Computationally efficient approximate likelihood procedures for the Boolean model

Abstract

The Boolean model is a simple nontrivial random closed set process. It consists of a Poisson point process (producing germs) coupled with an independent random shape process (grains). Origins of grains are translated to germs to produce an arrangement of overlapping (or interpenetrating) shapes. An accurate discrete approximation to the continuous Boolean model on the real line offers computationally efficient likelihood procedures including maximum-likelihood estimation and likelihood ratio tests. The discrete approximation allows covering probabilities to be calculated using recursive formulas, which approach continuous densities as the sampling rate increases. Inference for higher dimensional Boolean models can be handled by linear transects. Several 2D estimation examples demonstrate the efficacy of this method.