Stack filters and the mean absolute error criterion

A method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal, given noisy observations of this desired signal, is presented. Specifically, an optimal window-width-b stack filter can be determined with a linear program with O(b2^b) variables. This algorithm is efficient since the number of different inputs to a window-width-b filter is M^b if the filter has M-valued input and the number of stack filters grows faster than 2 raised to the 2^b/2 power. It is shown that optimal stack filtering under the mean-absolute-error criterion is analogous to optimal linear filtering under the mean-squared-error criterion: both linear filters and stack filters are defined by superposition properties, both classes are implementable, and both have tractable procedures for finding the optimal filter under an appropriate error criterion