Maximum-Likelihood Estimation for the Two-Dimensional Discrete Boolean Random Set and Function Models Using Multidimensional Linear Samples

Abstract

The Boolean model is a random set process in which random shapes are positioned according to the outcomes of an independent point process. In the discrete case, the point process is Bernoulli. Estimation is done on the two-dimensional discrete Boolean model by sampling the germ–grain model at widely spaced points. An observation using this procedure consists of jointly distributed horizontal and vertical runlengths. An approximate likelihood of each cross observation is computed. Since the observations are taken at widely spaced points, they are considered independent and are multiplied to form a likelihood function for the entire sampled process. Estimation for the two-dimensional process is done by maximizing the grand likelihood over the parameter space. Simulations on random-rectangle Boolean models show significant decrease in variance over the method using horizontal and vertical linear samples, each taken at independently selected points. Maximum-likelihood estimation can also be used to fit models to real textures. This method is generalized to estimate parameters of a class of Boolean random functions.