Rank conditioned rank selection filters for signal restoration

A class of nonlinear filters called rank conditioned rank selection (RCRS) filters is developed and analyzed in this paper. The RCRS filters are developed within the general framework of rank selection (RS) filters, which are filters constrained to output an order statistic from the observation set. Many previously proposed rank order based filters can be formulated as RS filters. The only difference between such filters is in the information used in deciding which order statistic to output. The information used by RCRS filters is the ranks of selected input samples, hence the name rank conditioned rank selection filters. The number of input sample ranks used is referred to as the order of the RCRS filter. The order can range from zero to the number of samples in the observation window, giving the filters valuable flexibility. Low-order filters can give good performance and are relatively simple to optimize and implement. If improved performance is demanded, the order can be increased but at the expense of filter simplicity. In this paper, many statistical and deterministic properties of the RCRS filters are presented. A procedure for optimizing over the class of RCRS filters is also presented. Finally, extensive computer simulation results that illustrate the performance of RCRS filters in comparison with other techniques in image restoration applications are presented.