Morphological PDE and dilation/erosion semigroups on length spaces

Abstract

This paper gives a survey of recent research on Hamilton-Jacobi partial differential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We first introduce the most general pair of dilation/ erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups.