Fuzzy mathematics (Q2769887)

The paper is a comprehensive overview of A. Rozenfeld's main research results in the area of fuzzy mathematics. Several main directions are discussed and the presentation is related to their importance and applications to pattern recognition and image processing. Fuzzy geometry plays an important role in image processing when it comes to measuring geometric properties of regions in an image and quantifying a number of properties such as connectedness and surroundedness, adjacency, convexity, area, perimeter and compactness. Owing to the continuous nature of images, a description of these properties lends directly itself to the language of fuzzy sets. Similarly, gray-level properties of images are conveniently captured in terms of fuzzy sets and fuzzy relations. The same applies to a variety of geometric shapes and results in descriptors such as fuzzy rectangles and fuzzy triangles. Digital topology is discussed in the context of image compression, enhancement and analysis and involves a number of generic concepts such as fuzzy connectedness, fuzzy components and convexity. Graph theory generalizes to fuzzy graphs by studying generalizations of paths, connectedness as well as expanding the notions of clusters, bridges, forests and trees to fuzzy graphs.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00020].