A design method of fuzzy weighted median filters

Stack filters are a class of nonlinear filters and include rank order filters, morphological filters, weighted median filters, and so on. The stack filter is defined by a Boolean function. The output of the Boolean function is restricted to two values (i.e., 0 or 1). We attempt to enlarge the class of stack filters by defining the output of the Boolean function from 0 to 1 continuously. We call this filter the fuzzy stack filter. Fuzzy center weighted median filters which are one type of fundamental fuzzy stack filters, have been proposed. We propose fuzzy weighted median (FWM) filters which are more general stack filters, and show the design method of those. The FWM filter is synthesized by two factors, the fuzzy Boolean function and the filter weights. We approximate the fuzzy Boolean function by a sigmoidal function. Furthermore, we approximate the weights of the proposed filter by the S-type function which is used in the field of fuzzy theory. As a result, the FWM filter is defined by only four parameters. The proposed filter can be derived by the LMS algorithm under the mean squared error criterion