Fuzzy Integrals and Smoothing Filters

Abstract

Fuzzy integrals can be considered as averaging operators. They can cover a wide variety of averaging operators. Smoothing filters have been treated in the field of digital filtering and the essence of filtering is averaging. Moving averaging and median filtering are typical examples of linear filter and non-linear filter. Some extensions of median filter were studied by many authors. There is another approach of non-linear filter from the field of mathematical morphology. These two fields, namely fuzzy integral and smoothing filter, have been developed independently, but there are many similarities between the two fields. In this paper, we investigate the relationship between the two fields. More precisely, we treat the relationship between fuzzy integral filters which are defined by using Choquet integral, Sugeno integral and opposite Sugeno integral and smoothing filters which are linear filter, order filter, stack filter and mathematical morphology filter. Finally, we propose new idempotent filters by using a cascade connection of two fuzzy integral filters i.e. Sugeno integral filter and opposite Sugeno integral filter.