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Robust watermarking method based on the Analytical Clifford Fourier Mellin Transform

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Abstract

In this paper, we intend to introduce a semi-blind watermarking algorithm for colour images according to the RGB standard, which takes into account the interdependence of the three planes. In this context, it is often desirable that these algorithms verify certain independency with respect to the pose difference and a robustness with respect to the attacks caused by the compression. On the other hand, the inversion properties and the compressive power of the algorithms are also required criteria. The approach based on the Analytical Fourier Mellin Transform (AFMT) confirms all these recommendations. However, its scalar character can only result in a separate watermark within the meaning of RGB planes. In a recent work, we were able to show that the Fourier transform defined on the Clifford algebra is suitable for joint watermarking of colour images. However, its numerical approximations presents a problem of convergence in neighbour the origin. This results in digital instability in the frame of the image data. Inspired by the work carried out on the Mellin Analytical Fourier transform (AFMT), we propose here the construction of an analytical extension of the Clifford transform which we propose to call the Analytical Mellin Clifford Transform (ACMT). An approximation is given guaranteeing its convergent and its numerical stability in the context of watermarking of images. Furthermore, the properties of invariance with respect to rigid geometric transformations and its robustness with respect to certain compression algorithm are proven by simulations. Thus, its good performances in terms of imperceptibility and robustness against JPEG compression and geometric attacks are demonstrated on the BSD300 and USC-SIPI databases.

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All analyzed data during this study are available on request.

Notes

  1. the set of equivalence classes of unitary irreducible representations of G

  2. Quadratically integrable function in \(\mathbb {R}^{2}\) with values in \(\mathbb {R}^{1}_{4,0}\)

  3. Clifford algebra of \(\mathbb {R}^{4}\)

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Authors and Affiliations

  1. GRIFT Group, CRISTAL Laboratory, Ecole Nationale des Sciences de l’Informatique, Campus Universitaire, Manouba, 2010, Manouba, Tunisia

    Maroua Affes Jemal & Faouzi Ghorbel

  2. CEA, The French Alternative Energies and Atomic Energy Commission, Evry, 91000, France

    Mallek Mziou Sallami

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  1. Maroua Affes Jemal
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  3. Faouzi Ghorbel
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Maroua Affes Jemal and Mallek Mziou Sallami contributed equally to this work.

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Jemal, M.A., Sallami, M.M. & Ghorbel, F. Robust watermarking method based on the Analytical Clifford Fourier Mellin Transform. Multimed Tools Appl 83, 25901–25922 (2024). https://doi.org/10.1007/s11042-023-16482-8

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