Full 4-D quaternion discrete Fourier transform based watermarking for color images

Among the few existing color watermarking schemes, some use quaternion discrete Fourier transform (QDFT). By modulating at least one component of QDFT coefficients, they spread the watermark over two or three of the RGB color channels. However, these schemes do not fully utilize the four-dimensional (4-D) QDFT frequency domain and some also suffer from a watermark energy loss directly at the embedding stage. In this paper, we first establish the links that exist between the OFT of the three RGB color channels and the components of QDFT coefficients while considering a general unit pure quaternion. Then, for different unit pure quaternions i, j, k or their linear combinations, we discuss the symmetry constraints one should follow when modifying QDFT coefficients in order to overcome the previous drawbacks. We also provide a general watermarking framework to illustrate the overall performance gain in terms of imperceptibility, capacity and robustness we can achieve compared to other QDFT based algorithms, i.e. when fully considering the 4-D QDFT domain. From this framework we derive three schemes, depending on whether i, j or k is used. Provided theoretical analysis and experimental results show that these algorithms offer better performance in terms of capacity and robustness to most common attacks, including JPEG compression, noise, cropping and filtering and so on, than other QDFT based algorithms for the same watermarked image quality. (C) 2014 Elsevier Inc. All rights reserved.