A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object representation

Abstract

In this paper, the generalized fuzzy mathematical morphology (GFMM) is proposed, based on a novel definition of the fuzzy inclusion indicator (FII). FII is a fuzzy set used as a measure of the inclusion of a fuzzy set into another, that is proposed to be a fuzzy set. It is proven that the FII obeys a set of axioms, which are proposed to be extensions of the known axioms that any inclusion indicator should obey, and which correspond to the desirable properties of any mathematical morphology operation. The GFMM provides a very powerful and flexible tool for morphological operations. The binary and grayscale mathematical morphologies can be considered as special cases of the proposed GFMM. An application for robust skeletonization and shape decomposition of two-dimensional (2-D) and three-dimensional (3-D) objects is presented. Simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the GFMM is better than by using the binary mathematical morphology in most cases. Furthermore, the use of the GFMM for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines.

On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering

“…Sinha and Dogherty [25] analyzed inclusion measures for general fuzzy sets based on particular axiom definitions. Chatzis and Pitas [26] proposed a new fuzzy inclusion indicator for morphological operations. Kehagins and Konstatinidou [27] introduced L-fuzzy valued inclusion measures and then explored the relationships between inclusion measures and fuzzy distance among general fuzzy sets.…”

Representations for Morphological Image Operators and Analogies with Linear Operators

“…The main disadvantage of these approaches is that composition of the operators from steps (1) and (2) is not guaranteed to be an algebraic opening or closing. In addition to the above approaches, there have been efforts to combine mathematical morphology and fuzzy logic or lattices and neuro-fuzzy systems by fuzzifying respectively the inclusion indicator or the partial ordering of the lattice, as done respectively in Chatzis and Pitas (2000) and Kaburlasos and Petridis (2000). In the field of pattern recognition, of relevance is also the work in Yang and Maragos (1995) on min-max classifiers that used max-min operations on vectors.…”

Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

“…The main disadvantage of these approaches is that composition of the operators from steps (1) and (2) is not guaranteed to be an algebraic opening or closing. In addition to the above approaches, there have been efforts to combine MM and fuzzy logic or lattices and neuro-fuzzy systems by fuzzifying respectively the inclusion indicator or the partial ordering of the lattice, as done respectively in [5] and [17].…”