Morphological PDE and dilation/erosion semigroups on length spaces

Jesus Angulo (1)

1 CMM - Centre de Morphologie Mathématique

This paper gives a survey of recent research on Hamilton-Jacobi partial dierential 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.

Mots clés: Hamilton--Jacobi PDE; length space; morphological PDE; morphological semigroup; Hamilton--Jacobi semigroup

Domaines: Traitement des images [eess.IV]; Equations aux dérivées partielles [math.AP]

Jesus Angulo. Morphological PDE and dilation/erosion semigroups on length spaces. 12th International Symposium on Mathematical Morphology, May 2015, Reykjavik, Iceland. pp.509-521, ⟨10.1007/978-3-319-18720-4_43⟩. ⟨hal-01108145v3⟩