Fuzzy Geometry and Shape Relations in Image Spaces

Abstract

This paper concerns the development of fuzzy geometry and shape relations in image space. Initially, fuzzy analog of gray tone image is described where the concepts like discrete fuzzy set and vector fuzzy set are introduced. Then properties like convex fuzzy set and convex hull, elongatedness and compactness as well as fuzzy perimeteres are defined. The problem of computing the distance between two fuzzy sets is reviewed while a new fuzzy Hausdorff distance is defined and its metric properties are established. Next, fuzzy geometric shapes like fuzzy point, fuzzy straight line, fuzzy circle, fuzzy ellipse and fuzzy polygon are defined. Also, a new concept called concave fuzzy set is introduced and its properties are established. Later on, the problem of matching two fuzzy shapes is considered and a metric shape distance is proposed. While these studies are done on fuzzy sets that are equivalent to gray tone objects, we next consider fuzzy features of two tone objects such as bigness, circularity and convexity, elongatedness and straightness etc. Also, fuzzy relations like middleness, leftness, lightness, surroundedness, betweenness etc are defined. Apart from theoretical advancement, this study can be used in object recognition, description and robotics applications.