General Definition of Fuzzy Mathematical Morphology Operations. Image and Non-image Applications

A New Approach to Fuzzy Arithmetic

“…Note that in this definition we can work with fuzzy numbers which do not contain 0 in their support, but this is the main practically useful case in the control systems when we measure imprecisely some physical quantities. It is not difficult to show directly that A + − B ⊆ A + B and A × − B ⊆ A × B in a fuzzy sense [10].…”

“…We have shown that fuzzy inner and outer arithmetical operations can be represented as morphological ones, and choosing an arbitrary t-norm we can obtain a large variety of different commutative outer and inner additions and multiplications (see also [10].) Also, we can define outer and inner versions of operations on fuzzy vectors like vector (cross) product and…”

“…We generalize this definition presenting a universal framework. Thus we can define naturally fuzzy geodesic morphological operations [10]. Also, this model is applicable to fuzzy arithmetic, built by…”

Rough Sets in Biomedical Informatics

“…Namely, the R-upper approximation is an analog of morphological dilation, while the R-lower approximation is an analog of morphological erosion. This fact is not surprising, since the relations between fuzzy sets and operations on them and morphology are well studied [1], and a relation between classical interval operations and morphological ones have been established. Moreover, it is evident that the rough approximation of a set is similar to an interval approximation of a real number.…”