Abstract
The stochastic representation of digital images through a two-dimensional autoregressive (2D-AR) model offers a proper way to approximate the empirical distribution of the eigenvalues coming from genuine images. By considering this model, we apply random matrix theory to analytically derive the asymptotic eigenvalue distribution of causal 2D-AR random fields that have undergone an upscaling operation with a particular interpolation kernel. This eigenvalue characterization is useful in developing new forensic techniques for image resampling detection since we can use theoretical bounds to drive the decision of detectors based on subspace decomposition. Moreover, experimental results with real images show that the obtained asymptotic limits turn out to be excellent approximations, even when working with images of small size.
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Notes
- 1.
The value of \(\phi \) is generally taken as \(\phi = \tfrac{1}{2}\left( \tfrac{M}{L}+1\right) \), as in imresize from MATLAB.
- 2.
There is an additional technical condition that applies in the cases considered in this paper, namely, that the set \(\left\{ \omega :d(\omega )=x\right\} \) has measure zero for all \(x \in {\mathbb R}\).
References
Bai, Z., Zhou, W.: Large sample covariance matrices without independence structures in columns. Statistica Sinica 18(2), 425–442 (2008)
Gloe, T., Böhme, R.: The Dresden image database for benchmarking digital image forensics. In: ACM SAC, pp. 1584–1590, March 2010
Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. University of California Press, Berkeley (1958)
Kirchner, M.: Fast and reliable resampling detection by spectral analysis of fixed linear predictor residue. In: ACM MM&Sec, pp. 11–20, September 2008
Kirchner, M.: Linear row and column predictors for the analysis of resized images. In: ACM MM&Sec, pp. 13–18, September 2010
Li, Y., Ding, Z.: Blind channel identification based on second order cyclostationary statistics. In: IEEE ICASSP. pp. 81–84, April 1993
Mahdian, B., Saic, S.: Blind authentication using periodic properties of interpolation. IEEE Trans. Inf. Forensics Secur. 3(3), 529–538 (2008)
Marčenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Math. USSR Sb. 1(4), 457–483 (1967)
Pfaffel, O., Schlemm, E.: Eigenvalue distribution of large sample covariance matrices of linear processes. Prob. Math. Stat. 31(2), 313–329 (2011)
Popescu, A.C., Farid, H.: Exposing digital forgeries by detecting traces of resampling. IEEE Trans. Signal Process. 53(2), 758–767 (2005)
Tao, T.: Topics in Random Matrix Theory. American Mathematical Society, Providence (2012)
Tulino, A.M., Verdú, S.: Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theor. 1(1), 1–182 (2004)
Vázquez-Padín, D., Comesaña, P., Pérez-González, F.: An SVD approach to forensic image resampling detection. In: EUSIPCO, pp. 2067–2071, September 2015
Vázquez-Padín, D., Pérez-González, F., Comesaña Alfaro, P.: Derivation of the asymptotic eigenvalue distribution for causal 2D-AR models under upscaling. arXiv:1704.05773 [cs.CR], April 2017
Vázquez-Padín, D., Pérez-González, F., Comesaña Alfaro, P.: A random matrix approach to the forensic analysis of upscaled images. IEEE Trans. Inf. Forensics Secur. 12(9), 2115–2130 (2017)
Acknowledgments
This work is funded by the Agencia Estatal de Investigación (Spain) and the European Regional Development Fund (ERDF) under project WINTER (TEC2016-76409-C2-2-R), and by the Xunta de Galicia and the ERDF under projects Agrupación Estratéxica Consolidada de Galicia accreditation 2016–2019 and Red Temática RedTEIC 2017–2018.
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Vázquez-Padín, D., Pérez-González, F., Comesaña-Alfaro, P. (2017). Random Matrix Theory for Modeling the Eigenvalue Distribution of Images Under Upscaling. In: Piva, A., Tinnirello, I., Morosi, S. (eds) Digital Communication. Towards a Smart and Secure Future Internet. TIWDC 2017. Communications in Computer and Information Science, vol 766. Springer, Cham. https://doi.org/10.1007/978-3-319-67639-5_10
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