# Two-Dimensional Histogram Shifting-Based Reversible Data Hiding for H.264/AVC Video
## 2 HS-based RDH Technique
### Proposed 2D HS-Based RDH Schemes
#### 3.2.2 2D HS without Using the \((0,0)\) Coefficient Pairs  
The proposed 2D HS scheme without using the \((0,0)\) coefficient pairs is illustrated in Figure 5. First, the points except \((0,0)\) are divided into several disjoint sets as follows. \begin{table}
\begin{tabular}{r r r r r r r r r} \hline \hline \(\mathbf{y}\)\(\mathbf{x}\) & \(\mathbf{-4}\) & \(\mathbf{-3}\) & \(\mathbf{-2}\) & \(\mathbf{-1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) \\ \hline \(\mathbf{-4}\) & 35 & 65 & 109 & 185 & 411 & 217 & 100 & 57 & 36 \\ \(\mathbf{-3}\) & 55 & 108 & 196 & 434 & 974 & 453 & 219 & 104 & 56 \\ \(\mathbf{-2}\) & 86 & 155 & 435 & 1272 & 3453 & 1332 & 495 & 198 & 99 \\ \(\mathbf{-1}\) & 164 & 371 & 1175 & 5263 & 24,321 & 5516 & 1298 & 417 & 168 \\ \(\mathbf{0}\) & 277 & 809 & 3274 & 26,301 & 251,897 & 26,346 & 3450 & 823 & 311 \\ \(\mathbf{1}\) & 160 & 390 & 1314 & 5472 & 24,283 & 5330 & 1262 & 370 & 158 \\ \(\mathbf{2}\) & 98 & 194 & 506 & 1357 & 3473 & 1257 & 494 & 184 & 86 \\ \(\mathbf{3}\) & 50 & 99 & 238 & 457 & 1021 & 432 & 199 & 95 & 54 \\ \(\mathbf{4}\) & 34 & 54 & 105 & 197 & 395 & 176 & 100 & 77 & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of different DCT coefficient pairs in the video ‘bus’. Figure 4: The proposed 2D HS using the \((0,0)\) coefficient pairs. \[S_{1}=\left\{(1,0)\right\},\ S_{2}=\left\{(0,1)\right\},\ S_{3}=\left\{(-1,0) \right\},\ S_{4}=\left\{(0,-1)\right\},\] \[S_{5}=\left\{(1,1)\right\},\ S_{6}=\left\{(-1,1)\right\},\ S_{7}= \left\{(-1,-1)\right\},\ S_{8}=\left\{(1,-1)\right\},\] \[S_{9}=\left\{(x,1)\mid x>1\right\},\ S_{10}=\left\{(x,1)\mid x<-1 \right\},\ S_{11}=\left\{(x,-1)\mid x<-1\right\},\ S_{12}=\left\{(x,-1)\mid x>1 \right\},\] \[S_{13}=\left\{(x,0)\mid x>1\right\},\ S_{14}=\left\{(x,0)\mid x<-1 \right\},\ S_{15}=\left\{(x,y)\mid y>1\right\},\ S_{16}=\left\{(x,y)\mid y<-1 \right\}.\]  
Then, the method of embedding data into the coefficient pairs belonging to different sets is described as below. If \((x,y)\in S_{1}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y)&\text{if }m_{i}=0,\\ (x,y+1)&\text{if }m_{i}m_{i+1}=10,\\ (x,y+1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{20}\]  
If \((x,y)\in S_{2}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y)&\text{if }m_{i}=0,\\ (x-1,y)&\text{if }m_{i}m_{i+1}=10,\\ (x,y-1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{21}\]  
If \((x,y)\in S_{4}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y)&\text{if }m_{i}=0,\\ (x,y-1)&\text{if }m_{i}m_{i+1}=10,\\ (x+1,y)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{22}\]  
If \((x,y)\in S_{5}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y)&\text{if }m_{i}=0,\\ (x,y-1)&\text{if }m_{i}m_{i+1}=10,\\ (x+1,y)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{23}\]  
If \((x,y)\in S_{5}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y+1)&\text{if }m_{i}=0,\\ (x+1,y)&\text{if }m_{i}m_{i+1}=10,\\ (x+1,y+1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{24}\]  
Figure 5: The proposed 2D HS without using the \((0,0)\) coefficient pairs. If \((x,y)\in S_{6}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y+1)&\text{if }m_{i}=0,\\ (x-1,y)&\text{if }m_{i}m_{i+1}=10,\\ (x-1,y+1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{25}\]  
If \((x,y)\in S_{7}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y-1)&\text{if }m_{i}=0,\\ (x-1,y)&\text{if }m_{i}m_{i+1}=10,\\ (x-1,y-1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{26}\]  
If \((x,y)\in S_{8}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})= \begin{cases}(x,y-1)&\text{if }m_{i}=0,\\ (x+1,y)&\text{if }m_{i}m_{i+1}=10,\\ (x+1,y-1)&\text{if }m_{i}m_{i+1}=11.\end{cases}  \tag{27}\]  
If \((x,y)\in S_{9}\), the marked coefficient pair will be  
\[(x^{\prime},y^{\prime})=