.NTEyMw.MjYwOTcy
on the punctual covariance, which is the moment of order two (probability of finding two points simultaneously in the chosen phase). Second 'example : the size measurement of one particular médium (chord distribution for example). G. Matheron proved that this kind of measurement dépends only on the probabi¬ lity that on an arbitrary point x, we can put a given convex set that is con- tained totally within the médium (infinité moment). The two steps above (first moments, and more complicated ones) are not sufficient for solving ail the problems. Let us consider for example the fol- lowing question : "Does the structure that we see, exhibit clustering?". The concept of a cluster is not defined on the same level as the perimeter, and needs, for its description, the reference to a random function with clusters (or without). Another reason for fitting a model cornes from the very large number of data provided by the moment measurements. Each size and shape of the structu- ring element corresponds to a différent probability. It seems often that the key parameters are marked by too much data. The model allows us to predict the theoretical value of every moment, and, as soon as it is correctly fitted, substitutes its own two orthree parameters into the moments. ^ 3 The last reason is that the stereological inference from R to R becomes easier when using models. Today, the library of models in stereology is not very large, and we do not prétend to give here an exhaustive list of them. The most developed are the point processes (4). Starting from the Poisson Point process, one can sophisti- cate it by varying the density, and introducing clusters around the Poisson Points. A second group of models consists of the tessellations of the space R (5). The basic one is thè Poisson flat process, studied by R.E. Miles and later by G. Matheron. In the same group of models, one can quote the doublet of Poisson- nian planes of Cauwe, and the Voronoï polyhedra (Meijering, Gilbert, Miles). A third group consists of the boolean schemes. We will présent later the basic prototype. The three groups probably represent the building blocks in model construc¬ tion, since the Poisson point processes, the Poisson flats and the boolean scheme are indefinitely divisible. THE BOOLEAN SCHEME. In order to illustrate the previous considérations, let us présent now, with more détails, the boolean scheme. . . n . The définition of the Poisson point process m R is well known. It con¬ sists of a random set of points which satisfies the two following properties : a) if B and B' are two sets such as B fl B' =0 , the numbers N(B) and N(B') of points falling in B and Bf are two independent random variables. b) the elementary volume dv contains one point with the probability 0(dv) and none with the probability 1-0(dv). The measure 0 is called density of the process. Here we will take 0 = cons¬ tant, because it leads to more geometrical results. By intégration, the two properties imply that for any bounded B the random integer N(B) satisfies the Poisson law with parameter B P{N(B)=n} - ij- e 0 (1) Let us take a realization of a Poisson process of constant density . This realization . This realization has points which may be indexed by the set I. We consider each point as the germ of a crystalline growth. If two crystals