Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 62-65). Also available via the Internet.

This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting

Aspects of fuzzy vector spaces and fuzzy groups are investigated, including linear independence, basis, dimension, group order, finitely generated groups and cyclic groups. It was necessary to consider cardinality of fuzzy sets and related issues, which included a question of ways in which to define functions between fuzzy sets. Among the results proved, are the additivity property of dimension for fuzzy vector spaces, Lagrange's Theorem for fuzzy groups ( the existing version of this theorem does not take fuzziness into account at all), a compactness property of finitely generated fuzzy groups and an extension of an earlier result on the order-homomorphisms. An open question is posed with regard to the existence of a basis for an arbitrary fuzzy vector space.

The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.

The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.

The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.

Wong Man-Lung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 99-103). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / List of Figures --- p.viii / List of Tables --- p.ix / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Objectives --- p.5 / Chapter 2 --- Cartesian Representation of Membership Function --- p.7 / Chapter 2.1 --- The Cartesian Representation --- p.8 / Chapter 2.2 --- Region of Well-defined Membership Functions --- p.10 / Chapter 2.3 --- Similarity Triangle Interpolation Method --- p.12 / Chapter 2.4 --- The Interpolation Example --- p.18 / Chapter 2.5 --- Further Issues --- p.23 / Chapter 2.6 --- Conclusions --- p.24 / Chapter 3 --- Membership Function as Elements in Function Space --- p.26 / Chapter 3.1 --- L2[0，2] Representation --- p.27 / Chapter 3.2 --- "The Inner Product Space of L2[0,2]" --- p.31 / Chapter 3.3 --- The Similarity Triangle Interpolation Method --- p.32 / Chapter 3.4 --- The Interpolation Example --- p.36 / Chapter 3.5 --- Conclusions --- p.48 / Chapter 4 --- Radius of Influence of Membership Functions --- p.50 / Chapter 4.1 --- Previous Works on Mountain Method --- p.51 / Chapter 4.2 --- Combining Mountain Method and Cartesian Representation --- p.56 / Chapter 4.3 --- Extensibility Function and Weighted-Sum-Averaging Equation --- p.61 / Chapter 4.4 --- Radius of Influence --- p.62 / Chapter 4.5 --- Combining Radius of Influence and Fuzzy Interpolation Technique --- p.64 / Chapter 4.6 --- Model Identification Example --- p.66 / Chapter 4.7 --- Eliminative Extraction --- p.67 / Chapter 4.8 --- Eliminative Extraction Example --- p.70 / Chapter 4.9 --- Conclusions --- p.71 / Chapter 5 --- Fuzzy Inferencing --- p.73 / Chapter 5.1 --- Fuzzy Inferencing and Interpolation in Cartesian Representation --- p.74 / Chapter 5.2 --- Sparse Rule Extraction via Radius of Influence and Elimination --- p.77 / Chapter 5.3 --- Single Input and Single Output Case --- p.78 / Chapter 5.4 --- Multiple Input and Single Output Case --- p.81 / Chapter 5.5 --- Application --- p.89 / Chapter 5.6 --- Conclusions --- p.94 / Chapter 6 --- Conclusions --- p.96 / Appendix --- p.99 / Bibliography --- p.99

Thesis (M. Phil.)--Chinese University of Hong Kong, 1998. / Description based on contents viewed Mar. 13, 2007; title from title screen. Includes bibliographical references (p. 99-103). Also available in print.

Thesis (Ph. D.) in Physics--University of Maine, 2003. / Includes vita. Includes bibliographical references (leaves 129-143).

Image segmentation is an important step in the image analysis process. Current image segmentation techniques, however, require that the user tune several parameters in order to obtain maximum segmentation accuracy, a computationally inefficient approach, especially when a large number of images must be processed sequentially in real time. Another major challenge, particularly with medical image analysis, is the discrepancy between objective measures for assessing and guiding the segmentation process, on the one hand, and the subjective perception of the end users (e.g., clinicians), on the other. Hence, the setting and adjustment of parameters for medical image segmentation should be performed in a manner that incorporates user feedback. Despite the substantial number of techniques proposed in recent years, accurate segmentation of digital images remains a challenging task for automated computer algorithms. Approaches based on machine learning hold particular promise in this regard because, in many applications, including medical image analysis, frequent user intervention can be assumed as a means of correcting the results, thereby generating valuable feedback for algorithmic learning. This thesis presents an investigation of the use of evolving fuzzy systems for designing a method that overcomes the problems associated with medical image segmentation. An evolving fuzzy system can be trained using a set of invariant features, along with their optimum parameters, which act as a target for the system. Evolving fuzzy systems are also capable of adjusting parameters based on online updates of their rule base. This thesis proposes three different approaches that employ an evolving fuzzy system for the continual adjustment of the parameters of any medical image segmentation technique. The first proposed approach is based on evolving fuzzy image segmentation (EFIS). EFIS can adjust the parameters of existing segmentation methods and switch between them or fuse their results. The evolving rules have been applied for breast ultrasound images, with EFIS being used to adjust the parameters of three segmentation methods: global thresholding, region growing, and statistical region merging. The results for ten independent experiments for each of the three methods show average increases in accuracy of 5\%, 12\% and 9\% respectively. A comparison of the EFIS results with those obtained using five other thresholding methods revealed improvements. On the other hand, EFIS has some weak points, such as some fixed parameters and an inefficient feature calculation process. The second approach proposed as a means of overcoming the problems with EFIS is a new version of EFIS, called self-configuring EFIS (SC-EFIS). SC-EFIS uses the available data to estimate all of the parameters that are fixed in EFIS and has a feature selection process that selects suitable features based on current data. SC-EFIS was evaluated using the same three methods as for EFIS. The results show that SC-EFIS is competitive with EFIS but provides a higher level of automation. In the third approach, SC-EFIS is used to dynamically adjust more than one parameter, for example, three parameters of the normalized cut (N-cut) segmentation technique. This method, called multi-parametric SC-EFIS (MSC-EFIS), was applied to magnetic resonance images (MRIs) of the bladder and to breast ultrasound images. The results show the ability of MSC-EFIS to adjust multiple parameters. For ten independent experiments for each of the bladder and the breast images, this approach produced average accuracies that are 8\% and 16\% higher respectively, compared with their default values. The experimental results indicate that the proposed algorithms show significant promise in enhancing image segmentation, especially for medical applications.