Study of fuzzy clustering algorithms and enhanced fuzzy reasoning application to texture based image segmentation

Abstract

c-means (k-means) is a popular algorithm for cluster analysis. Many variants of c-means algorithms are available. All these models are studied in depth and convergence of iterative solutions are verified, in this thesis. An example of texture based image segmentation is used to support this study of various clustering algorithms. In context of clustering points in a space, a cluster represents a set of elements. The set is created by studying the membership of each element within it. Conventionally there are two types of set theories: crisp and its extension fuzzy set theory. The extension of crisp sets to fuzzy sets in terms of membership functions, is alike to extension of the set of integers to the set of real numbers. But the development does not end here, the membership can be extended to a vector value. Clustering is significantly affected by the data dimensionality and the distance metric used during cluster formation. Distance between points and distance between clusters are the key attributes for an accurate cluster analysis. During analysis of fuzzy based clustering a need for a new distance metric was felt. This metric defines distance between fuzzy sets and also between elements and fuzzy sets. As a step to fulfil this requirement, in this work the fuzzy sets with vector memberships are defined and proposed. Basic set theoretic operations, such as complement, union and intersection are defined and discussed in axiomatic manner. This work also proposes a new distance function defined for points and sets, and the new function is proved to be a metric through systematic proofs.