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A Link Between the Multiplicative and Additive Functional Asplund’s Metrics

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Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11564))

Abstract

Functional Asplund’s metrics were recently introduced to perform pattern matching robust to lighting changes thanks to double-sided probing in the Logarithmic Image Processing (LIP) framework. Two metrics were defined, namely the LIP-multiplicative Asplund’s metric which is robust to variations of object thickness (or opacity) and the LIP-additive Asplund’s metric which is robust to variations of camera exposure-time (or light intensity). Maps of distances - i.e. maps of these metric values - were also computed between a reference template and an image. Recently, it was proven that the map of LIP-multiplicative Asplund’s distances corresponds to mathematical morphology operations. In this paper, the link between both metrics and between their corresponding distance maps will be demonstrated. It will be shown that the map of LIP-additive Asplund’s distances of an image can be computed from the map of the LIP-multiplicative Asplund’s distance of a transform of this image and vice-versa. Both maps will be related by the LIP isomorphism which will allow to pass from the image space of the LIP-additive distance map to the positive real function space of the LIP-multiplicative distance map. Experiments will illustrate this relation and the robustness of the LIP-additive Asplund’s metric to lighting changes.

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Authors and Affiliations

  1. University of Strathclyde, Institute of Global Public Health, Dardilly, Lyon, France

    Guillaume Noyel

  2. International Prevention Research Institute, Dardilly, Lyon, France

    Guillaume Noyel

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  1. Guillaume Noyel
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Correspondence to Guillaume Noyel .

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Editors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Bernhard Burgeth

  2. Forschungszentrum Jülich GmbH, Jülich, Germany

    Andreas Kleefeld

  3. Université de Strasbourg, Illkirch Cedex, France

    Benoît Naegel

  4. Université de Reims Champagne-Ardenne, Reims Cedex 2, France

    Nicolas Passat

  5. Université Paris-Est, Noisy le Grand Cedex, France

    Benjamin Perret

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Noyel, G. (2019). A Link Between the Multiplicative and Additive Functional Asplund’s Metrics. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_4

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  • DOI: https://doi.org/10.1007/978-3-030-20867-7_4

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