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.
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