There exist several methods to extend binary morphology to grey-scale images. One of these methods is based on fuzzy logic and fuzzy set theory. Another approach starts from the complete lattice framework for morphology and the theory of adjunctions. In this paper, both approaches are combined. The basic idea is to use (fuzzy) conjunctions and implications which are adjoint in the definition of dilations and erosions, respectively. This gives rise to a large class of morphological operators for grey-scale images. It turns out that this class includes the often used grey-scale Minkowski addition and subtraction.

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CWI. Probability, Networks and Algorithms [PNA]
Signals and Images

Deng, T.-Q., & Heijmans, H. (2000). Grey-scale morphology based on fuzzy logic. CWI. Probability, Networks and Algorithms [PNA]. CWI.