Boolean derivatives, weighted Chow parameters, and selection probabilities of stack filters

The theory of Boolean derivatives, the activities of the arguments of a Boolean function (BF), and Chow (1961) parameters are studied from the point of view of their application in the statistical analysis of a class of nonlinear filters-stack filters. The connection between the partial derivatives of a positive BF (PBF) and the selection probabilities of stack filters is established. The notions of the weighted activities of the variables of the PBF and weighted Chow parameters are introduced for the analysis, the computation of the joint selection probability matrix, and the sample selection probability vector of a continuous stack filter. Spectral approaches to the selection probabilities of stack filters are derived. In particular, spectral algorithms with computational complexity O(2^N), where N is the number of input samples within an input window, are given for the computation of sample selection probability vectors. The difference of the spectral algorithms presented from the nonspectral ones is that spectral algorithms are universal, i.e., their complexities are independent of the PBF, which is used as the base for stack filtering. They are also straightforward to implement, and fast spectral transforms exist