## 5 Experimental Results
### Embedding Efficiency  
Another measurement for undetectability is embedding efficiency [(21)]. The embedding efficiency is defined as the number of embedded random message bits per embedding change [(21)]. According to this definition, embedding efficiency (\(EF\)) can be expressed as follows:  
\[EF=\frac{\text{number of embedding bits}}{\text{number of embedding pixel changes}}. \tag{24}\]  
Methods with higher embedding efficiency are more undetectable, because under the same capacity, higher embedding efficiency methods will change less pixels than lower embedding efficiency methods. For \(k=3\), according to Equation (24), the estimated embedding efficiency of the proposed method (\(EF_{p}\)) can be calculated by Equation (25). \[EF_{p}=\frac{3}{0\times p_{0}+1\times p_{1}}, \tag{25}\]  
where \(p_{i}\) stands for the probability of \(i\) pixel changed in a block, here \(p_{0}=1/8\) and \(p_{1}=7/8\). The estimated embedding efficiency of [PERSON] et al. 's method (\(EF_{i}\)) can be calculated by Equation (26). \[EF_{i}=\frac{3}{0\times p_{0}+1\times p_{1}+2\times p_{2}+3\times p_{3}+4 \times p_{4}}, \tag{26}\]  
where \(p_{i}\) stands for the probability of \(i\) pixel changed in a block, here \(p_{0}=1/15\), \(p_{1}=p_{2}=p_{3}=4/15\), and \(p_{4}=2/15\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Method** & \multicolumn{3}{c}{**Proposed Method**} & \multicolumn{3}{c}{**[PERSON] et al.**} & \multicolumn{3}{c}{**[PERSON] et al.**} \\ \hline
**Palette** & \multicolumn{3}{c}{**Original**} & \multicolumn{3}{c}{**Original**} & \multicolumn{3}{c}{**Reordering**} \\ \hline \(k\) & 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 \\ \hline \multicolumn{11}{c}{Actual embedding rate (\%)} \\ \hline  & 0 & 23 & 46 & 69 & 0 & 23 & 46 & 69 & 0 & 23 & 46 & 69 \\ \hline \multicolumn{11}{c}{Estimated embedding rate (\%)} \\ \hline Aerial & 0 & -3 & -5 & 1 & 0 & 10 & 10 & 16 & \(-\)20 & 10 & 18 & 32 \\ Airplane & \(-\)16 & \(-\)1 & 2 & 8 & \(-\)16 & \(-\)37 & \(-\)56 & \(-\)80 & \(-\)9 & \(-\)12 & \(-\)5 & \(-\)9 \\ Balloon & \(-\)11 & \(-\)19 & \(-\)17 & \(-\)20 & \(-\)11 & \(-\)14 & \(-\)12 & \(-\)21 & 7 & 6 & 0 & 3 \\ Earth & \(-\)18 & \(-\)12 & \(-\)12 & \(-\)16 & \(-\)18 & \(-\)23 & \(-\)28 & \(-\)26 & 21 & 14 & 14 & \(-\)15 \\ Girl & 12 & \(-\)23 & \(-\)18 & \(-\)30 & 12 & 10 & \(-\)4 & \(-\)7 & \(-\)16 & \(-\)8 & \(-\)18 & \(-\)19 \\ Lena & 0 & 12 & 11 & 12 & 0 & 2 & \(-\)9 & \(-\)4 & 4 & 6 & 10 & 8 \\ Parrots & 11 & \(-\)1 & 1 & 2 & 11 & 12 & 11 & 15 & \(-\)9 & \(-\)6 & \(-\)7 & \(-\)6 \\ Pepper & \(-\)1 & 11 & 11 & 11 & \(-\)1 & \(-\)1 & \(-\)3 & \(-\)2 & 4 & 2 & 6 & \(-\)4 \\ Mandrill & 0 & 0 & 0 & \(-\)1 & 0 & 3 & 8 & \(-\)11 & 5 & 10 & 9 & 7 \\ Couple & \(-\)11 & \(-\)2 & \(-\)11 & \(-\)7 & \(-\)11 & 5 & \(-\)12 & \(-\)4 & \(-\)16 & 20 & 18 & 22 \\ Sailboat & 11 & \(-\)15 & \(-\)25 & \(-\)24 & 11 & 12 & 10 & 5 & \(-\)12 & \(-\)12 & \(-\)9 & \(-\)12 \\ Milkdrop & 15 & 10 & 10 & 13 & 15 & 14 & 18 & 14 & \(-\)1 & 2 & 0 & \(-\)1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of RS steganalysis for [PERSON] et al.’s [(16)] and the proposed methods with block size \(2\times 2\). The estimated embedding efficiency of [PERSON] et al. 's method (\(EF_{a}\)) can be calculated by Equation (27). \[EF_{a}=\frac{3}{(p_{0})\times(1\times p_{1}+2\times p_{2})+(1-p_{0})\times((1+1) \times p_{1}+(1+2)\times p_{2})}. \tag{27}\]  
where \(p_{0}\) stands for the probability of \(B_{0}\) unchanged, \(p_{i}\) stands for the probability of \(i\) pixels from \(\{B_{1}\), \(B_{2}\}\) changed, here \(p_{0}=1/2\), \(p_{1}=4/7\), and \(p_{2}=2/7\). Table 8 shows the estimated embedding efficiencies of the proposed, [PERSON] et al. 's [16], and [PERSON] et al. 's [17] methods obtained by using Equations (25)-(27). Table 9 shows part of experimental embedding efficiency calculated by applying Equation (24) to each stego image, which is obtained by one of [PERSON] et al. 's, [PERSON] et al. 's and the proposed methods.