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Condition for energy efficient watermarking without WSS assumption

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Abstract

Energy efficient watermarking preserves the watermark energy after linear attack as much as possible. We consider in this paper the non-stationary signal models and derive conditions for energy efficient watermarking under random vector model without wide sense stationary (WSS) assumption. We find that the covariance matrix of the energy efficient watermark should be proportional to the host covariance matrix to best resist the optimal linear removal attacks. For WSS process model, our result reduces to the well-known power spectrum condition. Intuitive geometric interpretations of the results in Hilbert space of random vectors are discussed, which also provides us simpler proof of the main results. Practical implementation of the covariance matrix shaped watermark for speech signal using linear prediction analysis (LPA) and image signal using eigen-value decomposition (EVD) are also presented and tested, showing improved performance as compared to lowpass and Su’s global watermark.

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Notes

  1. This is usually referred to as PCA (principal component analysis).

  2. KLT is based on EVD of the true covariance matrix while PCA is based on EVD of the sample covariance matrix [30].

  3. It is not difficult to show that after transforming the host using T  − 1, the covariance matrix will be I, an identity matrix. A diagonal covariance matrix indicates uncorrelation between components of a random vector, hence the name whitening transform.

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Acknowledgements

This work is supported by the project of National Natural Science Foundation of China (NSFC) under project grant number: 61272432 . The work of Bin Yan is also supported by Qingdao science and technology development plan (No. 12-1-4-6-(10)-jch). The work of Yin-Jing Guo is supported by the project of NSFC under project grant number: 61071087, and the natural science foundation of Shandong province(ZR2011FM018). The work of Xiao-Feng Liu is supported by the project of NSFC under project grant number: 60905060. The authors would like to thank the anonymous reviewers for their constructive comments and suggestions. We are indebted to the reviewers for their valuable time spent on the manuscript of this paper. The first author would like to thank Prof. Zhe-Ming Lu, Prof. Sheng-He Sun, Prof. Jeng-Shyang Pan and Prof. Xia-Mu Niu for their guidance and help. Their insight in watermarking research have significant influence on this work.

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Authors and Affiliations

  1. Department of Communication Engineering, School of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, 266510, Shandong Province, People’s Republic of China

    Bin Yan & Yin-Jing Guo

  2. College of Computer and Information Engineering, Changzhou Campus, Hohai University, 1 Xikang Road, Nanjing, 213022, People’s Republic of China

    Xiao-Feng Liu

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  1. Bin Yan
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  2. Yin-Jing Guo
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Correspondence to Bin Yan.

Appendix

Appendix

To prove the result in (17), we note that

$$ \mathbb{E} \left( \hat{\mathbf{w}}^T \hat{\mathbf{w}}\right) = \mathrm{tr} \left[ \mathbb{E} \left( \hat{\mathbf{w}} \hat{\mathbf{w}}^T\right) \right] $$
(61)
$$ = \mathrm{tr} \left\{\mathbb{E} \left[ \mathbf{H\left(x+w\right)}\left(\mathbf{x}^T +\mathbf{w}^T\right)\mathbf{H}^T \right] \right\} $$
(62)
$$ = \mathrm{tr} \left[ \mathbf{HC_x}\mathbf{H}^T + \mathbf{HC_w}\mathbf{H}^T\right],\label{eq_corr_EstWm_EstWm} $$
(63)

where we have used the fact that x and w are independent. Substituting into (63) the expression for Wiener filter \(\mathbf{H} = \mathbf{C_w \left(C_x+C_w \right)^{-1}}\), we have

$$ \mathbb{E} \left( \hat{\mathbf{w}}^T \hat{\mathbf{w}}\right) = \mathrm{tr} \left[ \mathbf{HC_x}\mathbf{H}^T + \mathbf{HC_w}\mathbf{H}^T\right] $$
(64)
$$ = \mathrm{tr} \left[\mathbf{C_w \left( C_x+C_w\right)^{-1} \left( C_x+C_w\right)\left( C_x+C_w\right)^{-1}C_w} \right] $$
(65)
$$ = \mathrm{tr} \left[\mathbf{C_w} \left( \mathbf{C_x+C_w}\right)^{-1}\mathbf{C_w}\right].\label{eq_corr_EstWm_EstWm_meet} $$
(66)

On the other hand we may obtain

$$ \mathbb{E} \left(\hat{\mathbf{w}}^T \mathbf{w}\right) = \mathrm{tr} \left[ \mathbb{E} \left(\mathbf{w} \hat{\mathbf{w}}^T \right)\right] $$
(67)
$$ = \mathrm{tr} \left\{ \mathbb{E} \left[\mathbf{w} \left(\mathbf{x}^T + \mathbf{w}^T \right)\mathbf{H}^T \right] \right\} $$
(68)
$$ = \mathrm{tr} \left( \mathbf{C_w} \mathbf{H}^T\right) $$
(69)
$$ = \mathrm{tr} \left[ \mathbf{C_w}\left(\mathbf{C_x}+\mathbf{C_w} \right)^{-1}\mathbf{C_w}\right], $$
(70)

which is equal to (66). So we have proved the result in (17).

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Yan, B., Guo, YJ. & Liu, XF. Condition for energy efficient watermarking without WSS assumption. Multimed Tools Appl 71, 1499–1528 (2014). https://doi.org/10.1007/s11042-012-1291-x

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