## 3 Proposed Method
### Recurrent Robust Reversible Data Hiding (Triple-RDH) Embedding Method  
Input: a gray-scale cover image \(I\). Output: a gray-scale stego image \(I^{\prime}\). Step 1. For the pixel in the embeddable area, the image \(I\) is sheared into a quotient image (i.e., higher significant bit-plane image \(I_{HISB}\)) and a remainder image (i.e., lower significant bit-plane image \(I_{LoSB}\)); that is, each embeddable pixel \(x_{i}\) is divided into a quotient image pixel \(x_{i}^{q}\) and a remainder image pixel \(x_{i}^{r}\), using the quotient and remainder operands, respectively, by Equation (6). \[x_{i,j}^{q}=x_{i,j}\text{ div }2^{n};\ x_{i,j}^{r}=x_{i,j}\text{ mod }2^{n}\text{ for }i=1,\ 2,\ \dots, \tag{6}\]  
for \(=1,\ 2,\ \dots,\ H\ and\ j=1,\ 2,\ \dots,\ W\), where \(\ x_{i,j}\in I\), \(x_{i,j}^{q}\in I_{HiSB}\), and \(x_{i,j}^{r}\in I_{LoSB}\); _div_ and _'mod'_ are the divide and modulo operations, respectively. Each quotient pixel \(x_{i,j}^{q}\) of the image \(I_{HiSB}\) is labeled into two image pixels of gray and white as in a chessboard-shaped chessboard. Step 2. For each pixel \(\ x_{i,j}^{q}\in I_{HISB}\) on the chessboard image \(I_{HiSB}\), let \(x_{c}=x_{i,j}^{q}\). Then, the top, bottom, left, and right of the pixel \(x_{c}\) are another-color pixel values that are different from the color of \(x_{i,j}^{q}\). The pixel values are, respectively, \(\left(x_{i-1,j}^{q},\ x_{i+1,j}^{q},\ x_{i,j-1}^{q},\ x_{i,j+1}^{q}\right)\in I _{HiSB}\). If \(x_{c}=x_{i,j}^{q}\) is gray, then \(\left(x_{i-1,j}^{q},\ x_{i+1,j}^{q},\ x_{i,j-1}^{q},\ x_{i,j+1}^{q}\right)\) is the four white neighboring pixels of \(x_{i,j}^{q}\); if \(x_{c}=x_{i,j}^{q}\) is white, then \(\left(x_{i-1,j}^{q},\ x_{i+1,j}^{q},\ x_{i,j-1}^{q},\ x_{i,j+1}^{q}\right)\) is the four gray neighboring pixels of \(x_{i,j}^{q}\). Step 3. The sort() function is used to sort \(\left(x_{i-1,j}^{q},\ x_{i+1,j}^{q},\ x_{i,j-1}^{q},\ x_{i,j+1}^{q}\right)\) in ascending order, and the result obtained is \(\left(x_{\pi(1)},x_{\pi(2)},x_{\pi(3)},x_{\pi(4)}\right)\). \[\left(x_{\pi(1)},x_{\pi(2)},x_{\pi(3)},x_{\pi(4)}\right)=\text{sort}\left(x_{ i-1,j}^{q},\ x_{i+1,j}^{q},\ x_{i,j-1}^{q},\ x_{i,j+1}^{q}\right) \tag{7}\]  
The sort() function \(\ \pi:\left\{1,\ 2,\ \dots,\ n\right\}\ \rightarrow\ \left\{1,\ 2,\ \dots,\ n\right\}\) is the unique one-to-one mapping. In the case of equal value, pixels are ordered by \(\ \pi\ (k)\ <\pi\ (\updownarrow)\), if \(x_{\pi(k)}=x_{\pi(\updownarrow)}\) and \(k<\updownarrow\). Step 4. The local complexity of pixel \(x_{c}\) is defined as the variance (\(\sigma_{c}^{2}\)) of the surrounding four color pixel values \(\left(x_{\pi(1)},x_{\pi(2)^{\prime}},x_{\pi(3)},x_{\pi(4)}\right)\), as shown in Equation (8). \[\sigma_{c}^{2}=\frac{1}{4}\sum_{k=1}^{4}\left(\mu_{c}-x_{\pi(k)}\right)^{2}, \tag{8}\]  
Figure 5: Image \(I\) is divided into \(I_{HiSB}\) and \(I_{LoSB}\). where \(\mu_{c}=\frac{1}{4}\ \sum_{k=1}^{4}x_{\pi(k)}\), which is the average value of the four color pixel values around pixel \(x_{c}\). Step 5. The recurrent round-trip embedding strategy (referred to as double R-TES strategy) is expressed. For the convenience of explanation, assume that \(x_{c}\) is a gray pixel at this time:  
\[x_{c}^{(out)}=\left\{ \begin{array}{ll}x_{c}+\varepsilon^{out},&\quad if\ \ e^{ out}=pp_{1}\\ x_{c}+1,&\quad if\ e^{ out}>pp_{1}\\ x_{c},&\quad if\ e^{ out}<pp_{1}\end{array} \right. \tag{9}\]  
\[x_{c}^{(back)}=\left\{ \begin{array}{ll}x_{c}^{(out)}-\varepsilon^{in},&\quad if \ \ e^{back}=pp_{2}\\ x_{c}^{(out)}-1,&\quad if\ e^{back}<pp_{2}\\ x_{c}^{(out)},&\quad if\ e^{back}>pp_{2}\end{array} \right. \tag{10}\]  
Step 6. \(pp_{1}\) and \(pp_{2}\) are two predetermined peak points. According to the experimental statistics of this research, setting \(pp_{1}=1\) and \(pp_{2}=-1\) will make the embedding capacity more effective. Step 7.