Minimum mean absolute error stack filtering with structural constraint and goals

Two approaches have been used in the past to design or choose a rank-order based filter to estimate a signal from a noise-corrupted observation of that signal: the structural approach and the estimation approach. In the structural approach, the goal is to find a filter which preserves those shapes that are part of the signal while removing those that are part of the noise. In the estimation approach, the goal is to find a filter which best estimates the desired signal given the noise-corrupted version of the signal as the filter's input. This paper develops a theory for the structural behavior of stack filters. This theory provides a test which can determine if a given stack filter has any root signals; a method for classifying the root signal behavior of any stack filter found to have roots; and, perhaps what is most important, a method for designing stack filters with specific root signals or other structural behavior. This theory of root signals for stack filters is then combined with the theory of minimum mean absolute error stack filtering. This new, unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior.