## 4 Simulation Results
At the decoder, after decoding with the 2-byte side information (\(a^{\prime}\), \(c^{\prime}\)), the block map **B** can be produced, and then both the cover image and user-defined information can be extracted. For verifying the reversibility of our algorithm, regarding to the image itself, we can see that all the mean square errors (MSE's) are 0.00, meaning that the recovered images are identical to their counterparts. Besides, for user-defined information, we observe the bit error rates (BER's) between the embedded and extracted ones are all 0.00%, meaning that they are identical. Therefore, from the data shown in the bottom two rows in Table 2, we prove that our data hiding algorithm can reach the goal of reversibility. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Block Size** & 512 \(\times\) 512 & 256 \(\times\) 256 & 128 \(\times\) 128 & 64 \(\times\) 64 & 32 \(\times\) 32 & 16 \(\times\) 16 & **Quadtree** \\ \hline Image quality (dB) & 53.54 & 51.22 & 51.03 & 51.02 & 50.36 & 50.14 & 51.13 \\ Capacity (bit/pixel) & 0.0113 & 0.0199 & 0.0280 & 0.0419 & 0.0552 & 0.0596 & 0.0546 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparisons of test image Lena with conventional scheme. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test image** & **Airplane** & **F-16** & **Lena** & **Pepper** & **Tank** & **Truck** \\ \hline Image size & 512 \(\times\) 512 & 512 \(\times\) 512 & 512 \(\times\) 512 & 512 \(\times\) 512 & 512 \(\times\) 512 & 512 \(\times\) 512 \\ Output image quality (dB) & 49.97 & 51.34 & 51.13 & 51.03 & 51.80 & 51.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons of performances of test images, in alphabetical order, with conventional scheme. Table 3 makes performance comparisons with the results in [10] and those with our scheme. Method in [10] also employs the histogram-based scheme for performing reversible data hiding. Suppose that the input images are identical, both are the test image Lena with the size of 512 \(\times\) 512. For the maximum number of capacity, they are 0.0204 and 0.0547 bit/pixel with the method in [10] and with our scheme, respectively, leading to the enhancement of \(\left\{ \begin{matrix}0.0546\\ 0.0204\end{matrix} -1\right\}\)\(\times\)100% = 168%. With this capacity, the output image with our scheme obtains the better quality of 2.93 dB in difference. Because the MSE and BER values between the two schemes are 0.0, it implies that reversibility can be retained with the two schemes. In short, our scheme outperforms that in [10] with the statistics in Table 3.  
Corresponding to the results represented in Table 2, we demonstrate the distortion-capacity performances in Figure 7 for the better understanding of the results. In Figure 7, the blue curve presents the distortion-capacity curve with the modified conventional scheme in [8], and the red dot denotes the result for quadtree decomposition with our scheme. We modify conventional scheme in [8] by using regular block sizes of 512 \(\times\) 512, 256 \(\times\) 256, 128 \(\times\) 128, 64 \(\times\) 64, 32 \(\times\) 32, or 16 \(\times\) 16, and treat each block as a small image for data embedding for making fair comparisons with the proposed scheme. In Figure 7(a,c-e) for airplane, Lena, pepper, and tank, respectively, proposed results outperform those with conventional scheme. For the F-16 and truck images in Figure 7(b,f), the modified conventional scheme performs a bit better than our scheme. This might be because the side information cancelled out the gain in the real capacity and corresponding output image quality. In short, the block map in quadtree decomposition is directly affected by the characteristics of cover image. The enhancement in capacity with our scheme might be balanced by the increase of size in block map. And we need to mention again that the output image quality among all the six images are all more than 48.13 dB, as we discussed earlier in Section 2.