Model of categories for image processing

Abstract

Theory of categories has various applications in technology. There are mathematical models that have been built using algebraic structures equipped with different tools of properties from disciplines inside mathematics even inside algebra. We are concentrated on the applications of algebra in image processing. We have seen that basic relations and operations between images, and other used in image compression, already have been described in terms of categories and homomorphism of modules over quantales. They include relations and operations between sets, set inclusion, union, intersection, used in mathematical morphology, and operations, order relation, t-norms in image compression. Focused on the way how these construction of categories of quantale modules are built up we want to show a new construction of an algebraic model through categories as an application of algebraic structures, involving issues from order theory, theory of modules and theory of categories in mathematical morphology and image compression. We start from a model on the theory of categories and theory of modules over a commutative ring. We think to combine the construction of a category from an object with homomorphism of modules over quantales using as a model the same issue over categories of modules and over the category of homomorphism of modules over rings. Our intention is to open the way the study of properties of some important modules and their homomorphism that are immediate objects in image processing, seeing them as categories on their own not as a simple object of a category. Through this we want to import the properties from the treatment existing now to investigate after other properties deriving from our point of view.