Type-$(2,k)$ Overlap Indices

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.