## 5 Experimental Results
### Comparison with Alternative Methods  
To further evaluate the performance of the proposed method, we compare it with methods [3; 11; 12; 14; 24] in terms of imperceptibility and robustness. It can be seen from Table 13 that the proposed method outperforms the imperceptibility of the schemes [14; 24], and [11] in terms of MRMS and MSDM. In addition, the imperceptibility performance in terms of the HD of the proposed technique is compared to the scheme [12]. From Table 14, it can be highlighted that the proposed method achieves good imperceptibility performance. The reported results in terms of MRMS and HD illustrates the invisibility of the proposed method. Moreover, our scheme outperforms [PERSON] et al. 's [3] scheme for MRMS and HD for the three objects Venus, Horse, and Rabbit. Table 15 sketches the imperceptibility comparison with [14] and [12] schemes in terms of the Hausdorff distance for Bunny, Venus, and Horse models. As depicted in Table 15, the proposed technique achieves good results since all the obtained HD values are less than 1.71. Moreover, the proposed method outperforms [PERSON] et al. 's [14] and [PERSON] et al. 's [12] schemes. In addition, as depicted in Table 16, the proposed method shows high robustness to different attacks including \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Subdivision** & **Flower** & **Vase** & **Cup** & **Ant** & **Cat** & **Bimba** \\ \hline Loop iter 1 & 1.0 & 1.0 & 1.0 & 0.98 & 0.98 & 0.99 \\ Midpoint iter 1 & 0.94 & 0.90 & 0.91 & 0.92 & 0.95 & 0.96 \\ Sqrt3 iter 1 & 0.99 & 0.97 & 0.96 & 0.99 & 1.0 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Watermark robustness against subdivision measured in terms of correlation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **MRMS (10\({}^{-3}\))** & **MSDM** \\ \hline
[14] & 3.17 & 0.3197 \\
[24] & 1.48 & 0.2992 \\
[11] & 2.90 & 0.3197 \\ Proposed method & 0.38 & 0.2254 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Imperceptibility comparison with [PERSON]’s [14], [PERSON]’s [24] and [PERSON]’s [11] schemes measured in terms of MRMS and MSDM for the Horse model. noise addition and smoothing, and outperforms the schemes in [14] and [12], except with respect to cropping attack, [PERSON] et al. 's scheme [12] shows high robustness compared to our method. We believe that this is due to the redundant insertion of the watermark in several patches. It can be seen in Figure 12 that the proposed scheme shows high robustness to several attacks including noise addition, quantization, and smoothing. Moreover, our methods achieves high robustness compared to [PERSON]'s method [14] in terms of normalized correlation. The robustness of our method in terms of normalized correlation (NC) was compared to [PERSON] et al. 's [3] method for noise addition, quantization, and smoothing attacks with different parameters. For noise addition, three amplitudes were used in the comparison. Quantization was applied with 9 bits, 8 bits, and 7 bits. Regarding smoothing attack, a watermarked rabbit has undergone this process using a fixed deformation factor \(\lambda=0.10\) with different iteration numbers. Figure 13 highlighted the obtained results for the Rabbit object. It can be observed from Figure 13 that the proposed method shows high robustness compared to [PERSON]'s method in terms of NC for noise addition, quantization, and smoothing. Moreover, as depicted in Table 17, the proposed method is able to withstand a smoothing attack, and the obtained results in terms of BER show the superiority of our method compared to [PERSON]'s method [11]. Tables 18-20 show the quality and robustness comparison with [PERSON] et al. 's method [3] against noise addition, quantization, and smoothing for Venus and Horse, respectively. The quality evaluation metrics used in the comparison are MRMS, HD, and MSDM, while the robustness is evaluated using NC. Regarding noise addition, Horse and Venus were chosen as comparison meshes, and three amplitudes were used in the comparison 0.05%, 0.25%, and 0.5%.