Unifying Quantitative, Semi-quantitative and Qualitative Spatial Relation Knowledge Representations Using Mathematical Morphology

One of the powerful features of mathematical morphology lies in its strong algebraic structure, that finds equivalents in set theoretical terms, fuzzy sets theory and logics. Moreover this theory is able to deal with global and structural information since several spatial relationships can be expressed in terms of morphological operations. The aim of this paper is to show that the framework of mathematical morphology allows to represent in a unified way spatial relationships in various settings: a purely quantitative one if objects are precisely defined, a semiquantitative one if objects are imprecise and represented as spatial fuzzy sets, and a qualitative one, for reasoning in a logical framework about space.