A theory for the structural behavior of stack filters is developed. 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 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 unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior. >