A Survey on Fuzzy Morphology

This paper considers various aspects of generalizing mathematical morphology towards a fuzzy discipline. Since mathematical morphology and fuzzy theory are both based on a set theory, there are many approaches for defining dilation and erosion operations of a fuzzy morphology. Among those approaches are: the alpha-Morphology; the triangular norm based approach of Bloch; the logical approach of Sinha and Dougherty and de Baets; and the fuzzy fusion based approaches. Received October 25, 2000 1 Mathematical Morphology comprises an important toolset for analyzing spatial structures in images [12, 19, 20, 23]. For binary images, the definitions of the fundamental morphological operations—dilation and erosion—can be related to the set-theoretic Minkowski addition and subtraction. The extension of those operations to grayscale images is strongly related to ranking operations and, therefore, to the concept of ordered sets. It has been considered for a long time how to extend mathematical morphology to the case of fuzzy sets (as was done in other image processing disciplines, e.g., see [18]). Although there was a simple idea to consider grayscale images as fuzzy versions of binary images, further works concentrated on the solicitation of a more reasonable and nontrivial concept of a fuzzy morphology [6, 8, 11, 13, 27]. Those works have culminated in the proposal of a new operation, using fuzzy structuring elements and basing on fuzzy level sets (referred to as alpha-morphology in the following). Now, it becomes more and more obvious that the intermixing of two concepts, which are primarily based on set theory (fuzzy logic and mathematical morphology), leads to a multitude of different approaches. There were also proposals of more and more operations, which also fuzzify the standard morphological operations, but differ entirely from alpha-morphology. This paper gives a survey on the various concepts of a fuzzy mathematical morphology; the most prominent fuzzy morphology, the alpha-morphology, is introduced. This approach is based on the level (or alpha-) sets of a fuzzy membership degree function. Since each level set is a classical set by itself, standard morphological operations can be applied to them. After doing so for each level, the dilated or eroded level sets can be reunified to a grayscale image. Bloch et al. provided a formula for simplifying this computation [6]. 1 This article was submitted by the authors in English. However, by investigating the use of triangular norms in this simplification, Bloch et al. discovered a further family of fuzzy morphologies [5, 6]. For the first time, such operations have a formal degree of freedom, by allowing for replacing the occurrence of a minimum operation within the analytical expression of the REPRESENTATION, PROCESSING, ANALYSIS, AND UNDERSTANDING OF IMAGES