## 5 Conclusions  
In this research, we propose a recurrent robust reversible data hiding (triple-RDH) method. The triple-RDH method embeds the secret message into the quotient pixel value of the gray-scale image, which can prevent malicious attacks and increase the robustness. We have developed a recurrent round-trip embedding strategy (referred to as a double R-TES), which allows the pixel value to shift to the right and then shift to the left after embedding the secret bits, resulting in the invariance of the pixel achieving the repetitive embedding strategy and effectively increasing the embedding capacity through the double R-TES embedding strategy. The triple-RDH method divides gray pixels and white pixels for the quotient image, adjusting the recursive parameter \(t_{S}\) and \(t_{w}\) to effectively improve the embedding capacity and maintain good image quality.  
The experimental results show that the proposed method has a great advantage in the embedding capacity, and the PSNR value tends to be flat when the embedding capacity is higher. The average maximum ECs of the TLE, SBDE, and IPVO and our proposed triple RDH methods are 179,587, 149,298, 33,465, and 261,466 bits, respectively. The experimental results show that our triple RDH method is superior to the TLE methodand the SBDE method in terms of embedding capacity and image quality. The average maximum embedding capacity of our proposed method is 261,466 bits, which is eight times that of IPVO method. It can also be seen from Figure 9a-j of the experimental results that the average PSNR value of the proposed triple RDH method is even higher than that of the TLE and SBDE methods.  
Obviously, the image quality of IPVO in the RDH method is quite excellent. When 20,000 bits are hidden, the PSNR of the image \"Lena\" for the IPVO method is 54.91 dB. This is just as described in the related literature, emphasizing that the IPVO method focuses on good image quality but the embedding ability is limited. Additionally, after hiding 20,000 bits, the image quality of our proposed method can reach a PSNR value of 43.94 dB and an SSIM value of 0.9779. Compared with the PSNR of the IPVO method, even if the PSNR value is less 10 dB, the high SSIM value of our method indicates that the stego-image has a perfect match with the original image, indicating that the third party will not be aware of the distortion of the image and therefore does not know the existence of secret information. Therefore, the proposed data hiding method helps to transmit data seamlessly and securely through communication channels.  
Conceptualization, C.-F.L.; methodology, C.-F.L.; software, H.-Z.W.; validation, C.-F.L. and H.-Z.W.; formal analysis, C.-F.L.; investigation, C.-F.L. and H.-Z.W.; resources, C.-F.L.; data curation, H.-Z.W.; writing--original draft preparation, H.-Z.W.; writing--review and editing, C.-F.L.; visualization, C.-F.L.; 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 research was partially supported by the Ministry of Science and Technology, Taiwan, Republic of China under the Grant under the Grants MOST 109-2221-E-324-022.  
Not applicable.  
We are grateful to all the reviewers' valuable comments for improving the quality of this research work.  
The authors declare no conflict of interest.