In the past decades, watermarking has become one of the most active research fields and has received a great deal of attention not only in the academic community but also in the industry for its supply of an effective solution to the intellectual copyright protection for digital media.

Watermarking strategy is accomplished by embedding digital information (watermark) in original multimedia content (host), mostly without affecting the perceptual quality of original data greatly.

Additive watermarking in the transform domain is the simplest and commonly used way, since it can enhance the robustness against attacks or control the perceptive quality.

Many approaches for additive watermarks implemented in Discrete Cosine Transformation (DCT) domain, which can concentrate the whole energy of the host signal in fewer coefficients, have been proposed in literature so far [1J.R. Hernández, M. Amado, and F. Pérez-González, "DCT-domain watermarking techniques for still images: detector performance analysis and a new structure", IEEE Trans. Image Process., vol. 9, no. 1, pp. 55-68, 2000.
[http://dx.doi.org/10.1109/83.817598] [PMID: 18255372] , 2A. Nikolaidis, and I. Pitas, "Asymptotically optimal detection for additive watermarking in the DCT and DWT domains", IEEE Trans. Image Process., vol. 12, no. 5, pp. 563-571, 2003.
[http://dx.doi.org/10.1109/TIP.2003.810586] [PMID: 18237932] ]. Many directional image representations have been proposed in the recent years, which include the dual tree complex wavelets, bandlets, contourlets and NonSubsampled Contourlet Transform (NSCT) [3I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury, "The dual-tree complex wavelet transform", IEEE Trans. Signal Process., vol. 22, no. 6, pp. 123-151, 2005.
[http://dx.doi.org/10.1109/MSP.2005.1550194] -6A.L. da Cunha, J. Zhou, and M.N. Do, "The nonsubsampled contourlet transform: theory, design, and applications", IEEE Trans. Image Process., vol. 15, no. 10, pp. 3089-3101, 2006.
[http://dx.doi.org/10.1109/TIP.2006.877507] [PMID: 17022272] ].

At the receiving end, watermark could be detected or extracted from watermarked (maybe attacked) images to declare the legal ownership. Blind watermark detection is a crucial part of a watermarking system and various optimal transform-based methods have been proposed in the literatures [7P. Comesaña, N. Merhav, and M. Barni, "Asymptotically optimum universal watermark embedding and detection in the high-SNR regime", IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2804-2815, 2010.
[http://dx.doi.org/10.1109/TIT.2010.2046223] -10H. Sadreazami, M.O. Ahmad, and M.N. Swamy, "A study of multiplicative watermark detection in the contourlet domain using alpha-stable distributions", IEEE Trans. Image Process., vol. 23, no. 10, pp. 4348-4360, 2014.
[http://dx.doi.org/10.1109/TIP.2014.2339633] [PMID: 25051554] ].

Various models have been developed to account for non-Gaussian behavior of image statistics. For the additive watermarking problem, Generalized Gaussian Distribution (GGD) [11Q. Cheng, and T.S. Huang, "An additive approach to transform-domain information hiding and optimum detection structure", IEEE Trans. Multimed., vol. 3, no. 3, pp. 273-284, 2001.
[http://dx.doi.org/10.1109/6046.944472] , 12R. Kwitt, P. Meerwald, and A. Uhl, "Blind dt-cwt domain additive spread-spectrum watermark detection", In: Proceedings of 2009 16^th International Conference on Digital Signal Processing, IEEE: Santorini, 2009, pp. 1-8.
[http://dx.doi.org/10.1109/ICDSP.2009.5201255] ] has been widely adopted in all these years by many researchers. Recently, Cauchy distribution, [13X. Yin, S. Peng, and X. Zhu, "Detection for multiplicative watermarking in DCT domain by cauchy model", Lect. Notes Comput. Sci., vol. 7043, pp. 173-183, 2011.
[http://dx.doi.org/10.1007/978-3-642-25243-3_14] , 14R. Kwitt, P. Meerwald, and A. Uhl, "Lightweight detection of additive watermarking in the DWT-domain", IEEE Trans. Image Process., vol. 20, no. 2, pp. 474-484, 2011.
[http://dx.doi.org/10.1109/TIP.2010.2064327] [PMID: 20699210] ] has been an alternative solution applicable to the same problem.

In this work, we first modeled the statistical property of NSCT coefficients using different distributions. Furthermore, we derived the Rao hypothesis test detector, which can give the better performance without any prior knowledge of the embedding strength factor. Finally, the robustness of the proposed watermarking scheme againstattacks was verified. The rest of the paper is organized as follows. In Section II, we give the overview of the model of watermarking from the view of the communication system and the asymptotically optimum detection is derived. We modeled the statistical modeling of the NSCT coefficients using different probability distribution functions in Section III. In Section IV, we derived the Rao statistics for Cauchy distribution and proposed detection statistics for additive watermarking. Section V is devoted to the evaluation of the performance of the proposed approach in terms of invisibility and robustness for various settings. Finally, the main contributions of this work are summarized in Section VI where some suggestions for future research are also outlined.

Non subsampled Contourlet Transform (NSCT) is a flexible multi-scale, multi-directional, and shift-invariant version for images representation originated from Contourlet Transform.

We discuss the statistical models of NSCT coefficients, and in particular, we take a closer look into the characteristic distributions which arise in some cases of natural images.

Fig. (2) illustrates the histogram of finest subband coefficients of NSCT. As we can see, the coefficients are distributed with sharp peak at zero point and heavy tails at each side of the peak. Fig. (2) also shows the corresponding p.d.f.s of fitted GGDs and Cauchy p.d.f.s to the same NSCT subband coefficients.

As illustrated by the visual GoF in Fig. (2), we conclude the Cauchy distribution can be the better choice for modeling the NSCT subband coefficients compared to the GGD.

To evaluate a comparative performance of various detectors for additive image watermarking in NSCT domain, a great deal of experiments with different images have been carried out to investigate the imperceptibility of the embedded watermark as well as the robustness of the proposed method against attacks. We embedded the watermark with the same size as the host images, which come from the standard image datasets of 256*256 bits. As a result, the watermark is 65536-bitlong.

We implement the following detectors combination for additive spread spectrum watermarking: the naming convention is that the first part of the name denotes the type of hypothesis test and the second part denotes the host signal statistical model.

First of all, we test the visual quality of the watermarked images, as listed in Table 1.

Table 1

Visual quality for various images.

As we can see, the PSNRs are rather high. The images are indistinguishable with high PSNR values, thus showing the effectiveness of the additive NSCT watermarking in terms of the invisibility of the watermark.

Secondly, we analyze the performance of our detectors in the absence of attacks. (see Fig. 3). We determine the experimental receiver operating characteristics (ROC) curves by running 1000 randomly generated watermarks (simulating the ₁ hypothesis).

We examined the performance of ROC, where probability of miss varies with the probability of false alarm. It should be noted that the probability of miss is expected to be kept at a relatively low level for a predefined probability of false alarm to increase the reliability of detection.

It is seen that the Rao detector with Cauchy distribution yields a performance which is much better than others as evidenced by a lower probability of miss for any given value of false alarm.

The image received by the watermark detector might have been subjected to image processing operations. Consequently, we analyzed the robustness of the detectors against various attacks using the same set of images. Similarly, we determine the experimental ROC curves by running 1000 randomly generated watermarks.

The robustness of the proposed detector under JPEG compression is now investigated and the watermarked images are compressed by JPEG with compression quality factor of 70. Fig. (4) shows the averaged ROC curves obtained using various detectors.

The figure shows that the proposed Rao-Cauchy detectors are slightly less robust than LRT-Cauchy and LRT-GGD at the low level of probability of miss, but more robust against JPEG compression than the others at much higher level probability of miss, especially when it is higher than 10^-3.

The performance against the random noise attack is also tested. Fig. (5) shows the averaged ROC curves for various detectors.

Fig. (3) Performance evaluation of ROC.

Fig. (4) Performance evaluation of ROC under JPEG compression attack with the quality of 70.

Fig. (5) Performance evaluation of ROC under random noise attack with 10.

Fig. (5) shows that the proposed Rao-Cauchy detectors are much more robust against random noise than the others by providing lower probability of miss also.

From all of results mentioned above, we conclude that the Rao hypothesis test for NSCT subband coefficients following Cauchy statistical distribution model might be more robust and a better choice for detecting the additive spread spectrum image watermarking.

In this paper, we proposed a blind detection approach for additive spread spectrum image watermarking scheme in the NSCT domain.

We first studied the suitability of the statistical distributions in modeling the NSCT coefficients of an image. The results show that the Cauchy distribution provides a more accurate fitting to the NSCT subband coefficients through a visual comparison to GGD.

Motivated by these modeling results, blind additive spread spectrum watermark detector in the NSCT domain following Cauchy distributions was designed.

The performance of the proposed detector has been evaluated in detail by conducting several experiments. The robustness of the proposed detector, using the Rao hypothesis and Cauchy p.d.f.s, against JPEG compression, random noise has been investigated and results show that the proposed detector is superior to that of other approaches.