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.
Statistical Assessment of Discrimination Capabilities of a Fractional Calculus Based Image Watermarking System for Gaussian Watermarks
