Recognizable and Rational Picture Series?

The theory of two-dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition and also concerns models of parallel computing. Here we investigate power series on pictures and assign weights to different devices, ranging from tiling systems to picture automata. We will prove that, for commutative semirings, the behaviours of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations and also coincide with the families of series characterized by weighted tiling or weighted domino systems. Thus we obtain a robust definition of recognizable picture series. The theory of two-dimensional languages is obtained when restricting to the boolean semiring. These new equivalent weighted picture devices can be used to model several interesting application-examples, e.g. the intensity of light of a picture (interpreting the alphabet as different levels of gray) or the maximal amplitude of a monochrome subpicture of a colored picture.