Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10225)


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 dendrograms from minimum spanning trees. This paper is a formal study of morphological operators for functions defined 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 ultrametric dilation and erosion are defined 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.


Ultrametric space Ultrametric semigroup Idempotent operator (max min)-convolution 


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CMM-Centre de Morphologie MathématiqueMINES ParisTech, PSL-Research UniversityFontainebleauFrance

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