Full 4-D quaternion discrete Fourier transform based watermarking for color images
Introduction
With the rapid development of communication and information technology, security has attracted much attention. For multimedia documents like images or videos, watermarking is a promising solution for broadcast monitoring, access control, copyright protection and so on. The key idea behind digital watermarking is to leave access to the document while maintaining it protected by means of an imperceptible watermark inserted or added to it. Watermark insertion is based on the principle of controlled perturbation of the host document. If the watermark should be unobtrusive, it can also be made robust to host document modifications [1].
Although many algorithms have been proposed to deal with grayscale images, only a few of them address color images. Nevertheless, by spreading the watermark over the three RGB (Red Green Blue) color channels rather than simply considering the image luminance, one may expect two advantages [2], [3]: the potential amount of embedded data can be greater; the watermark robustness can be increased while better preserving the fidelity/quality of the image.
Quaternions, which have been more and more used in color image processing in the past two decades, offer an interesting solution to achieve this goal. They represent an image by encoding its three color channels on the imaginary parts of quaternion numbers [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. The main advantage of such a representation is that a color image can be processed holistically as a vector field [5], [6], [7]. So does color image watermarking [8], [9], [10], [11], [12], [13], [14], [15], [16], which mainly makes use of the quaternion discrete Fourier transform (QDFT). By modulating at least one component of QDFT coefficients, the watermark is spread throughout two or three of the RGB channels [9].
QDFT watermarking was first introduced by Bas et al. in [8], where they compare the fidelity of color watermarked image considering two unit pure quaternions or in the case of message bit stream embedding by means of the quantization index modulation (QIM) [17]. It should be noticed that is the most common unit quaternion used in the quaternion based image processing literature. Since the preliminary work of Bas et al., Tsui et al. [10], [11] propose a non-blind algorithm (i.e. the original host image is required for watermark extraction) that inserts a sequence of bits into the QDFT domain derived from the components of the image. In fact, they modulate a color watermark pattern added to blocks of QDFT coefficients. On their side, Ma et al. [12] propose a semi-blind color watermarking scheme based on local QDFT. They first utilize an invariant feature transform so as to get some significant points or features of interest, and embed a binary watermark into QDFT coefficients of blocks centered on each of these points. Sun et al. [14] suggest embedding a binary image into the amplitude of QDFT AC coefficients of pixel blocks. Later, they combine QDFT and quaternion singular value decomposition into a semi-blind color image watermarking (original singular values of the host image are needed for watermark extraction) [15].
Recently, Jiang et al. [9] point out that, in [8], Bas et al. do not consider the precondition that all the real parts of the quaternion matrix which represents the image in the spatial domain are required to be equal to zero. Thus, applying the inverse QDFT (IQDFT) to QDFT coefficients that have been watermarked without care may lead to a quaternion matrix with non-null real parts. Then, only taking the three imaginary parts of this quaternion matrix so as to get the watermarked image induces a watermark energy loss. More clearly, even without attacking the image, the extracted watermark differs from the one embedded in the QDFT domain. Watermark detection is compromised. In the case of , Jiang et al. show that this issue can be solved by modifying QDFT coefficients symmetrically. Under this constraint, they insert a binary watermark by applying QIM to QDFT coefficients [9]. Notice that [10], [11], [12] also satisfy this constraint of symmetric distortion. The scheme of Jiang et al. has been extended into a non-blind watermarking scheme by Wang et al. [13] with the embedding of a grayscale image watermark. Wang et al. further improve this scheme making it robust to geometrical distortions with the help of least squares support vector machine and pseudo-Zernike moments [16].
Experimental results given in these papers show that quaternion-based algorithms can achieve a reasonable trade-off between fidelity and robustness. However, as we will see in Section 3, they do not fully take advantage of the four-dimensions (4-D) of the QDFT domain. Indeed, these previous works only consider from one to two components of QDFT coefficients but not all of them. Even though these algorithms can be used to cover independently all of the four components of QDFT coefficients, the above mentioned precondition as well as the image distortion may be more difficult to handle.
In this work, considering the case of different unit pure quaternions i, j, k or linear combinations of them, we demonstrate the conditions one must satisfy when modulating QDFT coefficients so as to: (1) avoid watermark energy loss at the embedding stage; and, (2) take full advantage of the 4-D QDFT domain in terms of watermark capacity and robustness.
The remaining of this paper is organized as follows. Section 2 comes back on the quaternion representation of color images, and on the definition of QDFT. In Section 3, considering the general definition of a unit pure quaternion (i.e. , , ), we explain how to avoid the watermark energy loss problem, i.e. the symmetry update constraints of QDFT coefficients, through establishing the relationships that exist between the different QDFT coefficient components and the conventional discrete Fourier transform (DFT) of the RGB color channels. We then analyze watermarking capacities one can expect using the different unit pure quaternions i, j or k compared with others (i.e. and ). After these preliminaries, Section 4 details the experimental framework we consider for evaluating the gain of performance in terms of watermark imperceptibility and robustness when the four dimensions of the QDFT domain are holistically used. This framework corresponds to a blind color watermarking system within which the modulations of Bas et al. [8] and of Jiang et al. [9] as well as of Wang et al. [16] have been included for fair comparison. It is important to notice that, in this work, we exploit an improved version of the solution proposed by Bas et al. to make it respect the above mentioned precondition. Indeed, this latter can be satisfied for if coefficients are modified accordingly. Experimental results are given and discussed in Section 5, where we further compare our approach with the method reported in [3], a more conventional, recent and efficient solution based on Schur's decomposition and which treats RGB channels independently. Section 6 concludes the paper.
Section snippets
Some preliminaries
In this section, we come back on the definition of quaternion before introducing the QDFT.
Watermarking of QDFT coefficients
In this section, considering the general pure unit quaternion μ defined above, we first demonstrate the relationships that exist between the different components of their QDFT coefficients with the conventional DFT of the RGB color channels so as to expose the constraints to satisfy in order to avoid watermark energy loss. We then analyze the watermarking capacities one must expect when modulating QDFT components under these constraints regarding the different unit pure quaternions i, j, k,
Experimental watermarking framework
In order to illustrate the gain of performance we achieve when considering the four dimensions of the QDFT domain holistically compared to other strategies, we propose to consider a general watermarking framework where our solution and the modulations of Jiang et al. [9], and Wang et al. [16] as well as the improved version of the one of Bas et al. [8] are implemented and tested for fair comparison. We describe thereafter the embedding and extraction procedure of this experimental framework
Experimental results and analysis
In order to evaluate the efficiency of the previous schemes, experiments have been conducted in terms of invisibility, capacity and robustness to different kinds of image modifications. We have also compared these five schemes with the one proposed by Su et al. [3]. This one treats RGB color channels independently, and modulates two coefficients of Schur decomposition of pixel block so as to insert one bit. Herein, for a fair comparison, Su's method has been extended so as to follow the
Conclusions
In this paper, by establishing the link between the components of QDFT coefficients with the DFT of color channels, we have shown how to avoid the watermark energy loss during its embedding, and specified the symmetry constraints one must respect when modulating QDFT coefficients in order to fully utilize the 4-D quaternion frequency domain. Considering a general embedding framework, we have demonstrated the overall gain of performance we can achieve for different unit pure quaternions.
Acknowledgements
This work was supported by the National Basic Research Program of China under Grant 2011CB707904, the NSFC under Grants 61271312, 61232016, 61103141, 61173141, 61105007 and 61272421, the Ministry of Education of China under Grant 20110092110023, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 13KJB520015, a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Science Research Foundation of
Beijing Chen received the B.S. degree in Mathematics and Applied Mathematics in 2003 from Jiangxi Normal University, the M.S. degree in Applied Mathematics in 2006 from Zhejiang University, and the Ph.D. degree in Computer Science in 2011 from Southeast University, China. Now he is a lecturer in the School of Computer & Software, Nanjing University of Information Science & Technology, China. His research interests include color image processing, information security and pattern recognition.
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Beijing Chen received the B.S. degree in Mathematics and Applied Mathematics in 2003 from Jiangxi Normal University, the M.S. degree in Applied Mathematics in 2006 from Zhejiang University, and the Ph.D. degree in Computer Science in 2011 from Southeast University, China. Now he is a lecturer in the School of Computer & Software, Nanjing University of Information Science & Technology, China. His research interests include color image processing, information security and pattern recognition.
Gouenou Coatrieux received the Ph.D. degree in Signal Processing and Telecommunication from the University of Rennes I, Rennes, France, in collaboration with the Ecole Nationale Supérieure des Télécommunications, Paris, France, in 2002. His Ph.D. focused on watermarking in medical imaging. He is currently an Associate Professor in the Information and Image Processing Department, Institut Mines-Telecom, Telecom Bretagne, Brest, France, and his research is conducted in the LaTIM Laboratory, INSERM U1101, Brest. His primary research interests concern medical information system security, watermarking, electronic patient records, and healthcare knowledge management.
Gang Chen received the B.S. degree from Anqing Teachers College in 1983 and the Ph.D. degree in the Department of Applied Mathematics at Zhejiang University, China, in 1994. Currently, he is an adjunct professor with the Department of Computer Science and Engineering, Southeast University, Nanjing, China. His research interests include applied mathematics, image processing, fractal geometry and computer graphics.
Xingming Sun received the B.S. degree in Mathematics from Hunan Normal University in 1984, the M.S. degree in Computer Science from Dalian University of Science and Technology in 1988, and the Ph.D. degree in Computer Science from Fudan University, China, in 2001. He is currently a Professor in the School of Computer & Software, Nanjing University of Information Science & Technology, China. His research interests include network and information security, digital watermarking, database security, and natural language processing.
Jean Louis Coatrieux received the Ph.D. and State Doctorate in Sciences in 1973 and 1983, respectively, from the University of Rennes 1, Rennes, France. Since 1986, he has been Director of Research at the National Institute for Health and Medical Research (INSERM), France, and since 1993 has been Professor at the New Jersey Institute of Technology, USA. He has been the Head of the Laboratoire Traitement du Signal et de l'Image, INSERM, up to 2003. His experience is related to 3D images, signal processing, pattern recognition, computational modeling and complex systems with applications in integrative biomedicine. He published more than 300 papers in journals and conferences and edited many books in these areas. He has served as the Editor-in-Chief of the IEEE Transactions on Biomedical Engineering (1996–2000) and is in the Boards of several journals. Dr. Coatrieux is a fellow member of IEEE.
Huazhong Shu received the B.S. degree in Applied Mathematics from Wuhan University, China, in 1987, and a Ph.D. degree in Numerical Analysis from the University of Rennes (France) in 1992. He is now a Professor of the Department of Computer Science and Engineering of Southeast University, China. His recent work concentrates on the image analysis, pattern recognition, and fast algorithms of digital signal processing.