Random Processes Simulations on the Texture Analyser Centre De Morphologie

The prototype Texture Analyser at Fontainebleau permits very swift simulations of many random processes, be they point processes or random set processes .We shall illustrate, with some examples of general processes, the possibilities of this apparatus. Three kinds of processes are presented here:-The Poisson point process-Cluster Processes ("shooting" scheme)-Hard-Core processes (repulsion models) With various combinations of these three fundamental processes, one may induce more complex realizations of random sets containing two or three phases. Poisson point process A Poisson point process, with intensity e, is very easy to simulate on a bounded set A. The simulation is divided in two stages : One chooses n, the number of points of the set A from a Poisson distri bution with mean e. x IA I. (IA I, Lebesgue measure of A). One distributes the n points at random over the set A, which in the present case, is the rectangular field of the texture analyser. It is possible, starting from a Poisson realization, to simulate random sets processes by centering in every point of the Poisson process one realization from a non-stationary random set, called the primary grain X'. Two particular realizations of such processes, the Boolean Schemes, are shown in figure 1. Cluster ProcesSes When the primary grain of a Boolean Scheme is a cluster of N points distributed independantly around each of the Poisson process points according to a distribution law F, one is simulating a cluster process, also called a "schooting scheme", ("Schema du Tireur"-Matheron G.) or a Neymann-Scoot process (fig. 2).