Geometrically Invariant NSCT–GPCET-Based Image Watermarking with Accurate Moment Calculation

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.

A Polar Complex Exponential Transform-Based Zero-Watermarking for Multiple Medical Images with High Discrimination

Zero-watermarking is one of the solutions for image copyright protection without tampering with images, and thus it is suitable for medical images, which commonly do not allow any distortion. Moment-based zero-watermarking is robust against both image processing and geometric attacks, but the discrimination of watermarks is often ignored by researchers, resulting in the high possibility that host images and fake host images cannot be distinguished by verifier. To this end, this paper proposes a PCET- (polar complex exponential transform-) based zero-watermarking scheme based on the stability of the relationships between moment magnitudes of the same order and stability of the relationships between moment magnitudes of the same repetition, which can handle multiple medical images simultaneously. The scheme first calculates the PCET moment magnitudes for each image in an image group. Then, the magnitudes of the same order and the magnitudes of the same repetition are compared to obtain the content-related features. All the image features are added together to obtain the features for the image group. Finally, the scheme extracts a robust feature vector with the chaos system and takes the bitwise XOR of the robust feature and a scrambled watermark to generate a zero-watermark. The scheme produces robust features with both resistance to various attacks and low similarity among different images. In addition, the one-to-many mapping between magnitudes and robust feature bits reduces the number of moments involved, which not only reduces the computation time but also further improves the robustness. The experimental results show that the proposed scheme meets the performance requirements of zero-watermarking on the robustness, discrimination, and capacity, and it outperforms the state-of-the-art methods in terms of robustness, discrimination, and computational time under the same payloads.