A general framework for the design of stack filters using the Lp norm objective function

It has been thought for some time now that the design of stack filters using the L_p norm is mathematically intractable. This paper, for the first time, addresses the problem of designing optimal stack filters by employing an L_p norm of the error between the desired signal and the estimated one. It is shown that the L_p norm can be expressed as a linear function of the decision errors at the binary levels of the filter. Thus, an L_p-optimal stack filter can be obtained as a solution of a linear program. The conventional design of using the mean absolute error (MAE), therefore, becomes a special case of the general, L_p norm based design, developed here. The conventional MAE design of an important subclass of stack filters, the weighted order statistic filters, is also extended to the L_p norm-based design. By considering a typical application of restoring images corrupted with impulsive noise, several design examples are presented, to illustrate that the L_p-optimal stack filters with p⩾2 can provide a far superior performance in terms of their capability of removing impulsive noise, compared to that achieved by using the conventional minimum MAE stack filters