RDWT domain statistical watermark detector using FRHFMs magnitudes and bivariate Cauchy-Rayleigh distribution

Abstract

Imperceptibility, robustness and data payload, which are complimentary to each other, are widely considered as the three main properties vital for any image watermarking systems. It is a challenging work to design a statistical model-based multiplicative watermarking scheme for achieving the tradeoff among three main properties. In this paper, we propose a novel statistical image watermarking scheme by modeling local redundant discrete wavelet transform (RDWT) and fast Radial harmonic Fourier moments (FRHFMs) magnitudes with bivariate Cauchy-Rayleigh distribution. Our image watermarking scheme consists of two parts, namely, embedding and detection. In the embedding process, RDWT is firstly performed on the host image and RDWT highpass subbands are divided into non-overlapping blocks. Then FRHFMs are computed on RDWT coefficient blocks. And finally, the watermark signal is inserted into robust RDWT-FRHFMs magnitudes through a non-linear multiplicative approach. In the detection process, robust local RDWT-FRHFMs magnitudes are firstly modeled by employing bivariate Cauchy-Rayleigh distribution, which can capture accurately both marginal distributions and strong dependencies of local RDWT-FRHFMs magnitudes. Statistical model parameters are then estimated effectively by the method of logarithmic cumulants (MoLC) approach. And finally, an image watermark detector for multiplicative watermarking is developed using bivariate Cauchy-Rayleigh model and locally most powerful (LMP) test. Also, we utilize the bivariate Cauchy-Rayleigh model to derive the closed-form expressions for the watermark detector. After performance testing and comparison with the experimental results of existing methods, the proposed statistical image watermarking method has achieved relatively ideal results in terms of robustness, imperceptibility and data payload.

Appendix

1.1 Appendix A. Variances and means of log-likelihood ratios under hypotheses H ₀ and H ₁

The performance of the LO detector can be assessed theoretically. In this section, we calculate the variances and means of the log-likelihood ratios under the H₀ and H₁ hypotheses, i.e., μ₀, μ₁, σ₀, σ₁. An expression for the mean μ₀ of the bivariate Cauchy-Rayleigh distribution under the H₀ hypothesis is derived by the following formula:

$$ {\displaystyle \begin{array}{c}{\mu}_0=E\left({l}_{LOD}(Y)|{H}_0\right)\\ {}\kern0.68em =\sum \limits_{i,j=1}^N\left(\frac{a_1}{2}\left({d}_{11}+\frac{d_{21}}{y_i}\left(1-\frac{5{y}_i^2}{\gamma^2+{y}_i^2+{y}_j^2}\right)\right)+\frac{a_1}{2}\left({d}_{12}+\frac{d_{22}}{y_i}\left(1-\frac{5{y}_i^2}{\gamma^2+{y}_i^2+{y}_j^2}\right)\right)\right)=0\end{array}} $$

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where d₁₁ and d₁₂ are the representations of d₁ when the embedded watermark is 1 and − 1, respectively. Meanwhile, d₂₁ and d₂₂ are the representations of d₂ when the embedded watermark is 1 and − 1, respectively.

Similarly, we also give the mean μ₁ of the log-likelihood ratio based on hypothesis ${H}_1:{y}_i={x}_i+{a}_1{a}_2^{x_i}{w}_i$ by

$$ {\displaystyle \begin{array}{c}{\mu}_1=E\left({l}_{LOD}(Y)|{H}_1\right)\\ {}=E\left[\sum \limits_{i,j=1}^N\left(\left({d}_1+\frac{d_2}{x_i+{a}_1{a}_2^{x_i}{w}_i}\left(1-\frac{5{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2}{\gamma^2+{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2+{y}_j^2}\right)\right)\cdotp {a}_1\right)\right]=\sum \limits_{i,j=1}^N\left({\alpha}_i+{\beta}_i\right)\end{array}} $$

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where ${\alpha}_i=\frac{a_1}{2}\left({d}_{11}+\frac{d_{21}}{x_i+{a}_1{a}_2^{x_i}}\left(1-\frac{5{\left({x}_i+{a}_1{a}_2^{x_i}\right)}^2}{\gamma^2+{\left({x}_i+{a}_1{a}_2^{x_i}\right)}^2+{y}_j^2}\right)\right)$, and ${\beta}_i=\frac{a_1}{2}\left({d}_{12}+\frac{d_{22}{a}_1}{x_i-{a}_1{a}_2^{x_i}}\left(1-\frac{5{\left({x}_i-{a}_1{a}_2^{x_i}\right)}^2}{\gamma^2+{\left({x}_i-{a}_1{a}_2^{x_i}\right)}^2+{y}_j^2}\right)\right)$.

The variance under hypothesis H₀ is

$$ {\displaystyle \begin{array}{c}{\sigma}_0^2= Var\left({l}_{LOD}(Y)|{H}_0\right)\\ {}\kern0.68em =E\left[{\left(\sum \limits_{i,j=1}^N\left({d}_1+\frac{d_2}{y_i}\left(1-\frac{5{y}_i^2}{\gamma^2+{y}_i^2+{y}_j^2}\right)\right)\cdotp {a}_1\right)}^2\right]\\ {}\kern0.68em =\sum \limits_{i,j=1}^N{\left(\left({d}_{11}+\frac{d_{21}}{y_i}\left(1-\frac{5{y}_i^2}{\gamma^2+{y}_i^2+{y}_j^2}\right)\right)\cdotp {a}_1\right)}^2\end{array}} $$

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The variance under hypothesis H₁ is given by

$$ {\sigma}_1^2= Var\left({l}_{LOD}(Y)|{H}_1\right)=E\left[{\left(\left({l}_{LOD}(Y)|{H}_1\right)-{\mu}_1\right)}^2\right]=\sum \limits_{i,j=1}^NE\left[{\left({d}_1+\frac{d_2{a}_1}{x_i+{a}_1{a}_2^{x_i}{w}_i}\left(1-\frac{5{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2}{\gamma^2+{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2+{y}_j^2}\right)-{\alpha}_i-{\beta}_i\right)}^2\right]+\sum \limits_l^N\sum \limits_{l\ne i,j}E\left[\left(\left({d}_1+\frac{d_2}{x_l+{a}_1{a}_2^{x_l}{w}_l}\left(1-\frac{5{\left({x}_l+{a}_1{a}_2^{x_l}{w}_l\right)}^2}{\gamma^2+{\left({x}_l+{a}_1{a}_2^{x_l}{w}_l\right)}^2+{y}_l^2}\right)\right)\cdotp {a}_1-{\alpha}_l-{\beta}_l\right)\cdotp \left(\left({d}_1+\frac{d_2}{x_i+{a}_1{a}_2^{x_i}{w}_i}\left(1-\frac{5{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2}{\gamma^2+{\left({x}_i+{a}_1{a}_2^{x_i}{w}_i\right)}^2+{y}_j^2}\right)\right)\cdotp {a}_1-{\alpha}_i-{\beta}_i\right)\right]=\sum \limits_{i,j=1}^N{\left({\alpha}_i-{\beta}_i\right)}^2 $$

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