Analysis and synthesis of textures through the inference of Boolean functions

Abstract

This work deals with Boolean functions of non-linear and linear basis. The Boolean random functions of non-linear basis were proposed by Serra (1988,1989). These functions are generated through a Poisson point process upon which a family of independent functions, called germ functions, are installed. This process of installation consists in taking the Sup (supremum), point to point, of the result of placing the germ functions upon the points of the Poisson process. Boolean functions of linear basis, which are defined and proposed in this paper, are generated in the same manner as the non-linear functions but with a modified installation process. Instead of taking the Sup point to point, the sum point to point is defined. So the process is then equivalent to the convolution of a Poisson train of deltas with a random pulse. The aim of this paper is to analyse textures through these two models, in order to infere their genetics through a given realisation of the process, i.e., to analyse the complete statistics of the germ functions and the density of the associated Poisson process in order to characterise a given texture. Experiments and results are provided which prove that the real textures can be understood as realisations of Boolean random functions (of linear and non-linear basis), and that it has been possible to infere the genetics of unidimensional Boolean random functions of linear basis with the algorithm proposed here. It has also been possible to do it with non-linear Boolean functions but only by imposing two restrictive conditions on the genetics of the realisation.