 in the context of a topological map-learning task proposed for benchmarking qualitative calculi. Our results confirm that our approach of adding a relative distance component to existing calculi improves the performance in realistic tasks when using algebraic closure for consistency checking.