Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

Jesus Angulo (1) , Santiago Velasco-Forero (1)

1 CMM - Centre de Morphologie Mathématique

Ultrametric spaces are the natural mathematical structure to deal with data embedded into a hierarchical representation. This kind of representations is ubiquitous in morphological image processing, from pyramids of nested partitions to more abstract dendograms from minimum spanning trees. This paper is a formal study of morphological operators for functions dened on ultrametric spaces. First, the notion of ultrametric structuring function is introduced. Then, using as basic ingredient the convolution in (max,min)-algebra, the multi-scale ultra-metric dilation and erosion are dened and their semigroup properties are stated. It is proved in particular that they are idempotent operators and consequently they are algebraically ultrametric closing and opening too. Some preliminary examples illustrate the behavior and practical interest of ultrametric dilations/erosions.

Mots clés: ultrametric space; ultrametric semigroup; idempotent operator; (max; min)-convolution

Domaines: Traitement des images [eess.IV]

Jesus Angulo, Santiago Velasco-Forero. Morphological Semigroups and Scale-Spaces on Ultrametric Spaces. 13th International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, May 2017, Fontainebleau, France. pp.28-39, ⟨10.1007/978-3-319-57240-6_3⟩. ⟨hal-01536368⟩