## 3 Results  
The proposed scheme was evaluated using the dataset described in Section 2.2, and the validation of the proposed scheme was performed as described in Section 2.3.5, by calculating the PSNR between the host and watermarked images and the NCC between the embedded logo image and the extracted image logo. The implementation of the approach was carried out in Python on a laptop computer equipped with an Intel Core i7 processor, with 32 GB RAM, and running on the Windows 10 operating system. To embed the watermark logo into the host image, the host image was decomposed using DTCWT. The filters \"near_sym_b\" and \"shift_b\" were empirically found to produce the best coefficients, and a decomposition level of four was chosen. The PSO algorithm described in Section 2.3.2 was utilized to select the best embeddable coefficients from the LL sub-band. The parameters of the PSO algorithm were optimized for computational efficiency using empirical settings. Specifically, the cognitive component (C1) and social component (C2) were both tested for a value which varied between 0.5 and 2.0. The inertia weight (\(\omega\)), which controls the search direction, was tested in the range of a maximum weight (\(W_{\text{max}}\) = 0.9) and a minimum weight (\(W_{\text{min}}\) = 0.4). Additionally, \(\varphi\)1 and \(\varphi\)2 were assigned uniformly distributed random values within [0, 1]. The number of swarms was analyzed in sizes ranging from 20 to 50, and the number of iterations was evaluated in the range of 50 to 200 iterations in order to discover the convergence point of our objective function. The findings of the PSO parameters' optimization underscore the importance of carefully selecting PSO parameters to enhance computational efficiency. Figure 5 demonstrates that \(\omega\) = 0.7 is the optimal inertia weight, achieving the best balance between computational efficiency and performance quality. At this \(\omega\) value, PSNR reaches its peak while computational cost is at its minimum, highlighting it as the ideal value. The red dashed line effectively emphasizes this point. Figure 6 shows that C1 = 1.5 is the optimal cognitive component, achieving the highest PSNR while maintaining a decreasing computational cost. The red dashed line emphasizes this ideal balance between performance and efficiency. The optimization results identified C2 = 1.5 as the optimal value, as illustrated in Figure 7. This value achieved the highest PSNR, while the computational cost increased gradually. The optimization process was also applied to the swarm size and number of iterations of PSO parameters to determine the optimal values that minimized the computational cost. The results of these optimization processes are presented in Figures 8 and 9. It should be noted that this insertion process may introduce artifacts and distortion into the watermarked image. To evaluate the impact of these distortions, the PSNR was measured for the watermarked image. A visual comparison between some standard cover images before and after the watermark was applied, along with the PSNR between the two images, is provided in Table 1.  
Figure 5: Impact of inertia weight (\(\omega\)) on computational cost and PSNR. As shown in Table 1, the proposed method achieved a high PSNR between the host and watermarked images, indicating that the watermark was embedded effectively. The shift-invariant properties of DTCWT were found to be crucial in achieving this high PSNR. These properties allow for robust watermark embedding while preserving the quality of the host image. To verify the embedding algorithm, we performed an extraction of the logo embedded in the images and measured the NCC between the original image logo and the extracted logo image. The results are provided in Table 2.  
Figure 6: Impact of cognitive component (C1) on computational cost and PSNR. Figure 7: Impact of social component (C2) on computational cost and PSNR. Figure 8: Impact of swarm size on computational cost and PSNR. Figure 9: Impact of number of iterations on computational cost and PSNR.