Lp norm design of stack filters
This paper addresses the problem of designing optimal stack filters by employing an Lp norm of the error between the desired signal and the estimated one. It is shown that the Lp norm can be expressed as a linear function of the decision errors at the binary levels of the filter. Thus, an Lp-optimal stack filter can be determined as the solution of a linear program. The conventional design of using the mean absolute error (MAE), therefore, becomes a special ease of the general Lp norm-based design developed here. Other special cases of the proposed approach, of particular interest in signal processing, are the problems of optimal mean square error (p=2) and minimax (p-->infinity) stack filtering. Since an Linfinity optimization is a combinatorial problem, with its complexity increasing faster than exponentially with the filter size, the proposed Lp norm approach to stack filter design offers an additional benefit of a sound mathematical framework to obtain a practical engineering approximation to the solution of the minimax optimization problem. The conventional MAE design of an important subclass of stack filters, the weighted order statistic filters, is also extended to the Lp norm-based design. By considering a typical application of restoring images corrupted with impulsive noise, several design examples are presented, to illustrate the performance of the Lp-optimal stack filters with different values of p. Simulation results show that the Lp-optimal stack filters with p=or>2 provide a better performance in terms of their capability in removing impulsive noise, compared to that achieved by using the conventional minimum MAE stack filters.