Locally optimum image watermark decoder by modeling NSCT domain difference coefficients with vector based Cauchy distribution

Abstract Improving the ability of imperceptibility, watermark capacity, and robustness at the same time still remains a challenge within the digital image watermarking community. By modeling the robust nonsubsampled Contourlet transform (NSCT) difference coefficients with vector based Cauchy distribution and employing locally most powerful (LMP) test, we propose a locally optimum image watermark decoder in NSCT domain. We first compute the difference coefficients according to the inter-scale dependency between NSCT coefficients, and investigate the robustness of the NSCT difference coefficients by subjective visual error and objective mean squared error (MSE) terms. We then embed the digital watermark into the significant NSCT difference subband with highest energy by modifying the robust NSCT difference coefficients. At the receiver, by combining the vector based Cauchy probability distribution and LMP test, we propose a locally optimum blind watermark decoder in the NSCT domain. Here, robust NSCT difference coefficients are firstly modeled by employing the vector based Cauchy probability density function (PDF), where the Cauchy marginal statistics and various strong dependencies of NSCT coefficients are incorporated. Then the statistical model parameters of vector based Cauchy PDF are estimated using second-kind statistics approach. And finally a blind image watermark decoder is developed using vector based Cauchy PDF and LMP decision rule. We conduct extensive experiments to evaluate the performance of the proposed blind watermark decoder, in which encouraging results validate the effectiveness of the proposed technique, in comparison with the state-of-the-art approaches recently proposed in the literature.