We will discuss the simplified implementation of Recursive Median (RM) filters. It will be shown that every RM filter an alternative implementation. This implies a fast algorithm, [O(1) per pixel on average], for the one-dimensional RM filter. We also consider the case when RM filters are applied in a cascade of increasing filter window lengths, that is, the RM sieve. We will show that the RM sieve can be implemented in constant time per scale by applying only 3-point median operations. Both of the above mentioned fast implementations are viewed in a new light by constructing the corresponding Finite State Machines (FSM), and observing the achievable state reduction. Radical reduction of complexity takes place by implementing standard state reduction techniques. FSM models also open new possibilities for the analysis of these systems. Finally we discuss the benefits of using the RM sieve instead of the RM filter. We consider the streaking problem of the RM filter. It is demonstrated that the RM filter is not in itself a reliable estimator of location. As the cascading element in the structure of the sieve, however, it is very useful. It turns out that the use of RM sieve reduces the streaking problem to manageable level.