Weighted order statistics filters-their classification, some properties, and conversion algorithm

Stack filters that are based on threshold logic with nonnegative weights and threshold value are called weighted order statistics (WOS) filters and are classified into four different types: type-0 (trivial), type-1 (decreasing), type-2 (increasing), and type-3 (mixed). A significant property of the threshold logic is that it only needs (n+1) tuples to represent its function representation if the input of this function has it variables. This associative representation is very useful in neural computing, nonlinear filtering, etc. The authors propose a significant classification of threshold logic such that the behaviors of WOS filters can be understood easily based on this classification. In the paper, all type-0, type-1, and type-2 WOS filters are shown to possess the convergence property. However, type-3 WOS filters do not necessarily possess the convergence property. Hence, the authors investigate the convergence behaviors of some type-3 WOS filters including symmetric weighted median filters and the type-3 filters proposed in Wendt [1990]. Finally, an efficient algorithm to determine whether a positive Boolean function corresponds to a weighted median filter is proposed. The authors use the property “all threshold logics are regular” to make the algorithm tractable