Minimum mean square error filtering over the class of extended threshold Boolean filters

A class of nonlinear digital filters, called the threshold Boolean filter (TBF), is introduced. The TBF is defined by a Boolean function on the binary domain and is a natural extension of stack filters. In this paper, the TBF is further extended to a larger class of filters, called the extended TBF (ETBF), which encompasses linear FIR and linear combination of order statistic (LOS) filters as well as the TBF. The optimal design of the TBF and ETBF under the mean square error (MSE) criterion is investigated. It is shown that the optimization of the TBF can be formulated as a quadratic zero-one programming, and that the optimization of the ETBF as a classical quadratic problem. Thus, linear FIR and median-type nonlinear filters can be analyzed and designed in a unified framework of the ETBF. The ETBF has been applied to enhance noisy images. The results show that the ETBF outperforms the TBF and the FIR Wiener filter