## 4 Conclusions  
In this paper, we present a novel approach for RDHEI using separable RRBE method with high ERs and error-free reversibility. While most existing RDHEI methods utilize the most or least significant bit planes to hide data, we propose exploiting the redundancy between adjacent pixels to calculate a prediction error image that is then processed to accommodate data in its embeddable bit planes. This image is subsequently converted into a new error image with positive pixel values, which undergoes quadtree decomposition for each bit plane to locate as many homogeneous blocks as possible with minimal auxiliary \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Method**} \\ \cline{2-13}
**DataSet** & \multicolumn{2}{c}{**Our**} & \multicolumn{2}{c}{[12]} & \multicolumn{2}{c}{[19]} & \multicolumn{2}{c}{[8]} & \multicolumn{2}{c}{[10]} & \multicolumn{2}{c}{[20]} \\ \cline{2-13}  & **ER** & **ER** & **Gain** & **ER** & **Gain** & **ER** & **Gain** & **ER** & **Gain** & **ER** & **Gain** \\ \hline BOSSbase & 3.813 & 0.966 & 294.7 & 2.561 & 48.8 & 3.361 & 13.4 & 3.498 & 9 & 2.556 & 49.1 \\ BOWS-2 & 3.696 & 0.968 & 281.8 & 2.519 & 46.7 & 3.246 & 13.8 & 3.393 & 8.9 & 2.530 & 46 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the average ERs of two image datasets between our method and five state-of-the-art methods. information. These blocks of varying sizes are then reordered to create space for embedding data. The proposed method enables error-free extraction of the secret message and lossless reconstruction of the original image. Experimental results demonstrate that our method is highly efficient, secure, and outperforms previous state-of-the-art methods. While our research has made significant advancements in the field of RDHEI, there are several avenues for future work and potential improvements. One avenue of future research entails further exploration of techniques aimed at maximizing embedding capacity while maintaining error-free reversibility. Additionally, we suggest an investigation of RDHEI's potential applications in emerging technologies such as blockchain, which could open doors to novel use cases and challenges. Conceptualization, A.M.; Software, M.A. ; Validation, M.A. ; Writing-original draft, M.A. ; Writing--review and editing, A.M.; Supervision, A.M. All authors have read and agreed to the published version of the manuscript. The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project under grant number RGP2/171/44. Not applicable. The datasets used in this paper are publicly available and their links are [[http://agents.fel.cvut.cz/boss/](http://agents.fel.cvut.cz/boss/)]([http://agents.fel.cvut.cz/boss/](http://agents.fel.cvut.cz/boss/)) (2021) and [[https://data.mendeley.com/datasets/kb3](https://data.mendeley.com/datasets/kb3) ngxfmjw/1]([https://data.mendeley.com/datasets/kb3](https://data.mendeley.com/datasets/kb3) ngxfmjw/1) (accessed on 21 January 2021). The authors declare no conflict of interest. The following abbreviations are used in this manuscript, Note: This table provides a brief overview, and readers are encouraged to refer to the corresponding sections in the paper for more detailed explanations. The predicted image  
The prediction error image  
The encrypted image  
A pseudo-random matrix generated with the PWLCM  
The image encryption key  
The data hiding key  
A binary label map. It is set to 1 for the overflow pixels and 0 otherwise  
A function that returns the neighboring pixel values, as depicted in Figure 3  
A function that takes a list of neighboring pixels \(L\) as input, and returns the predicted value based on the used predictor (MED, GAP,\(\cdots\))  
A function that accepts an image bit plane as input and returns a binary string, denoted as \(binaryStr\), which represents the quadtree decomposition of the input bit plane.