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The efficiency of PDE decomposition in images watermarking

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Abstract

The Discrete Cosine Transform (DCT) based watermarking scheme is a common and popular technique that has been used for long time for image watermarking. In this work, we aimed to further improve the commonly used DCT based watermarking method by combining the DCT with the PDE (Partial Differential Equation) method or with the FABEMD (Fast and Adaptive Bidimensional Empirical Mode Decomposition). Our experimental results and comparison analysis for the two proposed FABEMD-DCT and PDE-DCT methods demonstrated a better performance in terms of invisibility and the robustness of the watermark compared to some DCT based watermarking schemes. Furthermore, our method based on DCT and PDE model shows highest performance compared to other methods.

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Notes

  1. BV is the space of bounded variations functions. This space is widely used in image processing because it is a good candidate to modelize structures in images.

  2. the space of oscillating functions, which contains signals with large oscillations, and thus in particular textures

  3. E is a Banach space to model very oscillating patterns.

  4. The Legendre-Fenchel transform of F is given by \( {\mathrm{F}}^{*}\left(\mathrm{v}\right)={ \sup}_{\mathrm{u}}\left({\left\langle \mathrm{u},\mathrm{v}\right\rangle}_{{\mathrm{L}}^{2-\mathrm{F}\left(\mathrm{u}\right)\;}}\right) \) where \( {\left\langle .,.\right\rangle}_{{\mathrm{L}}^2} \) stands for L2 inner product

  5. \( \lambda {B}_G=\left\{fG/{\left\Vert f\right\Vert}_G\le \lambda \right\} \)

  6. \( \mu {B}_G=\left\{vG/{\left\Vert v\right\Vert}_G\le \mu \right\} \)

  7. \( \delta {B}_E=\left\{wG/{\left\Vert w\right\Vert}_E\le \delta \right\} \)

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Authors and Affiliations

  1. LIIAN, Department of Informatics, University Sidi Mohamed Ben Abdellah, B.P. 1796, Atlas-Fez, 30000, Fez, Morocco

    N. Aherrahrou & H. Tairi

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  1. N. Aherrahrou
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  2. H. Tairi
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Correspondence to N. Aherrahrou.

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Aherrahrou, N., Tairi, H. The efficiency of PDE decomposition in images watermarking. Multimed Tools Appl 75, 4593–4614 (2016). https://doi.org/10.1007/s11042-015-2494-8

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