## 6 Experimental Results  
In this section we show the robustness of the proposed scheme respect to signal processing operations and common attacks by applying the watermarking algorithm on different pictures taken from well-known data sets used for image processing [58; 59]. The experimental results were obtained using the reference gray-scale image \"Lena\" of size 512\(\times\)512\(\times\)8 bpp, which was partitioned into blocks of fixed size 16\(\times\)16 (512/\(n\times\)512/\(n\), with the optimal value for \(n=32\)). The size 16\(\times\)16 was driven experimentally by the following considerations:  
* to generate a sufficiently long sequence of features which allows to maintain the statistical properties required by the insertion/detection method;
* to have a high speed of computation of the SVD;
* to maintain a good visual quality of the watermarked image. The latter point is subjective, whilst the previous ones are controlled by the computing power and by the statistical properties. The watermark is a Gaussian pseudo-random sequence of size \(n^{2}=1024\). The values of the watermark represent a sequence of angles, in radians, which are used to rotate a specified pair of right singular vectors of the matrix \(V\) in every block. The parameters used through the embedding process are experimentally set to the values \(\beta=\pi\)/4 and \(\delta=4\), which represent a compromise between the strength of the inserted mark and the quality of the watermarked image. Increasing the value of \(\beta\) implies larger angle fluctuations, decreasing the image quality. To avoid the discontinuity in -\(\pi\) and \(\pi\) introduced by a large \(\beta\), a corresponding larger \(\delta\) is required (see step 5 in watermark embedding), and this fact implies large modifications to the original angle. By means of the secret key \(K\), the 6\({}^{\rm th}\) and 7\({}^{\rm th}\) right singular vectors are selected from each block, together with the indices \(k\) and \(l\) which, for simplicity, are set to the same values. Choosing other pairs of vectors (e.g., the 2\({}^{\rm nd}\) and 3\({}^{\rm rd}\)) can increase the robustness of the scheme against attacks but it negatively affects the visual quality of the watermarked image. The quality measure used to compute the amount of distortion introduced by the embedding process is the Peak Signal-to-Noise Ratio (PSNR). Using 100 images we computed the average PSNR for the proposed scheme obtaining a value higher than 37 dB. A sample of the embedding process is presented in Figure 2; as the reader can note, no visual artifacts can be observed in the watermarked image. The distribution of \(q\) under the hypothesis \(H_{0}\) and \(H_{l}\) was verified using the \"Lena\" image of size 512\(\times\)512. Under the hypothesis \(H_{0}\), we firstly generated 10,000 watermarks by setting 10,000 different seeds for the pseudo-random number generator; then, we checked the presence of these watermarks by correlating them with the original un-watermarked image. On the other hand, under the hypothesis \(H_{l}\), we embedded 10,000 watermarks, generated using 10,000 different seeds, into the \"Lena\" image, and we subsequently applied the detection process. The distribution of \(q\) for the statistic hypothesis test is presented in Figure 3.  
Figure 2: Original “Lena” image on the left; watermarked image on the right (PSNR = 37.6752 dB). We can observe from Figure 3 that the output statistic \(q\) in both hypothesis tests follows a Gaussian distribution, with mean \(m\) and variance approximately one, i.e., _N(m, var)_. Moreover, under the null hypothesis, the mean of the distribution of \(q\) is \(m_{{}_{H_{0}}}=0.03\) and \(var_{{}_{H_{0}}}=0.96\), whereas under the alternative hypothesis the distribution of \(q\) is characterized by \(m_{{}_{H_{1}}}=21.18\) and \(var_{{}_{H_{1}}}=0.98\). Note that the distributions of the output statistic \(q\) in both null and alternative hypothesis are well separated.