## 5 Results and Discussion
### Experimental Results  
The evaluation results of the proposed approach in comparison with four other watermarking methods in terms of the adopted criteria, which are presented in the previous section, are listed in Table 1. It can be noted in this table that the proposed approach (DWT+DCT+DTOSFS) was able to achieve the best values for the three cover images from Figure 5 when compared to the four other watermarking approaches. The other watermarking approaches included in the conducted experiment were the original DFT+SVD, the original DWT+DCT, the optimized DWT+DCT using the original SFS, and the optimized DWT+DCT optimized using the original DTO. The presented results show the effectiveness and superiority of the proposed approach. In the table, the PSNR value achieved by the proposed approach was 68.87, which was higher than the corresponding values achieved by the other approaches. Similarly, the values achieved using the proposed approach outperformed those of the competing approaches included in the experiment. On the other hand, a set of experiments were conducted to analyze the proposed approach's statistical significance. The statistical analysis was performed based on the results achieved by the proposed method and the four other competing approaches. Table 2 presents the results of the statistical analysis. In this table, the number of samples included in the statistical analysis was 10. The standard error of the mean was 0.07676, and the standard deviation was 0.2427 when using the proposed approach; these were better than \begin{table}
\begin{tabular}{r c c c c c} \hline
**Lena Image** & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** & **DWT+DCT+DTOSFS** \\ \hline PSNR & 36.18 & 44.67 & 49.66 & 53.78 & 63.87 \\ NCC & 0.0048 & 0.0037 & 0.0037 & 0.0025 & 0.002 \\ IF & \(-1.51233\)\(\times\)\(10^{-3}\) & \(-1.3554\)\(\times\)\(10^{-4}\) & \(-7.7164\)\(\times\)\(10^{-5}\) & \(-9.9887\)\(\times\)\(10^{-7}\) & \(-8.6357\)\(\times\)\(10^{-9}\) \\ \hline
**Cameraman Image** & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** & **DWT+DCT+DTOSFS** \\ \hline PSNR & 38.37 & 41.11 & 51.64 & 57.8 & 64.68 \\ NCC & 0.0048 & 0.0037 & 0.0037 & 0.0028 & 0.0023 \\ IF & \(-1.99187\)\(\times\)\(10^{-2}\) & \(-1.8836\)\(\times\)\(10^{-4}\) & \(-1.249\)\(\times\)\(10^{-5}\) & \(-2.5738\)\(\times\)\(10^{-7}\) & \(-9.12257\)\(\times\)\(10^{-10}\) \\ \hline
**Baboon Image** & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** & **DWT+DCT+DTOSFS** \\ \hline PSNR & 31.97 & 33.366 & 42.225 & 51.17 & 65.11 \\ NCC & 0.0041 & 0.0037 & 0.0031 & 0.0025 & 0.0019 \\ IF & \(-0.0071\) & \(-0.0059\) & \(-5.9966\)\(\times\)\(10^{-4}\) & \(-2.7456\)\(\times\)\(10^{-6}\) & \(-7.2137\)\(\times\)\(10^{-11}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Assessment of the results achieved by the proposed approach in comparison with those of the other competing approaches. the values achieved by the other approaches. The measured values in this table demonstrate the stability of the proposed approach. In addition, two tests were applied to study the significance of the proposed approach. These tests were the one-way analysis of variance (ANOVA) test and the Wilcoxon test. The results of these tests are presented in Tables 3 and 4, respectively. The difference between the proposed watermarking approach and each method of the competing watermarking methods was measured using the Wilcoxon test. The results of this test are shown in Table 4. In this table, the \(p\)-value is 0.002, which confirms the statistical difference in each watermarking method. As shown in Table 3, the measured \(p\)-value was less than 0.0001, which means that the proposed approach was statistically different from the other four watermarking methods. In addition, another experiment was conducted to focus on the study of the significance of the proposed approach using the linear regression test.