## 6 Experimental Results
\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \multirow{2}{*}{**Attack**} & \multirow{2}{*}{**Factor**} & \multicolumn{2}{c|}{**Detection Ratio**} \\ \cline{3-5}  & & & _Proposed_ & _[PERSON] et al._ \\ \hline \multirow{4}{*}{Sampling} & Down & 0.7 & 100\% & 0\% \\ \cline{2-5}  & & 0.8 & 100\% & 0\% \\ \cline{2-5}  & Up & 1.3 & 100\% & 100\% \\ \cline{2-5}  & & 1.75 & 100\% & 33\% \\ \hline \multirow{4}{*}{JPEG Compression} & \multicolumn{2}{c|}{QF = 90} & 100\% & 100\% \\ \cline{2-5}  & \multicolumn{2}{c|}{QF = 80} & 100\% & 86\% \\ \cline{2-5}  & \multicolumn{2}{c|}{QF = 70} & 100\% & 31\% \\ \cline{2-5}  & \multicolumn{2}{c|}{QF = 60} & 94\% & 11\% \\ \cline{2-5}  & \multicolumn{2}{c|}{1\%} & 100\% & 93\% \\ \cline{2-5}  & \multicolumn{2}{c|}{2\%} & 100\% & 21\% \\ \cline{2-5}  & \multicolumn{2}{c|}{3\%} & 100\% & 5\% \\ \hline \multirow{4}{*}{Salt \& Pepper Noise} & D = 0.001 & 100\% & 100\% \\ \cline{2-5}  & \multicolumn{2}{c|}{D = 0.003} & 100\% & 55\% \\ \cline{2-5}  & \multicolumn{2}{c|}{D = 0.005} & 100\% & 33\% \\ \hline \multirow{4}{*}{Row-Column Copying} & \multicolumn{2}{c|}{5-11} & 100\% & 97\% \\ \cline{2-5}  & \multicolumn{2}{c|}{35-78} & 100\% & 97\% \\ \cline{2-5}  & \multicolumn{2}{c|}{67-437} & 100\% & 96\% \\ \cline{2-5}  & \multicolumn{2}{c|}{3-16, 43-72} & 100\% & 91\% \\ \cline{2-5}  & \multicolumn{2}{c|}{139-126, 123-211} & 100\% & 91\% \\ \cline{2-5}  & \multicolumn{2}{c|}{333-164, 431-142} & 100\% & 92\% \\ \cline{2-5}  & \multicolumn{2}{c|}{14-26, 169-99, 119-192} & 100\% & 90\% \\ \cline{2-5}  & \multicolumn{2}{c|}{139-126, 123-211,43-72} & 100\% & 90\% \\ \cline{2-5}  & \multicolumn{2}{c|}{5-11, 35-78, 67-437} & 100\% & 92\% \\ \hline \end{tabular}
\end{table} 
Table 3: Performance of our scheme compared to the [PERSON] _et al_. [42] scheme. * **Security**. The security property of our watermarking scheme is granted by the capability to choose one or more couples of vectors among a set of many possible candidates. In addition, a further improvement to security is assured by selecting any pair of components of the chosen singular vectors. Instead, we can not say that the algorithm proposed by [PERSON] is secure because it always embeds the watermark by modifying the same components (\(2^{\text{nd}}\) and \(3^{\text{rd}}\)) of the first singular vector. * **Type of the watermark**. In our scheme, the watermark is a pseudo-random Gaussian sequence whereas in [PERSON]'s scheme it can be a significant watermark (i.e., binary or a gray-scale image). In general, it is better to use a significant mark in the watermarking process because of the possibility, in the detection step, to visually compare the original watermark with the inserted one [31]. Our method does not require the original image in the detection phase; however, we can improve the algorithm robustness by using the original image in the detection process. To test this approach, we modified the algorithm to perform a non-blind detection. Briefly, we used the original blocks to directly compute the angle difference in the singular vectors of the watermarked image, in order to extract the watermark and correlate it with the original one. In that case, we obtained a much higher detection rate, with an extremely small false positive probability. Therefore, even if the non-blind method cannot be used in some application contexts, such as copyright protection, we think that this variant of the algorithm could be used in other less demanding scenarios, e.g., image fingerprinting, due to its higher robustness.