Fuzzy Morphology and Fuzzy Distances: New Definitions and Links in both Euclidean and Geodesic Cases

The aim of this paper is to establish links between fuzzy morphology and fuzzy distances. We show how distances can be derived from fuzzy mathematical morphology, in particular fuzzy dilation, leading to interesting definitions involving membership information and spatial information. Conversely, we propose a way to define fuzzy morphological operations from fuzzy distances. We deal with both the Euclidean and the geodesic cases. In particular in the geodesic case, we propose a definition of fuzzy balls, from which fuzzy geodesic dilation and erosion are derived, based on fuzzy geodesic distances. These new operations enhance the set of fuzzy morphological operators, leading to transformations of a fuzzy set conditionally to another fuzzy set. The proposed definitions are valid in any dimension, in both Euclidean and geodesic cases.