BNPSIW: BRBS-based NSST-PZMs domain statistical image watermarking

Robustness, imperceptibility, and watermark capacity are three indispensable and contradictory properties for any image watermarking systems. It is a challenging work to achieve the balance among the three important properties. In this paper, by using bivariate Birnbaum–Saunders (BRBS) distribution model, we present a statistical image watermark scheme in nonsubsampled shearlet transform (NSST)-pseudo Zernike moments (PZMs) magnitude hybrid domain. The whole watermarking algorithm includes two parts: watermark embedding and extraction. NSST is firstly performed on host image to obtain the frequency subbands, and the NSST subbands are divided into non overlapping blocks. Then, the significant high-entropy NSST domain blocks are selected. Meanwhile, for each selected NSST coefficient block, PZMs are calculated to obtain the NSST-PZMs amplitude. Finally, watermark signals are inserted into the amplitude hybrid domain of NSST-PZMs. In order to decode accurately watermark signal, the statistical characteristics of NSST-PZMs magnitudes are analyzed in detail. Then, NSST-PZMs magnitudes are described statistically by BRBS distribution, which can simultaneously capture the marginal distribution and strong dependencies of NSST-PZMs magnitudes. Also, BRBS statistical model parameters are estimated accurately by modified closed-form maximum likelihood estimator (MML). Finally, a statistical watermark decoder based on BRBS distribution and maximum likelihood (ML) decision rule is developed in NSST-PZMS magnitude hybrid domain. Extensive experimental results show the superiority of the proposed image watermark decoder over some state-of-the-art statistical watermarking methods and deep learning approaches.