Mathematical morphology and spatial relationships: quantitative, semi-quantitative and symbolic settings

Basic mathematical morphology operations rely mainly on local information, based on the concept of structuring element. But mathematical morphology also deals with more global and structural information since several spatial relationships can be expressed in terms of morphological operations (mainly dilations). The aim of this paper is to show that this framework allows to represent in a unified way spatial relationships in various settings: a purely quantitative one if objects are precisely defined, a semi-quantitative one if objects are imprecise and represented as spatial fuzzy sets, and a qualitative one, for reasoning in a logical framework about space. This is made possible thanks to the strong algebraic structure of mathematical morphology, that finds equivalents in set theoretical terms, fuzzy operations and logical expressions.

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