Type-$(2,k)$ Overlap Indices
Antonio Francisco Roldán López de Hierro, Concepción Roldán, Miguel Ángel Tíscar, Zdenko Takáč, Regivan Santiago, Graçaliz Pereira Dimuro, Javier Fernández, Humberto Bustince
Abstract
Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">overlap indices</i> . They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">overlap indices</i> is also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">type-<inline-formula><tex-math notation="LaTeX">$(2,k)$</tex-math></inline-formula> overlap index</i> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k \in \lbrace 0,1,2\rbrace$</tex-math></inline-formula> ) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(2,k)$</tex-math></inline-formula> overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.