Most present-day media consumption and transmission occurs digitally, which introduces a major challenge regarding ownership. In short, how can content creators prove their ownership of a piece of digital media, such as a song, photograph, or book? Luckily, the nature of digital information usually requires storage of such media as a data matrix that is composed of numerical values, thereby allowing for the employment of mathematical methods in digital ownership protection. To this end, researchers have developed watermarking methods that embed identifying ownership information into a piece of media without causing any real perceptible change.
One must employ certain criteria when implementing and evaluating a watermarking scheme. First, the scheme should be imperceptible; embedding a watermark should not leave any significant observable impact on the medium itself. The scheme should also be secure; a third party should not be able to manipulate the scheme to establish themselves as the false owner of the medium. Finally, the item in question should still maintain some robustness against distortions; the owner could ideally still extract the watermark and reasonably prove ownership after the watermarked media undergoes some form of compression, distortion, or intentional attack.
Figure 1. Application of the Jain et al. watermark [6] to an image of a raccoon with varying values of the weighting parameter \(\alpha\). Courtesy of [4].
Several existing watermarking schemes utilize the many useful properties of the singular value decomposition (SVD), a matrix factorization with far-reaching applications in data analysis and statistics [3, 7, 12]. We employ the aforementioned criteria to investigate the performance of two such SVD-based watermarking schemes that were developed by Ruizhen Liu and Tieniu Tan [11] as well as Chirag Jain, Siddharth Arora, and Prasanta K. Panigrahi [6]. We also propose our own modification of the Jain et al. scheme to preserve the desirable security properties of the original version while simultaneously enhancing the imperceptibility.
The Jain et al. scheme is a modification of Liu and Tan’s scheme that significantly improves its security [6]. However, the embedding of watermarks is highly perceptible even with low weighting parameters (see Figure 1). We tested our modified scheme with the same method — we embedded a watermark (the image of a fox) into another image (the raccoon) and attempted to extract a phony watermark (the image of a husky) from the watermarked image. As evident in Figure 2, the extracted phony watermark is unidentifiable in comparison to the extracted true watermark. Our modified scheme therefore passes the security test. Furthermore, our watermark is significantly less perceptible than Jain et al.’s scheme. We have thus successfully developed a watermarking scheme that improves perceptibility while maintaining other advantages.
Figure 2. Security test on modified scheme. 2a. Watermarked image. 2b. Watermark. 2c. Extracted phony watermark. Figure 2a courtesy of [4], 2b courtesy of [2], and 2c courtesy of [13].
A cornerstone property of the SVD is that one can apply it to any matrix. The SVD-based watermarking schemes that we study—as well as our own proposed modification—are thus applicable to a broad range of media. We have seized on this potential by conducting numerical experiments with both audio and image watermarking.
For example, we convert audio samples to matrices with the Short-time Fourier transform [5, 10]. Here we embedded a piece of news audio as a watermark into a clip of the Bach Cello Suite No. 1 [1] to create a piece of watermarked audio.
We then performed two distortions by lowering the watermarked audio by an octave and adding reverberations to the audio.
In both cases, the extracted watermark audios still have identifiable melodies when compared to the original piece of news audio. This result suggests the robustness of our modified watermarking scheme.
Watermarking schemes have potential applications in the entertainment industry for trademarking and finding proof of plagiarism in a variety of media, from images to audio. Our work—particularly our proposed modification—also illustrate the way in which one can harness and manipulate the SVD’s powerful mathematical characteristics to both develop new applications and improve existing results.
This article serves as a brief introduction to our paper in SIAM Undergraduate Research Online [9], which originated from our project with David Melendez on Randomized Singular Value Decomposition and its Applications [8] at Summer@ICERM 2020: Fast Learning Algorithms for Numerical Computation and Data Analysis. The code that we used to generate the results is available online.
Jennifer Zheng and Katie Keegan presented this research during an undergraduate minisymposium at the 2021 SIAM Annual Meeting, which took place virtually in July.
Acknowledgments: We would like to thank our advisor, Minah Oh, for her guidance during the Summer@ICERM 2020 program. We also thank ICERM for sponsoring our project.
References
[1] Bach, J.S. (2005). Cello Suite No. 1, BWV 1007 [Suite recorded by Maria Kliegel]. On J.S. Bach Cello Suites (Complete). Naxos Digital Services. (Original work published 1717-1723.)
[2] Beaufort, J. Red Fox [Photograph]. PublicDomainPictures.net. CC0 Public Domain.
[3] Brunton, S.L., & Kutz, J.N. (2019). Data-driven science and engineering: Machine Learning, dynamical systems, and control. Cambridge, U.K.: Cambridge University Press.
[4] Buchanan, B., & U.S. Fish and Wildlife Service – Northeast Region. (2010). Young Raccoon in Crab Apple Tree [Photograph]. Public Domain Files.
[5] Jacobsen, E., & Lyons, R. (2003). The sliding DFT. IEEE Signal Process. Mag., 20(2), 74-80.
[6] Jain, C., Arora, S., & Panigrahi, P.K. (2008). A reliable SVD based watermarking schem. Preprint, arXiv:0808.0309.
[7] Kalman, D. (1996). A singularly valuable decomposition: The SVD of a matrix. Coll. Math. J., 27(1), 2-23.
[8] Keegan, K., Melendez, D., & Zheng, J. (2020). Randomized singular value decomposition and its applications. Brown University Summer@ICERM Project. Institute for Computational and Experimental Research in Mathematics.
[9] Keegan, K., Melendez, D., & Zheng, J. (2021). Media processing and a modified watermarking scheme based on the singular value decomposition. SIAM J. Undergrad. Res., 14.
[10] Kehtarnavaz, N. (2008). Frequency domain processing (pp. 175-196). Digital signal processing system design: LabVIEW-based hybrid programming (2nd Ed.). Burlington, MA: Academic Press.
[11] Liu, R., & Tan, T. (2002). An SVD-based watermarking scheme for protecting rightful ownership. IEEE Trans. Multimedia, 4(1), 121-128.
[12] Martin, C.D., & Porter, M.A. (2012). The extraordinary SVD. Am. Math. Mon., 119(10), 838-851.
[13] Ӧckel, A.S. Siberian Husky Dog Pet [Photograph]. Licensed under CC0 1.0. Retrieved from https://www.publicdomainpictures.net/en/view-image.php?image=223151&picture=red-fox
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Jennifer Zheng is an undergraduate senior at Emory University, where she is studying applied mathematics and statistics and double majoring in music. She is interested in numerical analysis and image processing. Zheng is working on an honors thesis under the direction of James Nagy and hopes to pursue computational mathematics in her graduate studies. |
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Katie Keegan is an undergraduate senior in applied mathematics at Mary Baldwin University. She is interested in the mathematics of data science, imaging, and scientific computing, and hopes to study computational mathematics in graduate school. Keegan began college at age 14 and will graduate in May 2022 at age 18. |