# A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering
## 4 Experimental Results and Analysis
### Robustness Evaluation and Analysis  
The security of watermarking methods is due to their robustness. If a watermarking method is robust, it is difficult to remove the watermark embedded into objects. This means the original provider easily protects the copyright of their products. To evaluate the robustness of the proposed method, we would like to analyze two aspects. The first aspect is the robustness of the watermarked 3D triangle meshes with geometric attacks, such as rotation, translation and scaling. As we explained in Section 2.3, the Menger curvature of a facet is robust to rotation and translation because these attacks only change the spatial location of 3D triangle meshes. With scaling attack, it changes the size of 3D triangle meshes. To re-scale, we only find the highest vertex and the lowest vertex on the original 3D triangle mesh and then calculate the distance between these vertices. With the scaled 3D triangle mesh, we also perform a similar calculation. Finally, we compare the distances to find the scale-ratio for the rescaling process. Regarding the second aspect, we evaluate and analyze the accuracy of the extracted watermark from 3D-printed objects comparing it to the original watermark. If the accuracy of the extracted watermark from 3D-printed objects is high, the robustness of the proposed method is high and otherwise low. To extract the watermark data from 3D-printed objects, we have to use a 3D scanner to scan 3D-printed objects and extract the watermark data from the scanned 3D triangle meshes. Due to the fact that geometric attacks, such as translation, rotation and scaling, do not affect the 3D scanning process and we always receive the scanned 3D triangle meshes, which are the same size as 3D-printed objects, we concluded that geometric attacks do not affect the scanned 3D triangle mesh. Moreover, the Menger curvature is robust to translation and rotation attacks. Because these attacks only change the spatial location of the 3D printing model, they do not change the shape of the 3D printing model. Consequently, the proposed method is robust to geometric attacks. Here, we used the XYZ Printing Pro 3 in1 3D printer [21] to print test models in Figure 5. Figure 7 shows the 3D-printed objects from test models. After the 3D printing process, we used the XYZ 3D Scanner [21] to scan 3D-printed objects and to construct the scanned 3D triangle meshes as shown in Figure 8. Due to the fact that the scanned 3D triangle mesh is affected by noise in the 3D scanning process, it is not perfectly the same as the original 3D triangle mesh. This means the number of facets in the scanned 3D triangle mesh is always different with the number of facets in the original 3D triangle  
Figure 6: Mean distance error according to the number of groups. mesh (see Tables 1 and 2). Therefore, to calculate the accuracy between the extracted watermark from 3D-printed objects and the original watermark, we used an expression as shown in Equation (13). \[\text{Accuracy}=\frac{\text{Extracted watermark}}{\text{Original watermark}}\times 100\% \tag{13}\]  
Here, we would like to explain one more time the difference between 3D model watermarking and 3D printing watermarking to show the novelty. In 3D model watermarking, watermark data are extracted from the watermarked 3D model. In 3D printing watermarking, watermark data are extracted from 3D-printed objects via the scanned 3D triangle mesh of 3D-printed objects. Table 2 shows the accuracy of the proposed method with test models. It is formed from 50.00%-74.42% with the 3D-printed objects in Figure 7. Overall, we can see that the accuracy of the proposed method is higher than 55%. The accuracy of the proposed method is very much dependent on the scanned 3D triangle meshes. However, the scanned 3D triangle meshes are dependent on the resolution of the 3D-printed object and the quality of the 3D scanner. Thus, we concluded that the accuracy of the proposed method is dependent on the quality of the 3D printer and 3D scanner.