Spectral design of weighted median Filters:A general iterative approach

A new design strategy for weighted median (WM) filters admitting real and complex valued weights is presented. The algorithms are derived from Mallows theory for nonlinear selection type smoothers, which states that the closest linear filter to a selection type smoother in the mean square error sense is the one having as coefficients the sample selection probabilities (SSPs) of the smoother. The new design method overcomes the severe limitations of previous approaches that require the construction of high order polynomial functions and high dimensional matrices. As such, previous approaches could only provide solutions for filters of very small sizes. The proposed method is based on a new closed-form function used to derive the SSPs of any WM smoother. This function allows for an iterative approach to WM filter design from the spectral profile of a linear filter. This method is initially applied to solve the median filter design problem in the real domain, and then, it is extended to the complex domain. The final optimization algorithm allows the design of robust weighted median filters of arbitrary size based on linear filters having arbitrary spectral characteristics.