## 5 Results and Discussion
### Experimental Results
The results of this experiment are presented in Table 5. In this test, if the \(p\)-value was smaller than 0.05, then the null hypothesis was rejected and it was concluded that the relationship between the proposed approach and the approach presented in the test was significant. The results presented in the table show that the \(p\)-values for all methods included in the experiment were less \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** & **DWT+DCT+DTOSFS** \\ \hline Theoretical median & 0 & 0 & 0 & 0 & 0 \\ Actual median & 36.18 & 44.67 & 49.66 & 53.78 & 63.87 \\ Number of values & 10 & 10 & 10 & 10 & 10 \\ Wilcoxon Signed Rank Test & & & & & \\ Sum of signed ranks (W) & 55 & 55 & 55 & 55 \\ Sum of positive ranks & 55 & 55 & 55 & 55 & 55 \\ Sum of negative ranks & 0 & 0 & 0 & 0 & 0 \\ \(P\) value (two-tailed) & 0.002 & 0.002 & 0.002 & 0.002 & 0.002 \\ Exact or estimate? & Exact & Exact & Exact & Exact & Exact \\ \(P\)-value summary & - & - & - & - & - \\ Significant (alpha = 0.05)? & Yes & Yes & Yes & Yes \\ How big is the discrepancy? & & & & & \\ Discrepancy & 36.18 & 44.67 & 49.66 & 53.78 & 63.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation of the Wilcoxon signed rank test based on the recorded results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** & **DWT+DCT+DTOSFS** \\ \hline Number of values & 10 & 10 & 10 & 10 & 10 \\ Std. Error of Mean & 0.2175 & 0.6155 & 0.2134 & 0.2333 & 0.07676 \\ Std. Deviation & 0.6877 & 1.947 & 0.6749 & 0.7379 & 0.2427 \\
25\% Percentile & 36.18 & 44.42 & 49.41 & 53.78 & 66.88 \\
75\% Percentile & 36.43 & 45.42 & 49.66 & 53.78 & 66.88 \\ Minimum & 35.98 & 41.67 & 47.66 & 51.78 & 66.13 \\ Maximum & 38.18 & 48.67 & 49.66 & 54.78 & 66.98 \\ Mean & 36.46 & 44.97 & 49.36 & 53.68 & 66.82 \\ Median & 36.18 & 44.67 & 49.66 & 53.78 & 66.88 \\ Range & 2.2 & 7 & 2 & 3 & 0.85 \\ Sum & 364.6 & 449.7 & 493.6 & 536.8 & 668.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical analysis of the proposed approach compared to other approaches. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **ANOVA** & **SS** & **DF** & **MS** & **F (DFn, DFd)** & \(p\) **Value** \\ \hline Treatment (between columns) & 4144 & 4 & 1036 & F (4, 45) = 984.5 & \(p\) < 0.0001 \\ Residual (within columns) & 47.36 & 45 & 1.052 & & \\ Total & 4192 & 49 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation of the ANOVA test based on the recorded results. than 0.05, and thus, there was a significant relationship between these methods and the proposed method. Moreover, an additional experiment was conducted to study the stability of the proposed watermarking approach. This experiment was based on the regression test. The results of this test are presented in Table 5. In this test, the results achieved by the proposed watermarking approach were analyzed with respect to the results achieved by the four other competing methods. The results shown in this table emphasize the stability of the proposed approach when compared to the other methods. The analysis of the area under the curve (AUC) was another experiment conducted to study the proposed approach's effectiveness. The results of this experiment are presented in Table 6. In this table, it can be noted that the %Area was 100%, reflecting the proposed approach's effectiveness. Moreover, the plots shown in Figure 7 illustrate the behavior of the residual, homoscedasticity, and quartile-quartile (QQ) of the results achieved by using the proposed DTOSFS approach. In these plots, it can be seen that the residual values were minor, reflecting the high accuracy of the proposed approach in efficiently recovering the watermark image compared to the other methods. Similarly, the behavior of the homoscedasticity and QQ was promising and showed the effectiveness of the proposed watermarking approach.