## 4 Discussion and Conclusions  
Using dual-tree complex wavelet transform (DTCWT) and particle swarm optimization (PSO), the watermarking method proposed revealed drastic improvements in terms of imperceptibility and resilience against attack, with a high capacity to embed large bits of messages into target images. Higher PSNR values indicate less degradation in the quality of host images due to watermarking operation. The technique shows good performance in preserving the image's quality up to a mean PSNR value of 80.50% across the dataset. Moreover, the average NCC value of 92.51% indicates high accuracy in extracting watermarks from an original image compared to its corresponding retrieved version (Table 2). This highlights the robustness of the proposed method in authenticating and verifying forgeries performed on watermarked images. Utilizing DTCWT for frequency domain transformation confers benefits such as shift invariance and directional selectivity, which are pivotal in obtaining high imperceptibility. The PSO algorithm's optimization of coefficient selection augmented the watermark's robustness against frequent image manipulation attacks. Overall, combining DTCWT and PSO in the proposed watermarking method delivers a highly effective solution for secure and superior image watermarking. To assess the performance of the proposed PSO-based watermarking method, a comparative analysis was conducted against other classical optimization approaches commonly used in watermark optimization. The comparison highlights that the proposed PSO-based watermarking method achieves the highest imperceptibility (PSNR of 80.5 dB) compared \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Method** & **PSNR (dB)** & **NCC** & **Computational Cost** \\ \hline GA-Based Method [12] & 47.81 & 97.84\% & High (requires DCNN-SVR correction) \\ \hline DE-Based Method [31] & 69.25 & 93.30\% & High (due to embedding additional data) \\ \hline ACO + LGBA Method [32] & 46.4–49.8 & 99\% for most attacks & Significantly improved with LGBA \\ \hline Proposed PSO–DTCWT & 80.5 & 92.51\% & Lower (due to parameter optimization) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the proposed PSO–DTCWT method and classical approaches. to the GA-based, DE-based, and ACO + LGBA methods. While its robustness (NCC of 92.51%) is slightly lower than the GA-based (97.84%) and ACO + LGBA methods (99%), it remains competitive. The PSO-based method also demonstrates significantly lower computational cost due to parameter optimization, making it an efficient and practical solution for watermarking applications. By leveraging PSO's ability to find optimal solutions in uncertain environments and DTCWT's capacity for multi-dimensional data analysis, the proposed approach can effectively address uncertainties, improve efficiency, and provide robust solutions across various industrial applications. The proposed watermarking method has limitations in terms of computational complexity, applicability to color images and advanced attack resistance. Future work must be considered to minimize the effect of these limitations and widen the use of the algorithm in other domains. Future research can be applied to improve such limitations in 3D gray scale and color images, optimize PSO parameters, and apply the method in real-time and IoT environments. Conceptualization, A.A.R.B. ; methodology, A.A.R.B. and A.B.I. ; software, A.B.I. ; validation, A.A.R.B. ; formal analysis, A.A.R.B. and A.B.I. ; investigation, A.A.R.B. and A.B.I. ; resources, A.A.R.B. ; data curation, A.B.I. and A.A.R.B. ; writing--original draft, A.B.I; writing--review and editing, A.A.R.B. ; data visualization, A.B.I; supervision, A.A.R.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. The authors declare no conflicts of interest.