Statistical image watermark decoder by modeling local NSST-PHFMs magnitudes with Morgenstern-type bivariate-generalized exponential distribution

For any image watermarking system, there are three indispensable and mutually constrained requirements, namely robustness, invisibility, and payload. Recently, to achieve the trade-off among three requirements, statistical watermarking schemes have gained a lot of attention. Despite their powerfulness and effectiveness, most existing statistical image watermarking approaches 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 statistical model cannot capture accurately the marginal distributions of the transform coefficients. Moreover, the significant coefficients dependencies are ignored. To deal with these issues, this paper introduces a new statistical image watermarking method in non-subsampled shearlet transform (NSST)-polar harmonic Fourier moments (PHFMs) magnitude domain, wherein a PDF based on the bivariate-generalized exponential distribution (MTBGED) is employed, in view of the fact that this PDF provides a better statistical match to the empirical PDF of the robust NSST-PHFMs magnitudes of the image. In watermark embedding, we first perform the NSST on the carrier image. We then select the maximum energy subband and divide it into blocks and compute the PHFMs for each block. Finally, we embed watermark in NSST-PHFMs magnitudes using multiplicative approach. In the decoding process, we first analyze the robustness and statistical characteristics of local NSST-PHFMs magnitudes. We then observe that, with a small number of parameters, the new MTBGED model can capture accurately the statistical distributions of the robust NSST-PHFMs magnitudes of the image. Meanwhile, statistical model parameters can be estimated effectively by using the method of logarithmic cumulants (MoLC). Motivated by our modeling results, we finally develop a new statistical image watermark decoder using the MTBGED distribution and maximum likelihood (ML) decision rule. Experimental results on extensive test images demonstrate that the proposed blind watermark decoder provides a performance better than that of most of the state-of-the-art statistical methods and deep learning approaches recently proposed in the literature.