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Fuzzy Sets, Near Sets, and Rough Sets for Your Computational Intelligence Toolbox

  • James F. Peters
Part of the Studies in Computational Intelligence book series (SCI, volume 202)

Summary

This chapter considers how one might utilize fuzzy sets, near sets, and rough sets, taken separately or taken together in hybridizations as part of a computational intelligence toolbox. These technologies offer set theoretic approaches to solving many types of problems where the discovery of similar perceptual granules and clusters of perceptual objects is important. Perceptual information systems (or, more concisely, perceptual systems) provide stepping stones leading to nearness relations and properties of near sets. This work has been motivated by an interest in finding a solution to the problem of discovering perceptual granules that are, in some sense, near each other. Fuzzy sets result from the introduction of a membership function that generalizes the traditional characteristic function. Near set theory provides a formal basis for observation, comparison and classification of perceptual granules. Near sets result from the introduction of a description-based approach to perceptual objects and a generalization of the traditional rough set approach to granulation that is independent of the notion of the boundary of a set approximation. Near set theory has strength by virtue of the strength it gains from rough set theory, starting with extensions of the traditional indiscernibility relation. This chapter has been written to establish a context for three forms of sets that are now part of the computational intelligence umbrella. By way of introduction to near sets, this chapter considers various nearness relations that define partitions of sets of perceptual objects that are near each other. Every perceptual granule is represented by a set of perceptual objects that have their origin in the physical world. Objects that have the same appearance are considered perceptually near each other, i.e., objects with matching descriptions. Pixels, pixel windows, and segmentations of digital images are given by way of illustration of sample near sets. This chapter also briefly considers fuzzy near sets and near fuzzy sets as well as rough sets that are near sets.The main contribution of this chapter is the introduction of a formal foundation for near sets considered in the context of fuzzy sets and rough sets.

Keywords

Description fuzzy sets near sets perceptual granule perceptual system rough sets 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • James F. Peters
    • 1
  1. 1.Computational Intelligence Laboratory, Department of Electrical & Computer EngineeringUniversity of ManitobaManitobaCanada

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