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Shape in Picture pp 177–195Cite as

On Information Contained in the Erosion Curve

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Part of the NATO ASI Series book series (NATO ASI F,volume 126)

Abstract

An erosion curve can be associated with any binary planar shape. This curve is the function which maps a given radius to the area of the shape eroded by a sphere with this radius. Note the analogy with the approach whereby a shape is quantified by granulometry. Under some regularity conditions the erosion curve of a given shape can be expressed as an integral of the quench function along the skeleton of this shape. This paper describes the relationship between sets with the same erosion curve. It is shown that this curve is not affected if the arcs of the skeleton are bent: the erosion curve quantifies “soft” shapes. In the generic case, there are five possible cases of behaviour of the second derivative of the erosion curve: each case corresponds to a different behaviour of the skeleton and the associated quench function. Finally, it is shown how to reconstruct the family of shapes with a given erosion curve.

Keywords

  • mathematical morphology
  • erosion curve
  • granulometry
  • skeleton
  • quench function
  • shape index
  • isoperimetrical deficiency index
  • elongation index
  • concavity index
  • stretching index
  • spectral function.

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  • DOI: 10.1007/978-3-662-03039-4_12
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Author information

Authors and Affiliations

  1. L.C.R, Thomson-CSF, Domaine de Corbeville, 91404, Orsay-Cedex, France

    Juliette Mattioli & Michel Schmitt

  2. CEREMADE, Université Paris Dauphine, 75775, Paris-Cedex, France

    Juliette Mattioli

Authors
  1. Juliette Mattioli
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  2. Michel Schmitt
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Editor information

Editors and Affiliations

  1. Centre for Mathematics and Computer Science (CWI), Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Ying-Lie O & Henk J. A. M. Heijmans & 

  2. Institute for Human Factors (TNO-IZF), Netherlands Organization for Applied Scientific Research, Kampweg 5, 3769 DE, Soesterberg, The Netherlands

    Alexander Toet

  3. Department of Communication and Neuroscience, University of Keele, ST5 5BG, Keele, Staffordshire, UK

    David Foster

  4. Department of Electrical and Computer Engineering, Rutgers University, 08855-0909, Piscataway, NJ, USA

    Peter Meer

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© 1994 Springer-Verlag Berlin Heidelberg

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Mattioli, J., Schmitt, M. (1994). On Information Contained in the Erosion Curve. In: O, YL., Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (eds) Shape in Picture. NATO ASI Series, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03039-4_12

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  • DOI: https://doi.org/10.1007/978-3-662-03039-4_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08188-0

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