Optimal and adaptive design of logical granulometric filters

Statistical design of filters for sieving granular random sets that consists of a union of disjoint compact grains are discussed. There is a signal random set S and a noise random set N. Both signal and noise are composed of disjoint grains, and signal and noise are disjoint. Granulometries, as applied to random sets, are based on the morphological opening operator. Given an input set and a structuring element (probe), which is a deterministic set, a point is in the output of the opening if and only if there exists a translate of the structuring element that contains the point and is also a subset of the input set. Points that do not satisfy this structural criterion are removed from the set. Single-parameter granulometric filters formed as unions of reconstructive openings, multiple-parameter granulometric filters formed as unions of reconstructive openings, single-parameter reconstructive granulometric bandpass filters formed as unions of differences of granulometric filters, and logical structural filters (LSFs), are studied.