## 5 Conclusions  
In this study, we developed and evaluated an optimized strategy for reversible data hiding (RDH) utilizing an enhanced exploiting modification direction (EMD) method, which significantly improved the efficacy of the data hiding technique. By integrating the foundational principles of EMD with the benefits of reversibility, we achieved an exceptional data hiding rate of approximately 1.10 bits per pixel while also preserving the images' high visual fidelity. Moreover, we greatly reduced the visual and technical anomalies in stego images, safeguarding the integrity and quality of the concealed data. A distinguishing characteristic of our methodology is its ability to enable the flawless restoration of the original image without the need for any supplementary payload, marking a significant advancement over the existing RDH practices. The efficacy of our proposed technique was verified through empirical testing, involving rigorous comparisons with several cutting-edge methods. Looking ahead, we aim to further refine our approach, ensuring it remains theoretically sound, practically viable, and provides an all-encompassing solution for RDH endeavors in digital imaging.  
The robustness of data hiding techniques in noisy environments is a critical research topic. This aspect is particularly useful for assessing resistance to various distortions that may occur due to external attacks or during the data transmission process. Therefore,  
Figure 8: The RS analysis for two stego images, \(\Delta I_{1}\) and \(\Delta I_{2}\), of Lena.  
future research should aim to evaluate the robustness of the method presented in this study against noise, while also exploring solutions to overcome related limitations.  
Each author discussed the details of the manuscript. [PERSON] designed and wrote the manuscript. [PERSON] implemented the proposed technique and provided the experimental results. K.-[PERSON] and [PERSON] reviewed and revised the article. [PERSON] drafted and revised the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (2019H1D3A1A01101687) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A3049788) (K.-H.J.). This research was supported by the National Natural Science Foundation of China (61866028), Technology Innovation Guidance Program Project (Special Project of Technology Cooperation, Science and Technology Department of Jiangxi Province) (20212 BDH81003) (L.L.).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
We thank the anonymous reviewers for their valuable suggestions that improved the quality of this article.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} DH & Data Hiding \\ RDH & Reversible Data Hiding \\ Dual RDH & Dual Reversible Data Hiding \\ EMD & Exploiting Modification Direction \\ \(I_{1}\) & Cover image (CI) 1 \\ \(I_{2}\) & Cover image (CI) 2 \\ \(\Delta I_{1}\) & Stego image (SI) 1 \\ \(\Delta I_{2}\) & Stego image (SI) 2 \\ \(x_{i}^{1}\) & a pixel of \(I_{1}\), i.e., \(x_{i}^{1}\in I_{1}\) \\ \(x_{i}^{2}\) & a pixel of \(I_{2}\), i.e., \(x_{i}^{2}\in I_{2}\) \\ \(m_{i}=(m_{1},m_{2},\dots,m_{n})\) & 5-ary secret data \\ \(N\times N\) & Image Size \\ \(f(\chi_{i},x_{2})\) & Extraction function, \(\sum_{i=1}^{N\times N}([x_{i}^{1}\cdot 1+x_{i}^{2}\cdot 2])\ mod\ 5\) \\ \(y_{i}^{1}\) & a pixel composed of SI 1 \\ \(y_{i}^{2}\) & a pixel composed of SI 2 \\ \(y||\) & Vector for \(y\) \\ \(p_{i}\) & Average of the two pixels, \(y_{i}^{1}\) and \(y_{i}^{2}\) \\ \(s(=f)\) & Extracted secret data \\ \end{tabular}