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Equivariance-Based Analysis of PDE Evolutions Related to Multivariate Medians

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Discrete Geometry and Mathematical Morphology (DGMM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13493))

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Abstract

For multivariate data there exist several concepts generalising the median, which differ by their equivariance properties w.r.t. transformations of the data space (e.g. Euclidean, affine). In earlier work on the asymptotic analysis of multivariate image filters built upon these concepts, it was observed that several affine equivariant median filters approximate the same system of partial differential equations (PDEs). In this paper we discuss the equivariance properties of multivariate medians and their associated PDEs in more detail. We discuss what equivariance concept is the preferable generalisation of the very strong equivariance of the scalar-valued median (sometimes also denoted as morphological equivariance) w.r.t. arbitrary monotone transformations. Moreover, we derive multivariate PDE evolutions systematically from equivariance properties. It turns out that the approximation of the same PDE system by different affine equivariant medians is no coincidence but a necessary implication of their equivariance properties. As a by-product, a more general class of multivariate PDE evolutions with favourable equivariance properties arises.

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Authors and Affiliations

  1. Institute of Biomedical Image Analysis, UMIT TIROL – Private University of Health Sciences, Medical Informatics and Technology, Eduard-Wallnöfer-Zentrum 1, 6060, Hall/Tyrol, Austria

    Martin Welk

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  1. Martin Welk
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Correspondence to Martin Welk .

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Editors and Affiliations

  1. University of Strasbourg, Strasbourg, France

    Étienne Baudrier

  2. University of Strasbourg, Strasbourg, France

    Benoît Naegel

  3. University of Strasbourg, Strasbourg, France

    Adrien Krähenbühl

  4. University of Strasbourg, Strasbourg, France

    Mohamed Tajine

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Welk, M. (2022). Equivariance-Based Analysis of PDE Evolutions Related to Multivariate Medians. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_16

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  • DOI: https://doi.org/10.1007/978-3-031-19897-7_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19896-0

  • Online ISBN: 978-3-031-19897-7

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