Solutions for optimal Boolean and stack filter design under a training framework

This paper introduces a training framework for the optimal nonlinear filter design problem. The problem to be solved within the present framework is the selection of the best filter under a data dependent criterion (rather than a model dependent criterion) in one class of nonlinear filters. A class of filters, namely Boolean filters is then considered, for which holds a decoupling property, allowing to transform the initial integer valued problem into the binary domain. The equivalence between the original criterion (in the integer signal domain) and a criterion expressed in the binary signal domain is shown then to hold. The procedures for obtaining the optimal solution for two classes of nonlinear filters, Boolean filters and stack filters, are derived under the new framework. Some numerical simulations are provided, in order to illustrate the effectiveness of the procedures in solving the noise rejection problem.