Minkowski operations and vector spaces

Mathematical morphology started as a set of tools for analysing images by the use of transformations based on set-theoretical operations which are the Minkowski sum and subtraction. It was first developed for the analysis of binary images. Its extension to grey-level images was a later development with the extension of the Minkowski operations to real-valued functions in terms of sup-convolution and inf-convolution. The purpose of this paper is to define a type of convolution between set-valued maps, to study its properties, and to establish some associated differential relations. This set-convolution map allows us to extend the Minkowski sum and substraction to multivalued functions and to functions with vectorial values.