Maximum-likelihood estimation for discrete Boolean models using linear samples

An observation of a Boolean model consists of a set of covered points. In the one-dimensional discrete case, the likelihood function of an observation can be expressed via the lengths of sequences of covered points and points not covered, called black and white runlengths, respectively. The black and white runlengths are independent random variables whose respective distributions determine the one-dimensional discrete Boolean model completely. Under certain conditions, a two-dimensional discrete Boolean model induces a one-dimensional discrete Boolean model, thereby allowing the likelihood function of a one-dimensional observation to be expressed in terms of the parameters of the two-dimensional model. This relationship enables maximum likelihood estimation to be performed on the two-dimensional model using linear samples. Examples are given including an application involving micrographs of toner particles.