A generalization of the concept of size

The size of a set is defined without ambiguity as a ratio of homothety with an elementary set having the same shape. In order to generalize this, we compare the object studied with the elements of the family {λB} of the homothetics of B. The size λ of a particular element B is going to act as a measure of the object. If the latter is made up of individualized elements, we can calculate the size of each one and construct the size histogram, assuming for example that the size of a connex component C according to B is the size of the largest B included in C. (Examples: inscribed circle radius, maximum intercept.) For any object (alveolar lung space, pore system), the size at each point x is defined as the largest λB included in the object and containing x. The set of points x of a given size λ is related to the opening with respect to λB. This geometrical transformation of opening has mathematical properties similar to those of the sieving process, and is the basis of the size distribution concept as generalized by mathematical morphology. In the plane, the texture analyser allows the above measurements.