## 2 Related Work
### Embedding Method of TLE  
The TLE method first divides the pixel value of an \(I_{HiSB}\) image into white and gray chessboards, as shown in Figure 1a. The secret information to be hidden is repeatedly embedded in the whole gray pixel values and then in the whole white pixel values. Figure 1a also shows that during the embedding process of the TLE method, the pixel values of the first column and the first row and the last column and last row of the \(I_{HiSB}\) image will not be changed. The TLE embedding procedure is as follows. Input: a gray-scale image \(I\). Output: a gray-scale stego image \(I^{\prime}\). Step 1. The binary representation of each pixel value in the gray-scale image \(I\) is \(\left(b_{7},b_{6},b_{5},b_{4},b_{3},b_{2},b_{1},b_{0}\right)\), \(b_{i}\in[0,\,1]\). The cover image is decomposed into an image (\(I_{HiSB}\)) and an image (\(I_{LoSB}\)), where the pixel value of each pixel in the \(I_{HiSB}\) image is \(\left\{x=\left(b_{7},\,\ldots,b_{n+2},b_{n+1},b_{n}\right)\ \mid\ 2^{n}\leq x\leq 2^{8}-1\right\}\). Step 2. For the convenience of description, the \(I_{HiSB}\) image be represented by the gray-scale pixel (\(x_{i,j}\)) in Figure 1b. In the \(I_{HiSB}\) image, it is used as the carrier for embedding the secret message of the embedding binary. Let \(x_{c}=x_{i,j}\); in Figure 1b, the white pixel values of the top, bottom, left and right of the pixel (\(x_{c}\)) are \(\left(x_{i-1,j},x_{i+1,j},x_{i,j-1},x_{i,j+1}\right)\). For the convenience of explanation, let \(\left(x_{u},x_{d},x_{l},x_{r}\right)=\left(x_{i-1,j},x_{i+1,j},x_{i-1},x_{i,j +1}\right)\). The vector \(\left(x_{u},x_{d},x_{l},x_{r}\right)\) is sorted according to the pixel value from small to large, and the result is \(\left(x_{\pi(1)},x_{\pi(2)},x_{\pi(3)},x_{\pi(4)}\right)\). Step 3. The local complexity of the pixel \(x_{c}\) is defined as its surrounding four white pixel values \(\left(x_{\pi(1)},x_{\pi(2)},x_{\pi(3)},x_{\pi(4)}\right)\), and the variance (\(\sigma_{c}^{2}\)), as shown in Equation (1). \[\sigma_{c}^{2}=\frac{1}{4}\sum\
olimits_{i=1}^{4}\left(\mu_{c}-x_{\pi(t)} \right)^{2}, \tag{1}\]  
Figure 1: Image scanning and embedding mode. (**a**) a zig-zag scan (**b**) the gray pixel \(\mathrm{x_{c}}\) and its surrounding four white pixels. where \(\mu_{c}=\frac{1}{4}\ \sum_{l=1}^{4}x_{\pi(t)}\), is the average value of the white pixel values around the pixel \(x_{c}\). Step 4. The calculation of predictor: Table 1 lists three predictor methods suggested by [PERSON] and [PERSON]. Each predictor will have a pair of predictors \((p_{1},\ p_{2})\) used as a prediction error. Predictor method 1 will use the largest and smallest pixel values in the ordered vector \((x_{\pi(1)},x_{\pi(2)},x_{\pi(3)},x_{\pi(4)})\), which is a pair of extreme values neighbors of pixel \(x_{c}\). The extreme value neighbors are \(x_{\pi(4)}\) and \(x_{\pi(1)}\) as predictors \(p_{1}\) and \(p_{2}\). The predictor method 2 is the most comprehensive predictor, which will use the four white neighbors of pixel \(x_{c}\). Each predictor \(p_{1}\) or \(p_{2}\) will use the average of the three white neighbors to make predictions. The design of predictor method 3 puts the predictors between the extreme predictor method 1 and the comprehensive predictor method 2. Step 5. Two-layer embedding (TLE) strategy: refers to the strategy that each pixel in the image (\(I_{H\!S\!B}\)) can embed the secret data twice. In the first layer embedding program, the TLE strategy uses Equation (2) to embed the secret bit \(s_{1}\) into the pixel \(x_{c}\), and the method is: first calculate the first prediction error \(e_{1}=x_{c}-p_{1}\), and then the to-be-embedded secret bit \(s_{1}\in\{0,1\}\), as in Equation (2):  
\[x_{c}^{(1)}=\left\{ \begin{array}{ll}x_{c}+s_{1},&\text{ if }e_{1}=1\\ x_{c}+1,&\text{ if }e_{1}>1\\ x_{c},&\text{ if }e_{1}<1\end{array} \right. \tag{2}\]  
where \(x_{c}^{(1)}\) is the intermediation value after embedding in the first layer. Step 6.