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Stability of the Tree of Shapes to Additive Noise

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Discrete Geometry and Mathematical Morphology (DGMM 2021)

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

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Abstract

The tree of shapes (ToS) is a famous self-dual hierarchical structure in mathematical morphology, which represents the inclusion relationship of the shapes (i.e. the interior of the level lines with holes filled) in a grayscale image. The ToS has already found numerous applications in image processing tasks, such as grain filtering, contour extraction, image simplification, and so on. Its structure consistency is bound to the cleanliness of the level lines, which are themselves deeply affected by the presence of noise within the image. However, according to our knowledge, no one has measured before how resistant to (additive) noise this hierarchical structure is. In this paper, we propose and compare several measures to evaluate the stability of the ToS structure to noise.

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Notes

  1. 1.

    We recall that the spectrum of the Laplacian of a graph does not depend on the enumeration of its nodes, which explains why we can establish measures on the spectra of two graphs to compute a “distance” between them.

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Authors and Affiliations

  1. EPITA Research and Development Laboratory (LRDE), Le Kremlin-Bicêtre, France

    Nicolas Boutry & Guillaume Tochon

Authors
  1. Nicolas Boutry
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  2. Guillaume Tochon
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Corresponding author

Correspondence to Nicolas Boutry .

Editor information

Editors and Affiliations

  1. Uppsala University, Uppsala, Sweden

    Joakim Lindblad

  2. Uppsala University, Uppsala, Sweden

    Filip Malmberg

  3. Uppsala University, Uppsala, Sweden

    Nataša Sladoje

A Appendix

A Appendix

1.1 A.1 Preservation of the Behavior of our Measures on Natural Images

Fig. 11.
figure 11

From left to right, the studied image and the computations of \(\mu _1\), \(\mu _2\) , \(\mu _3\), \(\mu _4\), \(\ell \), \(\mathcal {M} \) and \(\beta \) on three natural images.

Full size image

The main difference with synthetic images is that natural images show a stronger variance (see Fig. 11). Conversely, the behavior of our measures are preserved except for \(\mu _1\) which becomes relevant on natural images.

1.2 A.2 Preservation of the Behavior of our Measures on Natural Images

Fig. 12.
figure 12

Images and their depth, number of nodes, and maximal degrees as a function of the noise amplitude.

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Fig. 13.
figure 13

Ramifications appear in the tree of shapes as long as we add noise to the represented image.

Full size image

As we can observe in Figs. 12 and 13, elementary measures such as depth, numbers of nodes, and maximal degrees are not sufficient to measure the robustness of the ToS structure to noise.

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Boutry, N., Tochon, G. (2021). Stability of the Tree of Shapes to Additive Noise. 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_26

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  • DOI: https://doi.org/10.1007/978-3-030-76657-3_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-76656-6

  • Online ISBN: 978-3-030-76657-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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