## 3 Experimental Results
### Embedding Rate Comparisons  
In this section, the embedding rate performances of our proposed scheme and four state-of-the-art schemes, [PERSON] et al. [27], [PERSON] et al. [28], [PERSON] [29], [PERSON] et al. [30], [PERSON] et al. [31], and [PERSON] et al. [32], are compared. As scheme [32] also gathers every two vertices into one group, we first compare the efficiency of our proposed closest pair prediction and the relevant techniques in [32]. Table 3 lists the embedding rates of two schemes when \(\alpha=2\).  
Figure 5: Embedding rates of four test models under different \(\alpha\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**3D Models** & [32] & **The Proposed Scheme** & **Increment** \\ \hline Dog & 24.22 & 27.93 & 3.71 \\ Face & 23.05 & 26.86 & 3.81 \\ Spaceship & 28.99 & 31.10 & 2.11 \\ Car & 16.06 & 19.36 & 3.30 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Embedding Rates of Two Schemes When \(\alpha=2\).  
Obviously, the embedding rates of our proposed scheme are higher than those in [32], and the average increment achieves to 3.23 bpv. In [32], the majority voting method is used to predict binary three-dimensional coordinates of each usable vertex by its connected reference vertices. As the reference vertices connected to the currently useable vertex are not necessarily the closest to it, but each reference vertex has the same weight when majority voting, the result is some usable vertices in the edge area have poor prediction accuracy. For the CPP technique of our proposed scheme, we use only the closest reference vertex to predict corresponding usable vertex and only pay a little cost to record which the closest vertex is. Therefore, the prediction accuracy of our proposed scheme is higher and the embedding rates have superiority.  
Next, because our proposed multi-group partition technique will further improve the embedding rate, we adaptively chose a different value of \(\alpha\) for different 3D mesh models to achieve best performance for the rest comparisons. As shown in Figure 6, four test models were first used for comparison. In the first two schemes, homomorphic encryption was adopted, ensuring constant embedding rates. The performances of rest schemes were affected by the complexity of each 3D mesh models. The MSB prediction method was used in [30; 31; 32]. The scheme [30] only vacated the redundancy of the first MSB in each vertex, so that the embedding rates were constrained below 3 bpv. The majority voting method was used in [31; 32], and only the number of usable vertices was different in the two schemes. However, the majority voting method did not seem to be optimal for three-dimensional coordinate prediction, and the reference vertices seemed superfluous. In our proposed scheme we first improved the prediction method and gathered more vertices into one group to further vacate more spare room. Therefore, the embedding rate of our proposed scheme has evident superiority compared to other state-of-the-art schemes.  
We also evaluated the embedding rate on the whole PSR&AG dataset to test the practicality of our proposed scheme. As shown in Figure 7, the average embedding rates of our proposed scheme and state-of-the-art schemes are given. The embedding rates of only the proposed scheme exceed 35 bpv, and the increment rises to 10.31 bpv compared to the suboptimal scheme.  
Figure 6: Embedding rate comparison of four test models [27; 28; 29; 30; 31; 32].