A UNIFIED DESIGN METHOD FOR RANK ORDER, STACK, AND GENERALIZED STACK FILTERS BASED ON CLASSICAL BAYES DECISION

This paper presents a unified method for designing optimal rank order filters (ROF's), stack filters, and generalized stack filters (GSFs) under the mean absolute error (MAE) criterion. The method is based on classical Bayes minimum-cost decision. Both the a priori and the a posteriori approaches are considered. It is shown that designing the minimum MAE stack filters and GSF's is equivalent to the a priori Bayes decision. To solve the problem, linear programming (LP), whose complexity increases faster than exponentially as a function of the filter window width, is required. This renders the use of this approach extremely impractical. In this paper, we shall develop suboptimal routines to avoid the huge complexity of the LP for designing stack filters and GSF's. It is shown that the only required computations then reduce to data comparisons exclusively, and the number of comparisons needed increases only exponentially (for GSF's) or even slower than exponentially (for stack filters) as a function of the filter's window width. The most important feature of the design routines is perhaps the fact that, for most practical cases, they yield optimal solutions under the MAE criterion. When the a posteriori approach is employed, it is shown that the optimal solutions become ROF's with appropriately chosen orders that do not depend on the prior probability model of the input process. Moreover, it is shown that the a posteriori Bayes minimum-cost decision reduces to the median filter in frequent practical applications. The filters produced by the a priori and the a posteriori approaches are subjected to a sensitivity analysis to quantify their dependency upon the cost coefficients. Several design examples of ROF's, stack filters, and GSFs will be provided, and an application of stack filters and GSF's to image recovery from impulsive noise will be considered.