Comparison of secure spread-spectrum modulations applied to still image watermarking

Abstract

This article shows the results obtained when using secure spread-spectrum watermarking on gray-scale images in the watermark only attack (WOA) framework. Two secure modulations, natural watermarking (NW) and circular watermarking (CW), are compared with classical insecure modulations, spread spectrum (SS) and improved spread spectrum (ISS), from distortion, robustness, and security points of view. Implementations of CW and NW for still images are proposed: they use a wavelet transform and variable strength embedding with bounded distortion. Robustness of these schemes is assured by using JPEG compression and security is quantified by using a source separation technique: independent component analysis (ICA). Finally, tests are conducted on 2,000 natural images. They allow to distinguish between WOA security classes.

Appendix A: Distortion specifications

We want to link the target PSNR for embedding with the theoretical WCR, used in the formulae of the four modulations. We give proofs of Eqs. 23 and 27.

1.1 A.1 Constant strength embedding

The first point is that, thanks to the nice normalization of retroprojection (Eq. 24), distortion stays constant in the wavelet domain and in the projected space:

$$ \|\mathbf{w}_t\|^2=\|\mathbf{w}\|^2=d^2. $$

(31)

With renormalization against space dimensions, one gets:

$$ \sigma^2_{\mathbf{w}_t} = \frac{d^2}{N_t}, $$

(32)

$$ \sigma^2_{\mathbf{w}} = \frac{d^2}{N_v}. $$

(33)

So, we obtain:

$$ \sigma^2_{\mathbf{w}} = \frac{N_t}{N_v}\sigma^2_{\mathbf{w}_t}. $$

(34)

Mean square error in the spatial domain is:

$${\rm MSE} = \frac{N_t}{M\times N}\sigma^2_{\mathbf{w}_t}; $$

(35)

therefore, PSNR equals:

$${\rm PSNR}=10\log_{10}\left(\frac{255^2}{\frac{N_t}{M\times N}\sigma^2_{\mathbf{w}_t}}\right). $$

(36)

From the previous equation, one gets:

$$ \sigma^2_{\mathbf{w}} = 255^2\frac{M\times N}{N_v}10^{-\frac{{\rm PSNR}}{10}}, $$

(37)

which gives, once plugged into Eq. 3,

$${\rm WCR} = 10\log_{10}\left(\frac{255^2}{\sigma^2_{\mathbf{x}}}\times\frac{M\times N}{N_v}\right) - {\rm PSNR}. $$

(38)

1.2 A.2 Variable strength embedding

From [10], watermark signal varies with the absolute value of the current wavelet coefficient we want to watermark, assuming that x _t (i) is independent from w _t (i). We have:

$$ \|\mathbf{w'_t}\|^2 = \frac{1}{\mathbb{E}[|\mathbf{x_t}|]^2}\sum_{i=0}^{\mathit{N}_t-1}{|\mathbf{x_t}(i)|^2\mathbf{w_t}(i)^2} $$

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$$ {\kern4pc} \simeq \frac{1}{\mathbb{E}[|\mathbf{x_t}|]^2}\frac{1}{\mathit{N}_t}\sum_{i=0}^{\mathit{N}_t-1}{\mathbf{x_t}(i)^2}\sum_{i=0}^{\mathit{N}_t-1}{\mathbf{w_t}(i)^2} $$

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$$ {\kern4pc} = \frac{\mathbb{E}[\mathbf{x_t}^2]}{\mathbb{E}[|\mathbf{x_t}|]^2}\|\mathbf{w_t}\|^2. $$

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Equation 34 becomes:

$$ \sigma^2_{\mathbf{w}} = \frac{\mathbb{E}[\mathbf{x_t}^2]}{\mathbb{E}[|\mathbf{x_t}|]^2}\frac{N_t}{N_v}\sigma^2_{\mathbf{w'_t}}. $$

(42)

The same lines as above lead to the final equation for variable strength embedding:

$${\rm WCR} = 10\log_{10}\left(\frac{255^2}{\sigma^2_{\mathbf{x}}}\times\frac{M\times N}{N_v}\times \frac{\mathbb{E}[|\mathbf{x_t}|]^2}{\mathbb{E}[\mathbf{x_t}^2]}\right) - {\rm PSNR}. $$

(43)