ABSTRACT
The chapters in this book consider how one might utilize fuzzy sets, near sets, and rough sets, taken separately or taken together in hybridizations, in solving a variety of problems in image analysis. A brief consideration of Cantor sets (Cantor, 1883, 1932) provides a backdrop for an understanding of several recent types of sets useful in image analysis. Fuzzy, near and rough sets provide a wide spectrum of practical solutions to solving image analysis problems such as image understanding, image pattern recognition, image retrieval and image correspondence, mathematical morphology, perceptual tolerance relations in image analysis and segmentation evaluation. Fuzzy sets result from the introduction of a membership function that generalizes the traditional characteristic function. The notion of a fuzzy set was introduced by L. Zadeh in 1965 (Zadeh, 1965). Sixteen years later, rough sets were introduced by Z. Pawlak in 1981 (Pawlak, 1981a). A set is considered rough whenever the boundary between its lower and upper approximation is non-empty. Of the three forms of sets, near sets are newest, introduced in 2007 by J.F. Peters in a perception-based approach to the study of the nearness of observable objects in a physical continuum (Peters and Henry, 2006; Peters, 2007c,a; Peters, Skowron, and Stepaniuk, 2007; Henry and Peters, 2009b; Peters and Wasilewski, 2009; Peters, 2010).
