Translation invariant transformations of discrete random sets

Random modeling is an important technique for solving hard problems in coding, quantitative description and restoration of images. In the sixties. Matheron introduced the theory of Random Closed Sets (RACS), that permits the creation of models by morphological transformations of primitive processes. A limitation of this theory is that not all transformation of a RACS produces a new RACS. Recently Goutsias proposed a discrete version of Matheron´s theory that has not this limitation. However. A drawback of the discrete theory is the absence of general formulas for calculating a distribution functional of a Discrete Random Set (DRS) from a distribution functional of the corresponding primitive DRS. In this paper, we present some general formulas for characterizing a DRS generated by a translation invariant operator. Moreover, a new distribution functional for DRS characterization is introduced: the interval distribution functional