## 6 Experimental Results
\begin{table}
\begin{tabular}{|c|c|c|} \hline  & 5-11 & 19.65 \\ \cline{2-3}  & 35-78 & 20.07 \\  & 67-437 & 20.54 \\ \cline{2-3}  & 3-16, 43-72 & 19.12 \\ \cline{2-3} Row Column Blanking & 139-126, 123-211 & 20.81 \\ \cline{2-3}  & 333-164, 431-142 & 19.75 \\ \cline{2-3}  & 14-26, 169-99, 119-192 & 17.64 \\ \cline{2-3}  & 139-126, 123-211,43-72 & 17.27 \\ \cline{2-3}  & 5-9, 11–67, 437-500 & 17.60 \\ \hline  & \(\gamma=0.6\) & 10.31 \\ \cline{2-3}  & \(\gamma=0.8\) & 13.76 \\ \cline{2-3} Gamma Correction & \(\gamma=1.1\) & 14.04 \\ \cline{2-3}  & \(\gamma=1.2\) & 12.05 \\ \cline{2-3}  & \(\gamma=1.5\) & 10.58 \\ \hline  & 10\% & 17.12 \\ \cline{2-3}  & 20\% & 15.26 \\ \cline{2-3} Cropping & 25\% & 14.41 \\ \cline{2-3}  & 30\% & 10.34 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._that the SVD transform is less robust against aggressive noise-based attacks, such as additive white Gaussian noise or salt and pepper noise. In the row-column copying attack random rows and/or columns are copied to random locations of the image. To better study the behavior of the proposed scheme, we divided this attack into the following sub-classes:  
* 1-column copying (e.g., we copied the 5\({}^{\text{th}}\) column to the 11\({}^{\text{th}}\) column _or_ the 35\({}^{\text{th}}\) column to the 78\({}^{\text{th}}\) column _or_ the 67\({}^{\text{th}}\) column to the 437\({}^{\text{th}}\) column);
* 2-columns copying (e.g., we copied the 3\({}^{\text{rd}}\) column to the 16\({}^{\text{th}}\) column _and_ the 43\({}^{\text{rd}}\) column to the 72\({}^{\text{nd}}\) column, etc. );
* 3-columns copying (e.g., we copied the 14\({}^{\text{th}}\) column to the 26\({}^{\text{th}}\) column _and_ the 169\({}^{\text{th}}\) column to the 99\({}^{\text{th}}\) column _and_ the 119\({}^{\text{th}}\) column to the 192\({}^{\text{nd}}\) column, etc.). The same strategy was also adopted for the row-column blanking attack which selects random columns and rows and deletes them from each of the watermarked image (i.e., are replaced with zeros). Therefore, the sub-classes which correspond to this attack are:  
* 1-row 1-column blanking (e.g., we deleted the 5\({}^{\text{th}}\) row and the 11\({}^{\text{th}}\) column, etc. );
* 2-rows 2-columns blanking (e.g., we deleted the 3\({}^{\text{rd}}\) and 43\({}^{\text{rd}}\) rows and the 16\({}^{\text{th}}\) and 72\({}^{\text{nd}}\) columns, etc. );
* 3-rows 3-columns blanking (e.g., we deleted the 14\({}^{\text{th}}\), 169\({}^{\text{th}}\) and 119\({}^{\text{th}}\) rows and the 26\({}^{\text{th}}\), 99\({}^{\text{th}}\) and 192\({}^{\text{nd}}\) columns, etc.). For these common attacks the presence of the watermark sequence can be easily detected from attacked images. The gamma correction attack changes the brightness of pixels of the watermarked images by a specified factor \(\gamma=\{0.8,\,1.1,\,1.2,\,1.5\}\). In Table 2, the average of the values of \(q\) shows that the proposed scheme is robust against intensity modification of pixels. The cropping attack consists of cutting a portion of the watermarked image from borders, preserving however its main features. We performed this attack by varying the size of the cropped portion of each watermarked image up to 30% and we could still detect the presence of the watermark sequence \(W\) with high confidence. A sample of the attack is shown in Figure 4. Notice that the robustness against cropping can be further increased by embedding redundantly the watermark into the cover image. Regarding well known attacks such as rotation and translation, our algorithm cannot be considered sufficiently robust. Indeed, the values of \(q\) obtained during our tests for these attacks are smaller than the chosen threshold, besides being different from zero. As one of our aims was to minimize the false positive probability, we preferred not to lower the threshold, also considering the fact that the two previously cited attacks sensibly degrade the quality of the watermarked image. At this point a consideration deserves attention.