## 5 Results and Discussion
### Experimental Results
\begin{table}
\begin{tabular}{c c c c c} \hline  & **DFT+SVD** & **DWT+DCT** & **DWT+DCT+SFS** & **DWT+DCT+DTO** \\ \hline Best-fit values & & & & \\ Slope & 80 & \(-\)144.4 & \(-\)77.78 & \(-\)211.1 \\ X-intercept & 63.42 & 64.18 & 64.51 & 64.13 \\ Y-intercept & \(-\)5073 & 9271 & 5017 & 13,538 \\
1/slope & 0.0125 & \(-\)0.006923 & \(-\)0.01286 & \(-\)0.004737 \\ Slope & 71.49 & 211.5 & 70.27 & 35.14 \\ Std. Error & & & & \\ Y-intercept & 4566 & 13,512 & 4488 & 2244 \\
95\% Confidence Intervals & & & & \\ X-intercept & \(-\)infinity to 63.72 & 63.94 to \(+\)infinity & 64.08 to \(+\)infinity & 64.05 to 64.28 \\ Y-intercept & \(-\)15,603 to 5457 & \(-\)21,888 to 40,429 & \(-\)5333 to 15,367 & 8362 to 18,713 \\ Goodness of Fit & & & & \\ R square & 0.1353 & 0.05507 & 0.1328 & 0.8186 \\  & & & & \\  & & & & \\ Is the slope significantly non-zero? & & & & \\ F & 1.252 & 0.4662 & 1.225 & 36.1 \\  & & & & \\ DFn, Dfd & 1,8 & 1,8 & 1,8 & 1,8 \\  & & & & \\  & & & & \\ Deviation from zero? & Significant & Significant & Significant & Significant \\ Equation & Y = 80.00\(\times\)X - 5073 & Y = \(-\)144.4\(\times\) X + 9271 & Y = \(-\)77.78\(\times\) X + 5017 & Y = \(-\)211.1\(\times\) X + 13,538 \\ Data & & & & \\ Number of X values & 10 & 10 & 10 & 10 \\ Maximum number of Y replicates & 1 & 1 & 1 & 1 \\ Total number of values & 10 & 10 & 10 & 10 \\ Number of missing values & 0 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 5: The regression test between the proposed watermarking method and each one of the other competing watermarking methods. To emphasize the promising performance of the proposed approach, Figure 11 depicts the range of the PSNR values achieved with the proposed approach compared to those of the other four watermarking methods. In this figure, it is obvious that the proposed (DWT+DCT+DTOSFS) watermarking approach achieved the highest PSNR value, which strongly emphasized the proposed approach's promising performance in digital watermarking. Figure 11: The PSNR values achieved by the proposed approach in comparison with the other approaches. Figure 10: Histogram of the PSNR values achieved with the proposed approach. Figure 9: The ROC curve of the results achieved with the proposed approach. From the perspective of attacks on watermarked images, another experiment was conducted to study the effects of various types of attacks on the performance of the proposed approach when the Monkeypox was recovered. In this experiment, four types of attacks were considered: Gaussian filtering, median filtering, Wiener filtering, and average filtering. Table 7 presents the effect of each type of attack on the image recovered by the proposed watermarking approach in terms of the NCC and structural similarity index matrix (SSIM), along with their standard deviation (SD). As shown in the table, the quality of the recovered Monkeypox image was slightly affected by these attacks, especially when the Gaussian filtering attack was tested. In addition, the filter that significantly affected the recovered image was the average filter, for which the NCC value was 0.886 and the SSIM was 0.813. These results prove that the proposed watermarking approach was robust against certain types of attacks, such as Gaussian and median filtering attacks. Figure 12 shows the recovered Monkeypox image for each type of attack. On the other hand, the recovery time of the Monkeypox image using the proposed approach and the other four methods is listed in Table 8. In this table, it is shown that the proposed approach was able to recover/extract the Monkeypox image from the cover image faster than the other methods for the three cover images. These results clearly emphasize the superiority of the proposed watermarking approach. More comparison results are shown in Table A1 in Appendix A.  
Figure 12: The watermarking results achieved by the proposed approach when applied to the Lena image.