Fuzzy rough sets based on modular hememetrics

AbstractIn this paper, we introduce a notion of modular hemimetrics, an extension of hemimetrics and modular metrics, as the basic structure to define and study fuzzy rough sets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper and lower approximation operators and investigate their properties and interrelations. It is shown that upper definable sets and lower definable sets are equivalent. Definable sets form an Alexandrov fuzzy topology such that the upper and lower approximation operators are the closure and the interior operators respectively. At the end, an application of the modular hemimetrics based fuzzy rough set to the acquisition of color gradient images is proposed.