## 5 Experimental Results
### Robustness  
The resistance of the proposed watermarking method is tested under several attacks including element reordering, noise addition, smoothing, quantization, similarity transformations translation, rotation, and uniform scaling) and cropping. To this end, [PERSON] et al. 's benchmarking system has been used [23]. Figure 11 shows the Bimba object after several attacks. The robustness is measured using the normalized correlation \(NC\) between the extracted watermark bit and the original one. The noise addition attack aims to add a pseudo-random noise on vertex coordinates. The robustness against this attack is primordial since it simulates the artifacts induced during mesh transmission. We added random noise to each vertex of the original 3D meshes using several intensities 0.05%, 0.10%, 0.30%, and 0.50%. Table 5 sketches the robustness results in terms of NC for six 3D \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **MRMS (\(\mathbf{10^{-3}}\))** & **HD (\(\mathbf{10^{-3}}\))** & **MSDM** \\ \hline Flower & 0.84/**0.43** & 4.88/**3.23** & 0.78/**0.24** \\ Vase & 0.55/**0.32** & 4.67/**2.86** & 0.75/**0.34** \\ Cup & 0.09/**0.78** & 3.22/**2.58** & 0.81/**0.31** \\ Ant & 0.77/**0.41** & 3.36/**3.87** & 0.89/**0.38** \\ Cat & 0.95/**0.49** & 1.8/**0.78** & 0.28/**0.18** \\ Bimba & 0.74/**0.31** & 2.88/**1.41** & 0.61/**0.12** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Watermark imperceptibility without and with saliency measured in terms of MRMS, HD, and MSDM. Figure 10: Closeups of the original and watermarked meshes: (**a**) Original Bunny; (**b**) Watermarked Bunny; (**c**) Original Bimba; (**d**) Watermarked Bimba; (**e**) Original Cat; (**f**) Watermarked Cat. meshes. It can be seen from Table 5 that the proposed method is robust against noise addition for all the 3D test meshes. The robustness performance of the proposed method was also tested against smoothing attack. This is a common operation aiming to remove the noise caused by the generation of the mesh. The smoothing applied to the watermarked 3D meshes is Laplacian smoothing [15] with a fixed deformation factor (\(\lambda=0.1\)) with several amounts of iterations (5, 10, 30, and 50). Table 6 highlights the robustness evaluation of the proposed method in terms of NC. According to Table 6, it can be seen that our method can achieve high performance against this attack. Even with 50 iterations, the obtained NC values for the six objects are above 0.88. The robustness performance has been investigated for element reordering attack in which the vertex/ facets are reordered. It clearly appears from Table 7 that the method shows good robustness against element reordering manipulation for the three types of this attack presented in [PERSON] et al. 's benchmark [23]. Figure 11: Original Bimba and seven attack versions: (**a**) the original Bimba; (**b**) a noise addition of 0.50%; (**c**) a smoothing of \(\lambda=0.1\) with 5 iterations; (**d**) a quantization of 9 bits; (**e**) similarity transformation; and (**f**) a cropping ratio of 10.0. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Noise Intensity** & **Flower** & **Vase** & **Cup** & **Ant** & **Cat** & **Bimba** \\ \hline
0.05\% & 1.0 & 1.0 & 0.98 & 0.99 & 0.98 & 1.0 \\
0.10\% & 0.98 & 0.96 & 0.96 & 0.95 & 0.93 & 0.95 \\
0.30\% & 0.86 & 0.88 & 0.86 & 0.92 & 0.91 & 0.90 \\
0.50\% & 0.83 & 0.77 & 0.79 & 0.81 & 0.83 & 0.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Watermark robustness against additive noise measured in terms of correlation. Quantization is a common operation used mainly in mesh compression. The watermarked 3D meshes have undergone this attack with several parameters (11 bits, 10 bits, 9 bits, 8 bits, and 7 bits), and the robustness performance in terms of normalized correlation is presented in Table 8. According to these results, it can be concluded that our method is able to withstand quantization attack. Table 9 presents the robustness comparison between [PERSON]'s method and the proposed method in terms of bit error rate (BER).