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Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric

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Discrete Geometry and Mathematical Morphology (DGMM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12708))

Abstract

Mathematical morphology is a useful theory of nonlinear operators widely used for image processing and analysis. Despite the successful application of morphological operators for binary and gray-scale images, extending them to vector-valued images is not straightforward because there are no unambiguous orderings for vectors. Among the many approaches to multivalued mathematical morphology, those based on total orders are particularly promising. Morphological operators based on total orders do not produce the so-called false-colors. On the downside, they often introduce irregularities in the output image. Although the irregularity issue has a rigorous mathematical formulation, we are not aware of an efficient method to quantify it. In this paper, we propose to quantify the irregularity of a vector-valued morphological operator using the Wasserstein metric. The Wasserstein metric yields the minimal transport cost for transforming the input into the output image. We illustrate by examples how to quantify the irregularity of vector-valued morphological operators using the Wasserstein metric.

This work was supported in part by the São Paulo Research Foundation (FAPESP) under grant no 2019/02278-2 and the National Council for Scientific and Technological Development (CNPq) under grant no 310118/2017-4.

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Authors and Affiliations

  1. Universidade Estadual de Campinas, Campinas, SP, Brazil

    Marcos Eduardo Valle & Samuel Francisco

  2. Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, São Paulo, SP, Brazil

    Samuel Francisco & Marco Aurélio Granero

  3. Center of Mathematical Morphology, Mines ParisTech, PSL Research University, Paris, France

    Santiago Velasco-Forero

Authors
  1. Marcos Eduardo Valle
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  2. Samuel Francisco
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  3. Marco Aurélio Granero
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  4. Santiago Velasco-Forero
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Corresponding author

Correspondence to Marcos Eduardo Valle .

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Editors and Affiliations

  1. Uppsala University, Uppsala, Sweden

    Joakim Lindblad

  2. Uppsala University, Uppsala, Sweden

    Filip Malmberg

  3. Uppsala University, Uppsala, Sweden

    Nataša Sladoje

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Valle, M.E., Francisco, S., Granero, M.A., Velasco-Forero, S. (2021). Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2021. Lecture Notes in Computer Science(), vol 12708. Springer, Cham. https://doi.org/10.1007/978-3-030-76657-3_37

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  • DOI: https://doi.org/10.1007/978-3-030-76657-3_37

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-76656-6

  • Online ISBN: 978-3-030-76657-3

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