## 4 Conclusions  
This study proposes a robust semi-blind watermarking method for JPEG lossy compression, which uses a pre-designed embedder and decoder, combined with deep convolutional generative adversarial networks (DCGANs) for training, and finally, a well-trained generator model to restore (retrieve) the watermark. In addition, this study added a set of secret keys during the training process, which can ensure the security of the model and prevent third parties from copying a model that has a similar capability. Different from other watermarking methods based on deep learning, the training process of this method involves training a unique generator for the multiple protected images, and one generator corresponds to one watermark. Therefore, in the context of a third-party verification agency, the training process of this method is flexible and easy to manage. According to the experimental results, the method proposed in this study can restore (retrieve) the watermark well under the influence of different quality factors of JPEG compression. The image quality and error bit rate (BER) of the restored (retrieved) watermark are both good. An example of the excellent robustness of the method is that it produces PSNR values consisting of more than 49 dB, SSIM values of around 0.95, and BER values of less than 0.008, depending entirely on the JPEG quality factors. These parameters are the main character of an application that can outperform such distinguished conventional and new deep learning methods in retrieving watermark information under several conditions of compression. Symmetry determines the effectiveness of the proposed methodology. It brings improved resistance to the compression of pictures as well as distortion by watermark symmetry principles focused on embedding and decoding. An under-explored area, though, is the practical implementation and comparative gains of symmetry. Further quantitative or ablation study might demonstrate the impact of symmetry directly on robustness against different image distortions, including scaling, cropping, noise, etc. It could further add to the contribution of the method toward robust watermarking technologies. In this study, depthwise separable convolution (DSC) is used to reduce the amount of model parameters by about half. Although the image quality of the restored (retrieved) watermark is not ideal, the BER is still at a certain level. This confirms the feasibility of this direction. How to further improve the image quality while reducing the amount of model parameters is one of the important topics for future research. In addition, this method currently focuses on the robustness of JPEG compression; however, if it can resist other intentional image-processing attacks, the integrity of the method will be better. The model holds promise with datasets like USC-SIPI and Microsoft COCO; however, its scaling capacity to more diverse datasets as well as scenarios such as very high resolutions and extremely compressed versions remains an avenue to be explored. Future studies will be directed toward experimentation on these specific scenarios to ascertain the adaptability of the model and its further nuances to be applicable in real-world scenarios. Conceptualization, C.-F.L. and Z.-C.C. ; methodology, J.-J.S. and A.U.R. ; software, C.-F.L. ; and A.U.R. ; validation, C.-F.L., Z.-C.C. and J.-J.S. ; formal analysis, A.U.R. ; investigation, A.U.R. ; resources, C.-F.L. ; data curation, C.-F.L. and A.U.R. ; writing--original draft preparation, A.U.R. ; writing--review and editing, C.-F.L. and J.-J.S. ; visualization, A.U.R. ; supervision, C.-F.L. ; project administration, C.-F.L. ; funding acquisition, C.-F.L. All authors have read and agreed to the published version of the manuscript. This study was supported by the National Science and Technology Council, Taiwan, under contract number NSTC 113-2221-E-324-017-MY2. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.