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

the translated shapes. Indeed, this allows one to model overlapping objects. The random shape process is called The Boolean model is a random set process in which random shapes are positioned according to the outcomes of an indepenthe grain process and the point process is called the germ dent point process. In the discrete case, the point process is process. We call the union process a germ–grain model to Bernoulli. Estimation is done on the two-dimensional discrete distinguish the observed process from the generating set Boolean model by sampling the germ–grain model at widely and point processes. In the two-dimensional (2D) discrete spaced points. An observation using this procedure consists setting, we restrict the random shape process to a finite of jointly distributed horizontal and vertical runlengths. An collection of bounded subsets of the discrete plane. The approximate likelihood of each cross observation is computed. shapes will also be horizontally and vertically convex; that Since the observations are taken at widely spaced points, they is, a vertical or horizontal line intersecting the shape anyare considered independent and are multiplied to form a likeliwhere will produce only a single line segment. This is hood function for the entire sampled process. Estimation for required to use our previous work on the one-dimensional the two-dimensional process is done by maximizing the grand likelihood over the parameter space. Simulations on random(1D) model. Points covered by a random shape are called rectangle Boolean models show significant decrease in variance black, and white otherwise. over the method using horizontal and vertical linear samples, In previous work, we derived the likelihood function each taken at independently selected points. Maximum-likeliof the one-dimensional directional model. Under certain hood estimation can also be used to fit models to real textures. conditions described in [10], 2D models induce 1D models This method is generalized to estimate parameters of a class on intersecting lines. Observations of induced 1D models of Boolean random functions.  1997 Academic Press are observations of the 2D model and the 1D likelihood contains information about the 2D model [9]. These likelihood functions provide maximum-likelihood estimates via