Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

Morphological semigroups for functions on length spaces have recently studied [2], whose basic ingredients are the convolution in the (max,+)-algebra (or supremal convolution), the metric distance and a convex shape function. Links with Hamilton–Jacobi PDE on length spaces appears also naturally. These semigroups lead to powerful scale-space properties for multiscale filtering, regularization and feature extraction. Such operators on length spaces endowed with a Maslov measure are also related to idempotent analysis [3]. The goal of this paper is to consider a similar generalization of morphological semigroups to the case of functions on ultrametric spaces.