Root properties of median filters under three appending strategies

Three appending strategies are used to append some bits to both ends of one-dimensional finite input signals so that the median filter can perform a well-defined operation near the endpoints of the finite input signals. The three strategies are the first-and-last value carry on strategy, the constant-value carry-on strategy, and the circular strategy. Under all three, the cardinality of the root set of median filters is presented, using recurrence relations. A closed-form expression for the number of median roots with window width three under the constant-value carry-on strategy is derived