Algebraic Framework for Linear and Morphological Scale-Spaces

This paper proposes a general algebraic construction technique for image scale-spaces. The basic idea is to first downscale the image by some factor using an invertible scaling, then apply an image operator (linear or morphological) at a unit scale, and finally resize the image to its original scale. It is then required that the resulting one-parameter family of image operators satisfies the semigroup property. Such an approach encompasses linear as well as nonlinear (morphological) operators. Furthermore, there exists some freedom as to which semigroup operation on the scale (or time) axis is being chosen. Particular attention is given to additive and supremal semigroups. A large part of the paper is devoted to morphological scale-spaces, in particular to scale-spaces assoctiated with an erosion or an opening. In these cases, classical tools from convex analysis, such as the (Young-Frenchel) conjugate, play an important role.