Ordem supervisionada baseada em valores fuzzy para morfologia matemática multivalorada

Orientador: Marcos Eduardo Ribeiro do Valle MesquitaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Morfologia Matemática foi concebida como uma ferramenta para a análise e processamento de imagens binárias e foi subsequentemente generalizada para o uso em imagens em tons de cinza e imagens multivaloradas. Reticulados completos, que são conjuntos parcialmente ordenados em que todo subconjunto tem extremos bem definidos, servem como a base matemática para uma definição geral de morfologia matemática. Em contraste a imagens em tons de cinza, imagens multivaloradas não possuem uma ordem não-ambígua. Essa dissertação trata das chamadas ordens reduzidas para imagens multivaloradas. Ordens reduzidas são definidas por meio de uma relação binária que ordena os elementos de acordo com uma função h do conjunto de valores em um reticulado completo. Ordens reduzidas podem ser classificadas em ordens não-supervisionadas e ordens supervisionadas. Numa ordem supervisionada, o função de ordenação h depende de conjuntos de treinamento de valores de foreground e de background. Nesta dissertação, estudamos ordens supervisionadas da literatura. Também propomos uma ordem supervisionada baseada em valores fuzzy. Valores fuzzy generalizam cores fuzzy - conjuntos fuzzy que modelam o modo que humanos percebem as cores - para imagens multivaloradas. Em particular, revemos como construir o mapa de ordenação baseado em conjuntos fuzzy para o foreground e para o background. Também introduzimos uma função de pertinência baseada numa estrutura neuro-fuzzy e generalizamos a função de pertinência baseada no diagrama de Voronoi. Por fim, as ordens supervisionadas são avaliadas num experimento de segmentação de imagens hiperespectrais baseado num perfil morfológico modificadoAbstract: Mathematical morphology has been conceived initially as a tool for the analysis and processing of binary images and has been later generalized to grayscale and multivalued images. Complete lattices, which are partially ordered sets in whose every subset has well defined extrema, serve as the mathematical background for a general definition of mathematical morphology. In contrast to gray-scale images, however, there is no unambiguous ordering for multivalued images. This dissertation addresses the so-called reduced orderings for multi-valued images. Reduced orderings are defined by means of a binary relation which ranks elements according to a mapping h from the value set into a complete lattice. Reduced orderings can be classified as unsupervised and supervised ordering. In a supervised ordering, the mapping h depends on training sets of foreground and background values. In this dissertation, we study some relevant supervised orderings from the literature. We also propose a supervised ordering based on fuzzy values. Fuzzy values are a generalization of fuzzy colors - fuzzy sets that model how humans perceive colors - to multivalued images other than color images. In particular, we review how to construct the fuzzy ordering mapping based on fuzzy sets that model the foreground and the background. Also, we introduce a membership function based on a neuro-fuzzy framework and generalize the membership function based on Voronoi diagrams. The supervised orderings are evaluated in an experiment of hyperspectral image segmentation based on a modified morphological profileMestradoMatematica AplicadaMestre em Matemática Aplicada131635/2018-2CNP

Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications

Tesis por compendioMathematical models have extensively been used in problems related to engineering, computer sciences, economics, social, natural and medical sciences etc. It has become very common to use mathematical tools to solve, study the behavior and different aspects of a system and its different subsystems. Because of various uncertainties arising in real world situations, methods of classical mathematics may not be successfully applied to solve them. Thus, new mathematical theories such as probability theory and fuzzy set theory have been introduced by mathematicians and computer scientists to handle the problems associated with the uncertainties of a model. But there are certain deficiencies pertaining to the parametrization in fuzzy set theory. Soft set theory aims to provide enough tools in the form of parameters to deal with the uncertainty in a data and to represent it in a useful way. The distinguishing attribute of soft set theory is that unlike probability theory and fuzzy set theory, it does not uphold a precise quantity. This attribute has facilitated applications in decision making, demand analysis, forecasting, information sciences, mathematics and other disciplines. In this thesis we will discuss several algebraic and topological properties of soft sets and fuzzy soft sets. Since soft sets can be considered as setvalued maps, the study of fixed point theory for multivalued maps on soft topological spaces and on other related structures will be also explored. The contributions of the study carried out in this thesis can be summarized as follows: i) Revisit of basic operations in soft set theory and proving some new results based on these modifications which would certainly set a new dimension to explore this theory further and would help to extend its limits further in different directions. Our findings can be applied to develop and modify the existing literature on soft topological spaces ii) Defining some new classes of mappings and then proving the existence and uniqueness of such mappings which can be viewed as a positive contribution towards an advancement of metric fixed point theory iii) Initiative of soft fixed point theory in framework of soft metric spaces and proving the results lying at the intersection of soft set theory and fixed point theory which would help in establishing a bridge between these two flourishing areas of research. iv) This study is also a starting point for the future research in the area of fuzzy soft fixed point theory.Abbas, M. (2014). Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48470TESISCompendi

Metrics and T-Equalities

AbstractThe relationship between metrics and T-equalities is investigated; the latter are a special case of T-equivalences, a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of metrics from T-equalities triangular norms with an additive generator play a key role. Conversely, in the construction of T-equalities from metrics this role is played by triangular norms with a continuous additive generator or, equivalently, by continuous Archimedean triangular norms. These results are then applied to the biresidual operator ET of a triangular norm T. It is shown that ET is a T-equality on [0, 1] if and only if T is left-continuous. Furthermore, it is shown that to any left-continuous triangular norm T there correspond two particular T-equalities on F(X), the class of fuzzy sets in a given universe X; one of these T-equalities is obtained from the biresidual operator ETT by means of a natural extension procedure. These T-equalities then give rise to interesting metrics on F(X)

The Fuzzification of Classical Structures: A General View

The aim of this survey article, dedicated to the 50th anniversary of Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We believe that this paper will be a support for future research in this field

Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems