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Geometry, Morphology, and Computational Imaging pp 1–21Cite as

Flat Morphological Operators on Arbitrary Power Lattices

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2616))

Abstract

We give here the basis for a general theory of flat morphological operators for functions defined on a space E and taking their values in an arbitrary complete lattice V . Contrarily to Heijmans [4], [6], we make no assumption on the complete lattice V , and in contrast with Serra [18], we rely exclusively on the usual construction of flat operators by thresholding and stacking.

Some known properties of flat operators for numerical functions (V = \( \overline Z \) or \( \overline R \) ) extend to this general framework: flat dilations and erosions, flat extension of a union of operators or of a composition of an operator by a dilation. Others don’t, unless V is completely distributive: flat extension of an intersection or of a composition of operators; for these we give counterexamples with V being the non-distributive lattice of labels. In another paper [15], we will consider the commutation of flat operators with anamorphoses (contrast functions) and thresholdings, duality by inversion, as well as related questions of continuity.

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References

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Author information

Authors and Affiliations

  1. LSIIT UMR 7005 CNRS-ULP, Boulevard Sébastien Brant, 67400, Illkirch, France

    Christian Ronse

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  1. Christian Ronse
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Editor information

Editors and Affiliations

  1. JAIST, School of Information Science, 1-1 Asahidai, Tatsunokuchi, 923-1202, Ishikawa, Japan

    Tetsuo Asano

  2. Computer Science Dept. and CITR, University of Auckland, Tamaki Campus, Glen Innes, 1005, Auckland, New Zealand

    Reinhard Klette

  3. LSIIT UMR 7005 CNRS-ULP, Parc d’Innovation, Boulevard Sebastien Brant, BP 10413, 67412, Illkirch, France

    Chrisitan Ronse

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© 2003 Springer-Verlag Berlin Heidelberg

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Ronse, C. (2003). Flat Morphological Operators on Arbitrary Power Lattices. In: Asano, T., Klette, R., Ronse, C. (eds) Geometry, Morphology, and Computational Imaging. Lecture Notes in Computer Science, vol 2616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36586-9_1

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  • DOI: https://doi.org/10.1007/3-540-36586-9_1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00916-0

  • Online ISBN: 978-3-540-36586-0

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