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Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

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Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10225))

Abstract

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.

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Authors and Affiliations

  1. CMM-Centre de Morphologie Mathématique, MINES ParisTech, PSL-Research University, Fontainebleau, France

    Jesús Angulo & Santiago Velasco-Forero

Authors
  1. Jesús Angulo
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  2. Santiago Velasco-Forero
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Correspondence to Jesús Angulo .

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Editors and Affiliations

  1. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Jesús Angulo

  2. MINES ParisTech, PSL-Research University , Fontainebleau, France

    Santiago Velasco-Forero

  3. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Fernand Meyer

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Angulo, J., Velasco-Forero, S. (2017). Morphological Semigroups and Scale-Spaces on Ultrametric Spaces. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2017. Lecture Notes in Computer Science(), vol 10225. Springer, Cham. https://doi.org/10.1007/978-3-319-57240-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-57240-6_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57239-0

  • Online ISBN: 978-3-319-57240-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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