Granulometric Size Density for Segmented Random-Disk Models

Abstract

This design of optimal filters depends upon the random processes under consideration and the operator class from which a filter is to be chosen. Analytic derivation of an optimal filter is facilitated by finding partial descriptors of the random processes that facilitate analytic derivation of an optimal filter from the descriptors—for linear filtering, derivation of the optimal filter is via the covariance function. For nonlinear filtering of random sets (binary images), an analogous approach applies to the derivation of an optimal granulometric bandpass filter (GBF) by means of the granulometric size densities (GSDs) of the signal and noise. A GBF operates by passing or not passing components of a random set. For simply shaped grains having convenient analytic description, finding the GSD is not too difficult; however, when the disjoint union results from the segmentation of a non-disjoint union, the resulting process, and its GSD, can become very complex. This paper postulates a constrained overlap random-set model, and then finds the GSD for the segmented random set. The primary grain of the random model is a disk of random radius and segmentation is via the watershed transformation. Even for a union of disks, modest intersection greatly enhances the difficulty of finding the GSD, and therefore of designing optimal bandpass filters. Although application to disks is specialized, it gives quantitative insight into the effects of grain overlapping and segmentation. Such restriction to a tractable model is not uncommon in the literature of random sets when quantitative estimation results are desired.