Generic polar complex exponential transform (GPCET), as continuous orthogonal moment, has the advantages of computational simplicity, numerical stability, and resistance to geometric transforms, which make it suitable for watermarking. However, errors in kernel function discretization can degrade these advantages. To maximize the GPCET utilization in robust watermarking, this paper proposes a secondary grid-division (SGD)-based moment calculation method that divides each grid corresponding to one pixel into nonoverlapping subgrids and increases the number of sampling points. Using the accurate moment calculation method, a nonsubsampled contourlet transform (NSCT)–GPCET-based watermarking scheme with resistance to image processing and geometrical attacks is proposed. In this scheme, the accurate moment calculation can reduce the numerical error and geometrical error of the traditional methods, which is verified by an image reconstruction comparison. Additionally, NSCT and accurate GPCET are utilized to achieve watermark stability. Subsequent experiments test the proposed watermarking scheme for its invisibility and robustness, and verify that the robustness of the proposed scheme outperforms that of other schemes when its level of invisibility is significantly higher.

中文翻译：

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具有精确矩计算的基于几何不变的 NSCT-GPCET 图像水印

通用极坐标复指数变换（GPCET）作为连续正交矩，具有计算简单、数值稳定、抗几何变换等优点，适用于水印。然而，核函数离散化中的错误会降低这些优势。为了最大化 GPCET 在鲁棒水印中的利用率，本文提出了一种基于二次网格划分 (SGD) 的矩计算方法，该方法将每个像素对应的网格划分为不重叠的子网格，并增加采样点的数量。利用精确矩计算方法，提出了一种基于非下采样轮廓波变换(NSCT)-GPCET的抗图像处理和几何攻击的水印方案。在这个方案中，准确的矩计算可以减少传统方法的数值误差和几何误差，并通过图像重建对比得到验证。此外，NSCT 和准确的 GPCET 用于实现水印稳定性。随后的实验测试了所提出的水印方案的不可见性和鲁棒性，并验证了所提出的方案的鲁棒性在其不可见性水平显着较高时优于其他方案。