Non-linear statistical image watermark detector

Abstract 

Invisibility, robustness and payload are three indispensable and contradictory properties for any image watermarking systems. Recently, to achieve the tradeoff among above three requirements, statistical watermarking schemes have gained a lot of attention. Most existing approaches, however, often bear a number of drawbacks, in particular: (i) They all employ directly transform coefficients, which are always fragile to some attacks, for watermark embedding and statistical modeling; (ii) The adopted model cannot capture accurately the statistical distributions of the transform coefficients; (iii) Most of them simply use the linear function for watermark embedding, which often either miss higher capacity or may cause visible distortions. This has motivated us to introduce in this paper a novel non-linear statistical image watermark detector based on undecimated discrete wavelet transform (UDWT)-polar complex exponential transform (PCET) magnitude and the exponentiated Cauchy-Rayleigh distribution. We begin with a detailed study on the robustness and statistical characteristics of local UDWT-PCET magnitudes of natural images. This study reveals the strong robustness and highly non-Gaussian marginal statistics of local UDWT-PCET magnitudes. We also find that, with a small number of parameters, the new exponentiated Cauchy-Rayleigh model can capture accurately the statistical properties of the robust UDWT-PCET magnitudes of the image. Meanwhile, the statistical model parameters can be computed effectively by using the genetic simulated annealing (GSA)-based maximum likelihood (ML) estimation. Based on these findings, we finally develop a new non-linear statistical image watermark detector using the exponentiated Cauchy-Rayleigh PDF and locally most powerful (LMP) decision rule. Also, we use the exponentiated Cauchy-Rayleigh statistical model to derive the closed-form expressions for the watermark detector. Extensive experimental results show the superiority of the proposed blind watermark detector over most of the state-of-the-art methods recently proposed in the literature.

Appendix A: Appendix

Under $H_{0}$ hypothesis, the mean $\mu_{0}$ and variance $\sigma_{0}^{2}$ of the LO statistic $l_{LO} ({\mathbf{y}})$ can be respectively derived as

$$\begin{gathered} \mu_{0} = E[l_{LO} ({\mathbf{y}})|H_{0} ] = E[l_{LO} ({\mathbf{x}})] \hfill \\ \;\;\;\;\; = E\left[ {\sum\limits_{i = 1}^{L} { - ab^{{x_{i} }} w_{i} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right)} } \right] \hfill \\ \;\;\;\;\; = \sum\limits_{i = 1}^{L} {\left( {E\left( { - ab^{{x_{i} }} w_{i} } \right)E\left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right)} \right)} \hfill \\ \begin{array}{*{20}c} {} & { = 0} \\ \end{array} \hfill \\ \end{gathered}$$

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and

$$\begin{gathered} \sigma_{{0}}^{{2}} = {\text{var}} [l_{LO} ({\varvec{y}})|H_{0} ] = {\text{var}} [l_{LO} |{\varvec{x}}] \hfill \\ \;\;\;\;\; = E\left[ {\left( {\sum\limits_{i = 1}^{L} { - ab^{{x_{i} }} w_{i} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right)} } \right)^{2} } \right] \hfill \\ \begin{array}{*{20}c} {} & { = \sum\limits_{i} {E\left[ {\left( { - ab^{{x_{i} }} w_{i} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right)} \right)^{2} } \right]} } \\ \end{array} \hfill \\ \; + \sum\limits_{l} {\sum\limits_{l \ne i} {E\left[ \begin{gathered} - ab^{{x_{i} }} w_{l} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{l}^{2} } \right)^{\frac{3}{2}} + \lambda x_{l}^{2} \left( {\beta { - }1} \right)}}{{x_{l} \left( {\lambda^{2} + x_{l}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{l}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{l} - \log b \cdot \left( {\lambda^{2} + x_{l}^{2} } \right)}}{{\lambda^{2} + x_{l}^{2} }}} \right) \cdot \hfill \\ - ab^{{x_{i} }} w_{i} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right) \hfill \\ \end{gathered} \right]} } \hfill \\ \;\;\;\;\; = \sum\limits_{i}^{{}} {\left( {ab^{{x_{i} }} } \right)^{2} \cdot \left( {\frac{{\left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} + \lambda x_{i}^{2} \left( {\beta { - }1} \right)}}{{x_{i} \left( {\lambda^{2} + x_{i}^{2} } \right)^{\frac{3}{2}} }}\left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + x_{i}^{2} } }}} \right]^{{{ - }1}} - \frac{{3x_{i} - \log b \cdot \left( {\lambda^{2} + x_{i}^{2} } \right)}}{{\lambda^{2} + x_{i}^{2} }}} \right)^{2} } \hfill \\ \end{gathered}$$

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Under $H_{1}$ hypothesis, the mean $\mu_{1}$ and variance $\sigma_{1}^{2}$ of the LO statistic $l_{LO} ({\mathbf{y}})$ can be respectively derived as

$$\begin{gathered} \mu_{1} = E[l_{LO} ({\mathbf{y}})|H_{1} ] = E[l_{LO} |{\mathbf{x}} + {\mathbf{w}}^{\prime}] \hfill \\ \;\;\;\; = E\left[ {\sum\limits_{i = 1}^{L} { - ab^{{x_{i} + w_{i} ^{\prime}}} w_{i} \cdot \left( \begin{gathered} \frac{{\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{i} + w_{i} ^{\prime}} \right)\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } }}} \right]^{{{ - }1}} \hfill \\ - \frac{{3\left( {x_{i} + w_{i} ^{\prime}} \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} }} \hfill \\ \end{gathered} \right)} } \right] \hfill \\ \;\;\;\; = \sum\limits_{i = 1}^{L} {\left( {c_{i} + d_{i} } \right)} \hfill \\ \end{gathered}$$

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where

$$\begin{gathered} c_{i} = \left( {\frac{{ - ab^{{x_{i} + ab^{{x_{i} }} }} }}{2}} \right) \cdot \hfill \\ \left( {\frac{{\left( {\lambda^{2} + \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{i} + ab^{{x_{i} }} } \right)\left( {\lambda^{2} + \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} } }}} \right]^{{{ - }1}} - \frac{{3\left( {x_{i} + ab^{{x_{i} }} } \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{i} + ab^{{x_{i} }} } \right)^{2} }}} \right)\;\;\;\;\;\;\;\;\;\; \hfill \\ d_{i} = \left( {\frac{{ab^{{x_{i} { - }ab^{{x_{i} }} }} }}{2}} \right) \cdot \hfill \\ \left( {\frac{{\left( {\lambda^{2} + \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{i} - ab^{{x_{i} }} } \right)\left( {\lambda^{2} + \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} } }}} \right]^{{{ - }1}} - \frac{{3\left( {x_{i} - ab^{{x_{i} }} } \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{i} - ab^{{x_{i} }} } \right)^{2} }}} \right) \hfill \\ \end{gathered}$$

and

$$\begin{gathered} \sigma_{{1}}^{{2}} = E[((l_{LO} ({\varvec{y}})|H_{1} ) - m_{1} )^{2} ] \hfill \\ \;\;\;\;\; = \sum\limits_{i}^{{}} {E\left[ {\left( { - ab^{{x_{i} + w_{i} ^{\prime}}} w_{i} \cdot \left( \begin{gathered} \frac{{\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{i} + w_{i} ^{\prime}} \right)\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } }}} \right]^{{{ - }1}} \hfill \\ - \frac{{3\left( {x_{i} + w_{i} ^{\prime}} \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} }} \hfill \\ \end{gathered} \right) - c_{i} - d_{i} } \right)^{2} } \right]} \hfill \\ \hfill \\ \begin{array}{*{20}c} {} & + \\ \end{array} \sum\limits_{l} {\sum\limits_{l \ne i} {E\left[ \begin{gathered} \left( { - ab^{{x_{l} + w_{l} ^{\prime}}} w_{l} \cdot \left( \begin{gathered} \frac{{\left( {\lambda^{2} + \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{l} + w_{l} ^{\prime}} \right)\left( {\lambda^{2} + \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} } }}} \right]^{{{ - }1}} \hfill \\ - \frac{{3\left( {x_{l} + w_{l} ^{\prime}} \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{l} + w_{l} ^{\prime}} \right)^{2} }} \hfill \\ \end{gathered} \right) - c_{l} - d_{l} } \right) \cdot \hfill \\ \left( { - ab^{{x_{i} + w_{i} ^{\prime}}} w_{i} \cdot \left( \begin{gathered} \frac{{\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} + \lambda \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} \left( {\beta { - }1} \right)}}{{\left( {x_{i} + w_{i} ^{\prime}} \right)\left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)^{\frac{3}{2}} }} \cdot \left[ {1 - \frac{\lambda }{{\sqrt {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } }}} \right]^{{{ - }1}} \hfill \\ - \frac{{3\left( {x_{i} + w_{i} ^{\prime}} \right) - \log b \cdot \left( {\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} } \right)}}{{\lambda^{2} + \left( {x_{i} + w_{i} ^{\prime}} \right)^{2} }} \hfill \\ \end{gathered} \right) - c_{i} - d_{i} } \right) \hfill \\ \end{gathered} \right]} } \hfill \\ \;\;\;\;\; = \sum\limits_{i}^{{}} {\left( {c_{i} - d_{i} } \right)^{2} } \hfill \\ \end{gathered}$$

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