## 5 Experimental Results
### Robustness
It can be seen from Table 9 that the proposed method outperforms the scheme in [11]. Similarity transformation is a common attack that every robust technique should be able to resist. It includes rotation, translation, and uniform scaling and their combination while keeping the mesh shape intact. Table 10 sketches the NC values after carrying out three types of similarity transformations. It can be remarked that the watermark is robust against these attacks regardless of the used 3D mesh. Subdivision is one of the most used geometric attacks that objects can undergo. It consists of adding vertices and edges to the 3D mesh in order to obtain a modified version of a higher visual quality containing more details. The 3D test meshes were exposed to three typical subdivision schemes using one iteration. It includes a loop scheme, a midpoint scheme, and a \(\sqrt{3}\) scheme. Table 11 presents the obtained results in terms of NC after applying the schemes mentioned before. According to Table 11, it can be observed that our method is quite robust to subdivision attacks for the three scheme and for all the 3D meshes used in the experiments. In addition, all the NC values are above 0.9. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number of Iterations** & **Flower** & **Vase** & **Cup** & **Ant** & **Cat** & **Bimba** \\ \hline
5 & 1.0 & 0.99 & 1.0 & 0.99 & 1.0 & 1.0 \\
10 & 1.09 & 0.99 & 0.98 & 1.0 & 0.98 & 0.98 \\
30 & 0.99 & 0.97 & 0.93 & 0.95 & 0.94 & 0.94 \\
50 & 0.93 & 0.92 & 0.91 & 0.88 & 0.90 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Watermark robustness against Laplacian smoothing (\(\lambda=0.1\)) measured in terms of correlation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Elements Reordering** & **Flower** & **Vase** & **Cup** & **Ant** & **Cat** & **Bimba** \\ \hline Element Reordering 1 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ Element Reordering 2 & 0.99 & 0.98 & 1.0 & 0.97 & 1.0 & 0.99 \\ Element Reordering 3 & 1.0 & 1.0 & 0.97 & 1.0 & 1.0 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Watermark robustness against element reordering measured in terms of correlation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Quantization** & **Flower** & **Vase** & **Cup** & **Ant** & **Cat** & **Bimba** \\ \hline
11 bits & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
10 bits & 1.0 & 1.0 & 0.99 & 1.0 & 0.97 & 1.0 \\
9 bits & 0.99 & 0.99 & 0.97 & 0.99 & 1.0 & 0.98 \\
8 bits & 0.95 & 0.94 & 0.93 & 0.94 & 0.96 & 0.96 \\
7 bits & 0.83 & 0.83 & 0.81 & 0.82 & 0.82 & 0.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Watermark robustness against quantization measured in terms of correlation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Quantization** & **[11]** & **Our Method** \\ \hline
7 bit & 0.23 & 0.14 \\
8 bit & 0.15 & 0.09 \\
9 bit & 0.14 & 0.06 \\
10 bit & 0.03 & 0.02 \\
11 bit & 0.01 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of comparison with the robustness of [PERSON] et al.’s [11] method measured in terms of BER against quantization. Cropping attack is considered one of the most severe attacks that 3D can suffer from. This manipulation consists of cutting one part or several parts of the 3D mesh. This attack has been applied to the 3D objects using different ratios (10, 30, and 50). As depicted in Table 12, it can be concluded that the resistance of the proposed method against this attack shows relative weakness. We believe that this is due to the fact that the cropped regions could contain salient points that have been used to choose the wavelet coefficients to be quantized.