NSIWD: new statistical image watermark detector

Abstract

For any image watermarking algorithm, how to achieve the trade-off among robustness, imperceptibility, and watermark capacity is a challenging problem because of their mutually constrained relationship. In this paper, we design a statistical-based image watermarking detection system to solve the trade-off problem. In embedding process, consider the imperceptibility and the robustness, we inventively combine the undecimated discrete wavelet transforms (UDWT) difference domain and the polar harmonic Fourier moments (PHFMs) to obtain the UDWT difference domain PHFMs magnitudes as the watermark carriers, and we embed watermark signals in multiplicative manner. In modeling phase, by analyzing the statistical property and the strong inter-orientation correlations of the UDWT difference domain PHFMs magnitudes in horizontal direction and vertical direction, we model the magnitude coefficients with the bivariate generalized exponential distribution so that we can capture accurately the marginal characteristics and the strong inter-orientation dependencies at the same time. Moreover, we obtain the model parameters with the modified maximum likelihood estimation. Benefit from the reliable modeling result, we finally employ the locally most powerful decision rule to derive a novel specific locally optimum image watermark detector with a closed-form expression to blindly detect the existence of the watermarks. Extensive experiment results declare the designed statistical image watermarking system can accurately detect the existence of the watermarks, and it achieves the better balance among imperceptibility, robustness, and payload.

Appendix 1

Means and variances of log-likelihood ratios under hypotheses $H_{0}$ and $H_{1}$ .

In this paper, the image watermark detection is considered as a binary hypothesis problem as illustrated in Sect. 6.1. Here, we derive the means and variances of the log-likelihood ratios under hypotheses $H_{0}$ and $H_{1}$. Let N denote the total numbers of the coefficients employed to insert the watermark signals. Then the mean value $\mu_{0}$ and the variance $\sigma_{0}^{2}$ under $H_{0} :{\mathbf{y}} = {\mathbf{x}}$ are given as

$$\begin{gathered} \mu_{0} = E\left[ {\left. {l_{{{\text{LOD}}}} \left( Y \right)} \right|H_{0} } \right] \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = \sum\limits_{i,j = 1}^{N} {\left( {\frac{\lambda }{2}\left( {d_{11} + d_{21} \cdot h_{{{\text{LO}}}} \left( {x_{i} ,x_{j} } \right)} \right) + \frac{\lambda }{2}\left( {d_{12} + d_{22} \cdot h_{{{\text{LO}}}} \left( {x_{i} ,x_{j} } \right)} \right)} \right)} = 0 \hfill \\ \end{gathered}$$

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where $h_{{{\text{LO}}}} \left( {x_{i} ,x_{j} } \right)$ indicates the “locally optimal nonlinearity,” $d_{11}$ and $d_{12}$ are the representations of $d_{1}$ when the embedded watermark is 1 and − 1, respectively. Meanwhile, $d_{21}$ and $d_{22}$ are the representations of $d_{2}$ when the embedded watermark is 1 and − 1, respectively.

$$\begin{gathered} \sigma_{{0}}^{{2}} = {\text{Var}}\left( {\left. {l_{{{\text{LOD}}}} \left( Y \right)} \right|H_{0} } \right) \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = E\left[ {\left( {\sum\limits_{i,j = 1}^{N} {\left( {d_{1} + d_{2} \cdot h_{{{\text{LO}}}} \left( {x_{i} ,x_{j} } \right)} \right)} \cdot \lambda } \right)^{2} } \right] \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = \sum\limits_{i,j = 1}^{N} {\left( {\left( {d_{11} + d_{21} \cdot h_{{{\text{LO}}}} \left( {x_{i} ,x_{j} } \right)} \right) \cdot \lambda } \right)^{2} } \hfill \\ \end{gathered}$$

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Besides, the mean value $\mu_{1}$ and the variance $\sigma_{1}^{2}$ under $H_{1} :{\mathbf{y}} = {\mathbf{x}} \cdot (1 + \lambda {\mathbf{w}})$ can be written as

$$\begin{gathered} \mu_{1} = E\left[ {l_{{{\text{LOD}}}} (Y)\left| {H_{1} } \right.} \right] \hfill \\ \begin{array}{*{20}c} {} & = \\ \end{array} E\left[ {\sum\limits_{i,j = 1}^{N} {\left( {\left( {d_{1} + d_{2} \cdot h_{{{\text{LO}}}} \left( {x_{i} \cdot (1 + \lambda \omega_{i} ),y_{j} } \right)} \right) \cdot \lambda } \right)} } \right] = \sum\limits_{i,j = 1}^{N} {\left( {\alpha_{i} + \beta_{i} } \right)} \hfill \\ \end{gathered}$$

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where $\alpha_{i} = \frac{\lambda }{2}\left( {d_{11} + d_{21} \cdot \left( {\frac{\alpha }{{1 - \alpha \cdot e^{{ - (\lambda_{1} x_{i} \cdot (1 + \lambda ) + \lambda_{2} y_{j} )}} }} - \frac{{e^{{ - (\lambda_{1} x_{i} \cdot (1 + \lambda ) + \lambda_{2} y_{j} )}} \cdot (1 - \alpha ) + 1}}{{e^{{ - (\lambda_{1} x_{i} \cdot (1 + \lambda ) + \lambda_{2} y_{j} )}} \cdot (1 - e^{{ - (\lambda_{1} x_{i} \cdot (1 + \lambda ) + \lambda_{2} y_{j} )}} )}}} \right)} \right)$, $\beta_{i} = \frac{\lambda }{2}\left( {d_{12} + d_{22} \cdot \left( {\frac{\alpha }{{1 - \alpha \cdot e^{{ - (\lambda_{1} x_{i} \cdot (1 - \lambda ) + \lambda_{2} y_{j} )}} }} - \frac{{e^{{ - (\lambda_{1} x_{i} \cdot (1 - \lambda ) + \lambda_{2} y_{j} )}} \cdot (1 - \alpha ) + 1}}{{e^{{ - (\lambda_{1} x_{i} \cdot (1 - \lambda ) + \lambda_{2} y_{j} )}} \cdot (1 - e^{{ - (\lambda_{1} x_{i} \cdot (1 - \lambda ) + \lambda_{2} y_{j} )}} )}}} \right)} \right)$.

$$\begin{gathered} \sigma_{1}^{2} = Var\left( {l_{{{\text{LOD}}}} \left( Y \right)\left| {H_{1} } \right.} \right) \hfill \\ \quad \quad \quad = E\left[ {\left( {\left( {l_{{{\text{LOD}}}} (Y)\left| {H_{1} } \right.} \right) - \mu_{1} } \right)^{2} } \right] \hfill \\ \quad \quad \quad = \sum\limits_{i,j = 1}^{N} {E\left[ {\left( {\left( {d_{1} + d_{2} \cdot h_{{{\text{LO}}}} \left( {x_{i} \cdot (1 + \lambda \omega_{i} ),y_{j} } \right)} \right) \cdot \lambda - \alpha_{i} - \beta_{i} } \right)^{2} } \right]} \hfill \\ \quad \quad \quad \quad + \sum\limits_{l}^{N} {\sum\limits_{l \ne i,j} {E\left[ {\left( {\left( {d_{1} + d_{2} \cdot h_{{{\text{LO}}}} \left( {x_{l} \cdot (1 + \lambda \omega_{l} ),y_{l} } \right)} \right) \cdot \lambda - \alpha_{l} - \beta_{l} } \right) \cdot \left( {\left( {d_{1} + d_{2} \cdot h_{{{\text{LO}}}} \left( {x_{i} \cdot (1 + \lambda \omega_{i} ),y_{j} } \right)} \right) \cdot \lambda - \alpha_{i} - \beta_{i} } \right)} \right]} } \hfill \\ \quad \quad \quad = \sum\limits_{i,j = 1}^{N} {\left( {\alpha_{i} - \beta_{i} } \right)^{2} } \hfill \\ \end{gathered}$$

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