A new statistical image watermark detector in RHFMs domain using beta-exponential distribution

Abstract

The detection of watermarks can be achieved by statistical approaches. How to select robust modeling object, appropriate statistical model, and decision rules is one of the major issues in statistical image watermark detection. In this paper, we propose a new image watermark detector in robust fast radial harmonic Fourier moments (FRHFMs) magnitudes domain, wherein the Beta-exponential distribution model and locally most powerful (LMP) decision rule are used. We first investigate the statistical modeling of the robust FRHFMs magnitudes by the Beta-exponential distribution. It is shown that the Beta-exponential distribution model fits the empirical data more accurately than the formerly employed statistical distributions, such as the Cauchy, Weibull, BKF, and Exponential, do. Motivated by the statistical modeling results, we design a blind image watermark detector in FRHFMs magnitudes domain by using Beta-exponential distribution and LMP test. Also, we utilize the Beta-exponential model to derive the closed-form expressions for the watermark detector. We provide comparative experimental results to alternative approaches to demonstrate the advantages of the proposed image watermark detector.

Appendix A: Variance and mean of log-likelihood ratio under hypotheses H 0 and H 1.

In this section, the likelihood ratio provided can be regarded to obey the Beta-exponential distribution conditioned on each of the H₀ and H₁ hypotheses. We can calculate the variance and mean under the two hypotheses, i.e., $\sigma_{0}$, $\sigma_{1}$, $\mu_{0}$, $\mu_{1}$. An expression for the mean $\mu_{0}$ under the H₀ hypothesis is derived by

$$ \begin{aligned} \mu_{0} = E(T_{LOD} ({\mathbf{y}}){|}H_{0} ) = E(T_{LOD} {\text{|x}}) \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = E\left[ {\sum\limits_{i = 1}^{L} {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - kx_{i} }} }}{{1 - e^{{ - kx_{i} }} }}} \right)} \cdot \lambda x_{i} w_{i} } \right] \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = \sum\limits_{i = 1}^{L} {\left( {\frac{{\lambda x_{i} }}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - kx_{i} }} }}{{1 - e^{{ - kx_{i} }} }}} \right)} + \left( { - \frac{{\lambda x_{i} }}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - kx_{i} }} }}{{1 - e^{{ - kx_{i} }} }}} \right) = 0 \hfill \\ \end{aligned} $$

(A.1)

Similarly, we also give the mean $\mu_{1}$ of the log-likelihood ratio based on hypothesis $H_{1} :{\text{y}} = {\text{x}}(1 + \lambda {\text{w}})$ by

$$ \begin{aligned} \mu_{1} = E(T_{LOD} ({\mathbf{y}})|H_{1} ) = E(T_{LOD} |{\mathbf{x}} + \lambda {\mathbf{xw}}) \hfill \\ \;\;\;\; = E\left[ {\sum\limits_{i = 1}^{L} {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k\left( {x_{i} + \lambda x_{i} w_{i} } \right)}} }}{{1 - e^{{ - k\left( {x_{i} + \lambda x_{i} w_{i} } \right)}} }}} \right) \cdot \lambda \left( {x_{i} + \lambda x_{i} w_{i} } \right)w_{i} } } \right] \hfill \\ \;\;\;\; = \sum\limits_{i = 1}^{L} {\left( {\left( {\frac{{\lambda \left( {x_{i} + \lambda x_{i} } \right)}}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k\left( {x_{i} + \lambda x_{i} } \right)}} }}{{1 - e^{{ - k\left( {x_{i} + \lambda x_{i} } \right)}} }}} \right) + \left( { - \frac{{\lambda \left( {x_{i} - \lambda x_{i} } \right)}}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k\left( {x_{i} - \lambda x_{i} } \right)}} }}{{1 - e^{{ - k\left( {x_{i} - \lambda x_{i} } \right)}} }}} \right)} \right)} \hfill \\ \;\;\;\; = \sum\limits_{i = 1}^{L} {(\omega_{i} + \upsilon_{i} )} \hfill \\ \end{aligned} $$

(A.2)

where $\omega_{i} = \left( {\frac{{\lambda \left( {x_{i} + \lambda x_{i} } \right)}}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k\left( {x_{i} + \lambda x_{i} } \right)}} }}{{1 - e^{{ - k\left( {x_{i} + \lambda x_{i} } \right)}} }}} \right),$ $\upsilon_{i} = \left( { - \frac{{\lambda \left( {x_{i} - \lambda x_{i} } \right)}}{2}} \right)\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k\left( {x_{i} - \lambda x_{i} } \right)}} }}{{1 - e^{{ - k\left( {x_{i} - \lambda x_{i} } \right)}} }}} \right)$. The variance under hypothesis H₀ is expressed by

$$ \begin{gathered} \sigma_{{0}}^{{2}} = Var(T_{LOD} ({\mathbf{y}})|H_{0} ) = Var(T_{LOD} ({\mathbf{y}})|{\text{x}}) \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = E\left[ {\left( {\sum\limits_{i = 1}^{L} {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - kx_{i} }} }}{{1 - e^{{ - kx_{i} }} }}} \right) \cdot \lambda x_{i} w_{i} } } \right)^{2} } \right] \hfill \\ \;\;\;\;{\kern 1pt} {\kern 1pt} = \sum\limits_{i = 1}^{L} {\left( {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - kx_{i} }} }}{{1 - e^{{ - kx_{i} }} }}} \right) \cdot \lambda x_{i} } \right)^{2} } \hfill \\ \end{gathered} $$

(A.3)

The variance under hypothesis H₁ is given by

$$ \begin{gathered} \sigma_{{1}}^{{2}} = Var(T_{LOD} ({\text{y}})|H_{1} ) = E[\left( {(T_{LOD} ({\text{y}})|H_{1} ) - \mu_{1} } \right)^{2} ] \hfill \\ \;\;\;\;\; = \sum\limits_{i = 1}^{L} {E\left[ {\left( {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k(x_{i} + \lambda x_{i} w_{i} )}} }}{{1 - e^{{ - k\left( {x_{i} + \lambda x_{i} w_{i} } \right)}} }}} \right) \cdot \lambda \left( {x_{i} + \lambda x_{i} w_{i} } \right)w_{i} - \omega_{i} - \upsilon_{i} } \right)^{2} } \right]} \hfill \\ \;\;\;\;\; + \sum\limits_{l}^{L} {\sum\limits_{l \ne i} {E\left[ {\left( {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k(x_{l} + \lambda x_{l} w_{l} )}} }}{{1 - e^{{ - k(x_{l} + \lambda x_{l} w_{l} )}} }}} \right) \cdot \lambda (x_{l} + \lambda x_{l} w_{l} )w_{l} - \omega_{l} - \upsilon_{l} } \right) \cdot } \right.} } \hfill \\ \;\;\;\;\;\left. {\left( {\left( {\beta k - \tfrac{{(\alpha - 1) \cdot ke^{{ - k(x_{i} + \lambda x_{i} w_{i} )}} }}{{1 - e^{{ - k(x_{i} + \lambda x_{i} w_{i} )}} }}} \right) \cdot \lambda (x_{i} + \lambda x_{i} w_{i} )w_{i} - \omega_{i} - \upsilon_{i} } \right)} \right] \hfill \\ \;\;\;\;\; = \sum\limits_{i = 1}^{L} {\left( {\omega_{i} - \upsilon_{i} } \right)^{2} } \hfill \\ \end{gathered} $$

(A.4)