## 3 Proposed Method
### Recurrent Robust Reversible Data Hiding (Triple-RDH) Embedding Method
Then, determine whether \(x_{c}^{(back)}\) can be used a carrier to embed information: compare the gray pixel \(x_{c}\) of the original cover image with the gray pixel \(x_{c}^{(back)}\) that has been embedded back and forth \(t_{w}\). times (that is, at most \(2t_{w}\) bits of information have been hidden). If \(x_{c}^{(back)}=x_{c}\), this means that the process goes back to Step 6, and the recurrent round-trip embedding strategy can be used again and again to let \(x_{c}^{(back)}\) be the carrier and have the opportunity to embed more than two secret data bits. If \(x_{c}^{(q)}=x_{c}^{(back)}\), then go to Step 8. Step 8. When the last embeddable gray pixel \(x_{c}=x_{i,j}^{q}\) of \(I_{HiSB}\), i.e., \(i=H-1\) and \(j=W-1\), has been processed, the procedure in a zig-zag scanning manner starts to proceed to the first embeddable white pixel \(x_{i,j}^{q}\), i.e., \(i=\)2 and \(j=\)3 of \(I_{HiSB}\), until all white pixels are processed to embed secret data. Step 9. The final process is to merge the quotient image pixels (\(8-n\) HiSBs) and the remainder image pixels (\(n\) LoSBs) back to the gray-scale stego image pixels by Equation (11) and return the stego image \(I^{\prime}\). \[x_{c}^{\prime}=x_{c}^{\prime\ (q)}\ \text{mul}2^{n}+x_{i}^{\prime}, \tag{11}\]  
where '\(mul\)' stands for the multiplier operator. This research will generate some extra information: the location map \(LM_{i,j}^{(t)}\) and the parameters for hiding data bits. The lossless compression technology \"Arithmetic Coding\" is used to compress the location map to a length of (\(\updownarrow_{CLM}\)) \(<<t\times H\times W\), and the least significant bit (LSB) substitution method is used with other additional information to hide the end of the image. The function of location map \(LM_{i,j}^{(t)}\) has the following two situations:  
CASE 1: When the recursive parameter \(t=1\), the location map \(LM_{i,j}^{(1)}\) is used to avoid overflow/underflow problems and lossless extraction of secret message to restore the image. \[x_{i,j}=\left\{ \begin{array}{ll}x_{i,j}-1,&\quad if\ x_{i,j}=MAX\\ x_{i,j}+1,&\quad if\ x_{i,j}=MIN\quad and\\ x_{i,j},&\quad otherwise\end{array} \right. \tag{12}\]  
\[LM_{i,j}^{(1)}=\left\{ \begin{array}{ll}0,if\ x_{i,j}=MAX\ or\ MIN\\ 1,otherwise\end{array} \right. \tag{13}\]  
CASE 2: When the recursive parameter \(t>1\), the location map \(LM_{i,j}^{(t)}\) is used to allow the receiving end to recover smoothly in the process of extracting the information, and every pixel must be recorded how many times the triple-RDH embedding method have been carried out on the pixel; every time it is carried out, it hides a 2-bit data with \(LM_{i,j}^{(t)}=1\), otherwise \(LM_{i,j}^{(t)}=0\). In addition to the location map, the parameters \(n,t_{g},\ t_{w},\ S_{cmd}\) are required as additional information, where \(n\{1,2,\ 3,\ 4\}\), using two bits to encode the \(n\) value and \(t_{g},\ t_{w}\{1,2,\ 3,\ 4\}\) using four bits to encode the \((t_{g},\ t_{w})\) values. The index value \(S_{cmd}\) is recorded to distinguish the boundary of extra information and the secret message \(S\). The position of the last embedded pixel value is found out when extracting the extra information. The extra information items and the required bit sizes are shown in Table 2. It is required to embed the overhead information at the end of the cover image by exploiting the LSB method. The size of extra information is \(\left(6+\lceil log_{2}(H\times W)\rceil+\sum_{k}(t_{g}+t_{w})\ \mathit{l_{ clm}}^{k}\right)\) bits, and the LSB, which has been replaced by the extra information, is represented as \(H_{LSB}\).