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A preventive and curative watermarking scheme for an industrial solution

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Abstract

Watermarking for identity images printed on a plastic card support is still challenging. In this application, the scheme must be robust against a combination of geometric and signal processing attacks related to the print/scan process. In addition, the scheme must deal with all the possible aggressions that a smart card can encounter during its lifetime. This paper investigates a robust watermarking solution in the Fourier domain in the continuity of our earlier developments reported in this field. It includes both preventive and curative stages that are complementary. The solution is preventive as only a small set of selected bits presenting a small variance are concerned the watermarking process. This property increases the chances of watermark detection rate while maintaining the same level of security. The curative part consists in pre-processing the watermarked image before the detection process. It comprises two specific and accurate counterattacks: the first one deals with blurring correction under a dedicated Wiener filter and the second one focuses on color corrections. The hybrid scheme is highly efficient and has a low computational cost. The new watermarking scheme clearly outperforms competitive approaches and is compatible with industrial constraints.

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Authors and Affiliations

  1. FPO-ERMAM, Ibn Zohr University, Ouarzazate, Morocco

    Rabia Riad

  2. PRISME, University of Orleans, Orleans, France

    Frédéric Ros & Rachid Harba

  3. IRF-SIC, Ibn Zohr University, Agadir, Morocco

    Khadija Gourrame, Mohamed El Hajji & Hassan Douzi

Authors
  1. Rabia Riad
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  2. Frédéric Ros
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  3. Khadija Gourrame
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  4. Mohamed El Hajji
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  5. Hassan Douzi
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  6. Rachid Harba
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Correspondence to Rabia Riad.

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Appendix: A

Appendix: A

1.1 A.1 Proof 1:

In the case when the image was watermarked using a watermark W, the correlation coefficient between the extracted vector X = X0 + α × W and W can be written as:

$$ C=\frac{\mathbb{E}\left[(X_{0}+\alpha W-\mu_{X_{0}}-\alpha\mu_{W})(W-\mu_{W})\right]}{\sqrt{\mathbb{E}\left[(X_{0}+\alpha W-\mu_{X_{0}}-\alpha\mu_{W})^{2}]\mathbb{E}[(W-\mu_{W})^{2}\right]}}. $$
(14)

where \(\mu _{X_{0}}\) and μW are the mean of X0 and W respectively, and \(\mathbb {E}\) is the expectation operator. After some simple arithmetic, one obtains:

$$ C=\frac{\mathbb{E}\left[(X_{0}+\mu_{X_{0}})(W-\mu_{W})\right]+\alpha\mathbb{E}\left[(W-\mu_{W})^{2}\right]}{\sqrt{\left( \mathbb{E}\left[(X_{0}-\mu_{X_{0}})^{2}\right]+2\alpha\mathbb{E}\left[(X_{0}-\mu_{X_{0}})(W-\mu_{W})\right]+\alpha^{2}\mathbb{E}\left[(W-\mu_{W})^{2}\right]\right)\mathbb{E}\left[(W-\mu_{W})^{2}\right]}}. $$
(15)

As X0 and W are statistically independent, this this gives:

\(\mathbb {E}\left [(X_{0}-\mu _{X_{0}})(W-\mu _{W})\right ]=0\), then:

$$ C=\frac{\alpha\mathbb{E}\left[(W-\mu_{W})^{2}\right]}{(\sqrt{\mathbb{E}\left[(X_{0}-\mu_{X_{0}})^{2}\right]+\alpha^{2}\mathbb{E}\left[(W-\mu_{W})^{2}\right])\mathbb{E}\left[(W-\mu_{W})^{2}\right]}}. $$
(16)

As \({\sigma _{Y}^{2}}=\mathbb {E}[(Y-\mu _{Y})^{2}]\), the correlation coefficient can be reformulated as:

$$ C=\frac{{\alpha\sigma_{W}^{2}}}{\sqrt{\left( \sigma_{X_{0}}^{2}+\alpha^{2}{\sigma_{W}^{2}}\right){\sigma_{W}^{2}}}}=\frac{1}{\sqrt{\frac{\sigma_{X_{0}}^{2}}{\alpha^{2}{\times\sigma_{W}^{2}}}+1}}. $$
(17)

1.2 A.2 Proof 2:

In the case when the image was watermarked using a watermark W∗ different from the one used by the decoder, the correlation coefficient between the extracted vector X = X0 + α × W∗ and W can be written as follows:

$$ C^{*}=\frac{\mathbb{E}\left[\left( X_{0}+\alpha W^{*}-\mu_{X_{0}}-\alpha\mu_{W^{*}}\right)\left( W-\mu_{W}\right)\right]}{\sqrt{\mathbb{E}\left[(X_{0}+\alpha W^{*}-\mu_{X_{0}}-\alpha\mu_{W^{*}})^{2}\right]\mathbb{E}\left[(W-\mu_{W})^{2}\right]}}. $$
(18)

After some simple arithmetic operations, we obtain (19):

$$ C^{*}=\frac{\mathbb{E}[(X_{0}+\mu_{X_{0}})(W-\mu_{W})]+\alpha\mathbb{E}[(W^{*}-\mu_{W^{*}})(W-\mu_{W})]}{\sqrt{\left( \mathbb{E}\left[(X_{0}-\mu_{X_{0}})^{2}\right]+2\alpha\mathbb{E}\left[(X_{0}-\mu_{X_{0}})(W^{*}-\mu_{W^{*}})\right]+\alpha^{2}\mathbb{E}\left[(W^{*}-\mu_{W^{*}})^{2}\right]\right)\mathbb{E}\left[(W-\mu_{W})^{2}\right]}}. $$
(19)

As X0, W, W∗ are statistically independent. This yields:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[(X_{0}+\mu_{X_{0}})(W-\mu_{W})]=0,\\ \mathbb{E}[(X_{0}+\mu_{X_{0}})(W^{*}-\mu_{W^{*}})]=0,\\ \mathbb{E}[(W^{*}-\mu_{W^{*}})(W-\mu_{W})]=0. \end{array} $$
(20)

Thus, the correlation coefficient is theoretically null, then C∗ = 0.

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Riad, R., Ros, F., Gourrame, K. et al. A preventive and curative watermarking scheme for an industrial solution. Multimed Tools Appl 82, 651–679 (2023). https://doi.org/10.1007/s11042-022-13268-2

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