NSST Domain Statistical Watermark Decoder Using Local Low-Order PZMs Magnitudes and Student’s-t Mixture Model

The primary evaluation criteria for image watermarking methods encompass imperceptibility, robustness, and capacity. However, there exists a mutual dependence and constraint relationship among these three factors, making it of paramount importance to investigate how to strike a balance between them. Statistical watermarking methods that leverage the statistical characteristics of the carrier and human visual features hold promise in achieving such equilibrium among imperceptibility, robustness, and capacity. Nevertheless, statistical watermarking methods face certain issues, including poor robustness in the embedding domain and a deficiency of decoding methods, etc. In this paper, we introduce a novel approach aimed at enhancing robustness in the embedding domain by utilizing the nonsubsampled shearlet transform (NSST) domain local pseudo-Zernike moments (PZMs) magnitudes. And we designed a two-component student’s-t mixture model (SMM) to represent the NSST-PZMs magnitudes, taking into account their “peak-tail” characteristics. Finally, by applying the SMM in conjunction with maximum-likelihood criteria, we derived a closed-form decoding expression. Our extensive experimental results indicate that when embedding a 1024-bit watermark sequence, our method exhibits outstanding performance, with a peak signal-to-noise ratio exceeding 58 dB and a bit error rate lower than 0.033 on the SIPI, CVG-UGR, and BOSSbase datasets. Compared to some advanced methods, our approach demonstrates significant advantages in terms of imperceptibility and robustness.