On the role of complete lattices in mathematical morphology: From tool to uncertainty model

Mathematical morphology has a rich history. Originally introduced for binary images, it was quite soon extended to grayscale images, leading to grayscale morphology with the threshold approach and the umbra approach. Later on, different models based on fuzzy set theory were introduced. These models were based on the observation that, from a formal point of view, grayscale images and fuzzy sets are modeled in the same way. Consequently, techniques from fuzzy set theory could be applied in the context of mathematical morphology, and fuzzy mathematical morphology was born. In that framework, fuzzy set theory was only a tool to construct morphological models, and was not employed to model any fuzziness or uncertainty. Quite recently however, new extensions have led to the construction of fuzzy interval-valued and fuzzy intuitionistic mathematical morphologies. Here, extensions of fuzzy set theory actually take into account the uncertainty that comes along with image capture, specifically regarding the grayscale values, which in some cases is also related to the uncertainty regarding the spatial position of an object in an image. In this framework, (extended) fuzzy set theory not only serves as a tool to deal with grayscale images, but also as a model for uncertainty. This paper sketches this evolution of fuzzy set theory in the field of mathematical morphology, and also points out some directions for future research.