Fuzzy rough sets based on modular hemimetrics

In this paper, we introduce a novel notion of modular hemimetrics and use it as the basic structure to define a pair of fuzzy upper and lower approximation operators by using the usual addition and subtraction of real numbers. Then we investigate properties and interrelations of the two approximation operators. We demonstrate that upper definable sets, lower definable sets and 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. Finally, we consider an application of the modular hemimetrics based fuzzy rough sets to the acquisition of gray gradient images in order to show how the fuzzy rough set model based on modular hemimetrics can be successfully applied to solve image processing problems, such as image coding and decoding problems and image morphological processing.