On the Structure of Idempotent Monotone Boolean Functions
Monotone Boolean functions have been extensively studied in the area of nonlinear digital filtering, specifically stack and morphological filtering. In fact, any Stack Filter of window-width n is uniquely specified by a monotone Boolean function of n variables. Similarly, the Stacking Property obeyed by all stack filters is the monotonicity of these Boolean functions . In this paper, we focus on idempotent monotone Boolean functions and develop some interesting properties related to their structure as well as give several necessary conditions for idempotent functions with two minimal primes. The idempotence property implies that a root signal is obtained in one pass. That is, subsequent filter passes do not alter the signal. By developing some structural properties of these functions, we pave the way toward a characterization of the structure of this class of functions. Such a characterization would prove to be most useful in the theory of optimal stack filtering and would facilitate the search for optimal idempotent stack filters. To conserve space, we will omit the proofs of some propositions and lemmas which are relatively straightforward to construct.
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