Fuzzy integrals as a generalized class of order filters
Michel Grabisch · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
We introduce a new large class of filters similar to order filters, namely fuzzy integrals. Fuzzy integrals are a generalization of Lebesgue integral, and are defined with respect to a nonadditive measure. Fuzzy integrals can also be viewed as function processing filters, and appear to include a large class of known filters. We show that fuzzy integrals includes as particular case all linear filters, all order filters, and thus rank filters, the median, erosion, and dilation, whose properties of noise filtering are well known. In a second part, we give more insight into properties related to morphological filters, in an attempt to generalize results known for rank filters. Considering a gray level image as a fuzzy set, we extend the usual definition of dual filters with respect to complementation to any function processing filter. Based on this, we show that the dual of a fuzzy integral filter is the fuzzy integral filter with respect to the dual measure.