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Distributed arithmetic (DA)-based implementation of linear filters relies on the linear nature of this operation and has been suggested as a multiplication free solution. In this work, we introduce a nonlinear extension of linear filters optimizing under mean-square error criterion the memory function [(MF) multivariate Boolean function with not only binary output] which is in the core of DA based implementation. Such an extension will improve the filtering of noise which may contain non-Gaussian components without increasing the complexity of implementation. Experiments on real images have shown the superiority of the proposed filters over the optimal linear filters. Different versions of these filters are also considered for an impulsive noise removal, faster processing, and filtering using large input data windows.