# A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering
## 4 Experimental Results and Analysis
### Performance Evaluation
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name Model** & **of Facets** & **of Groups** & **Accuracy (\%)** \\ \hline Scanned Orient Tube & 112 & 19 & 68.42 \\ Scanned Orient Holder & 126 & 21 & 52.38 \\ Scanned Number 7 & 130 & 21 & 74.42 \\ Scanned Fidget & 274 & 31 & 51.61 \\ Scanned Valve Tube & 830 & 64 & 50.00 \\ Scanned Ptico & 1859 & 232 & 54.31 \\ Scanned Diamond Grip & 2434 & 277 & 50.90 \\ Scanned Holder & 3196 & 309 & 51.78 \\ Scanned Lion & 4682 & 384 & 52.34 \\ Scanned 3D Printer & 14,733 & 887 & 50.62 \\ \hline \hline \end{tabular}  
Average accuracy \textgreater{}55%.  
\end{table} 
Table 2: Accuracy of the extracted watermark. and two previous methods. Figure 9 shows the performance of the proposed method compared to the two previous methods of 3D printing watermarking. Consequently, the proposed method is better than the two methods of [PERSON] and [PERSON]. To compare the proposed method with the conventional works of 3D triangle mesh model watermarking, we also compare the accuracy between the methods. This accuracy is the accuracy of the extracted watermark from the watermarked 3D triangle mesh. In [PERSON]'s method [9], he embedded the watermark by modifying the selected vertices based on the topology of the 3D triangle mesh. He experimented with two 3D models and explained that his method can extract the embedded watermark when the watermarked 3D model is simplified by less than 5%. This means the maximum accuracy of [PERSON]'s method is 95%. The robustness of [PERSON]'s method is affected by geometric attacks (rotation, translation and scaling) and the perturbation of the order of facets, because it used vertices to embed watermark bits. In [PERSON]'s method [10], he also embedded watermark bits by modifying the vertex positions along the radial directions. He experimented on 13 models with noise of 0.1%, 1% and 5%, respectively. This method is also affected by geometric attacks and the perturbation of the order of facets. The maximum accuracy of [PERSON]'s method is 98%. In [PERSON]'s method [12], he performed the watermark embedding process by modifying the input model according to the changes of the histogram of the x-y plane projected face normal vector from the input model. He experimented with four 3D triangle meshes, and the average bit error rate (BER) is approximate 4%. This means the average accuracy of [PERSON]'s method is 96%. [PERSON]'s method is affected by rotation, scaling and the perturbation of the order of facets. In our method, the watermark embedding process is based on changing the mean Menger curvature of groups of facets in the 3D triangle mesh. The Menger curvature of a facet is robust to geometric attacks and independent of the order of facets. This means the perturbation of facets does not affect the Menger facet curvatures. This leads to the clustering \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method No.**} & **Watermarking** & **Number of** & **Length of** & **Accuracy (\%)** \\  & **Domain** & **Test Models** & **Watermark Bit** & **Accuracy (\%)** \\ \hline [PERSON] method & Frequency & 3 & 256 & 40 \\ [PERSON]’s method & Spatial & 2 & 64 & \(\sim\)0 \\ Proposed method & Spatial & 10 & Flexible & 74.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between methods. Figure 9: Performance of the proposed method compared to the two previous methods. process not being affected by the perturbation of facets. Thus, the embedded watermark is robust to the perturbation. Here, we conclude that our method is robust to geometric attacks and perturbation. To compare to the conventional works of 3D model watermarking, we extracted the embedded watermark from the watermarked 3D triangle meshes after the watermark embedding process (refer to Figure 2). The average accuracy of the extracted watermark from the watermarked 3D triangle meshes is greater than 98.5%. This means the accuracy of our method is better than the accuracy of the conventional works of 3D model watermarking.