Optimal and adaptive design of logical granulometric filters

Abstract

Publisher Summary This chapter discusses the statistical design of filters for sieving granular random sets. The random set consists of a union of disjoint compact grains, and some of the grains are to be passed, while others are to be eliminated. There is a signal random set and a noise random set. Both signal and noise are composed of disjoint grains, and signal and noise are disjoint. In fact, if the random sets originate as binary images, then the captured images may not fit the model; however, upon application of some segmentation algorithm, they fit the model. Given an input set and a structuring element, a deterministic set, a point is in the output of the opening only if there exists a translate of the structuring element that contains the point and is a subset of the input set. Points that do not satisfy this structural criterion are removed from the set. If a grain is not sufficiently large or appropriately shaped to contain translate of the structuring element, then the entire grain is removed.