On Estimating Granulometric Discrete Size Distributions of Random Sets

Abstract

Morphological granulometries, and the associated size distributions and densities, are important shape/size summaries for random sets. They have been successfully employed in a number of image processing and analysis tasks, including shape analysis, multiscale shape representation, texture classification, and noise filtering. For most random set models however it is not possible to analytically compute the size distribution. In this contribution, we investigate the problem of estimating the granulometric (discrete) size distribution and size density of a discrete random set. We propose a Monte Carlo estimator and compare its properties with that of an empirical estimator. Theoretical and experimental results demonstrate superiority of the Monte Carlo estimation approach. The Monte Carlo estimator is then used to demonstrate existence of phase transitions in a popular discrete random set model known as a binary Markov random field, as well as a tool for designing “optimal” filters for binary image restoration.