# A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. 3D Model Watermarking

#### 2.2. 3D Printing Watermarking

#### 2.3. Menger Curvature-Based 3D Printing Watermarking

## 3. The Proposed Algorithm

#### 3.1. Overview

#### 3.2. Watermark Embedding

**M**| is the number of facets in a 3D triangle mesh

**M**, and ${F}_{i}$ is the ${i}^{th}$ facet. Each facet contains three vertices (three points), ${F}_{i}=\left\{{v}_{ij}\right|j\in \left[1,3\right]\}$ and a normal vector ${n}_{i}(n{x}_{i},n{y}_{i},n{z}_{i})$. The Menger curvature ${K}_{i}$ of each facet ${F}_{i}$ is computed by its vertices and corresponding area as shown in Equation (2).

**M**| facets in the 3D triangle mesh

**M**are divided into G groups;$G=\left\{{m}_{g}\right|g\in \left[1,\left|G\right|\right]\}$ based on the value of the Menger curvature. Figure 3 shows the result of the facet clustering of the bunny triangle mesh based on Menger curvature. Facets in the same group will have the same color.

**M**| facets into G groups, we find the maximum Menger curvature and minimum Menger curvature of each group and calculate the mean Menger curvature of each group. Assume that ${K}_{max}^{{m}_{g}},{K}_{min}^{{m}_{g}}\mathrm{and}\text{}{K}_{mean}^{{m}_{g}}$ are the maximum Menger curvature, minimum Menger curvature and mean Menger curvature of the group ${m}_{g}$, respectively. The mean Menger curvature ${K}_{mean}^{{m}_{g}}$ of the group ${m}_{g}$ is the average value of all Menger curvatures in the group ${m}_{g}$ and calculated as shown in Equation (3) with $\left|{m}_{g}\right|$ the number of facets in the group ${m}_{g}$.

#### 3.3. Watermark Extracting

**M**’ to compute the Menger facet curvatures. After that, we classify them into groups by the K-mean clustering algorithm based on the value of Menger curvatures. The watermark key is re-used for the clustering process. For each group ${m}_{g}$, we find the maximum Menger curvature ${K}_{max}^{{{m}_{g}}^{\prime}}$ and the minimum Menger curvature ${K}_{min}^{{{m}_{g}}^{\prime}}$ and calculate the mean Menger curvature ${K}_{mean}^{{{m}_{g}}^{\prime}}$ similar to Equation (3). ${\Delta}_{mg}^{\prime}=({K}_{min}^{{{m}_{g}}^{\prime}}+{K}_{max}^{{{m}_{g}}^{\prime}})/2$ is the average value of ${K}_{min}^{{{m}_{g}}^{\prime}}\text{}\mathrm{and}{K}_{max}^{{{m}_{g}}^{\prime}}$. Finally, the watermark bit ${\omega}_{g}$ can be extracted by comparing the mean Menger curvature ${K}_{mean}^{{{m}_{g}}^{\prime}}$ with the average value ${\Delta}_{mg}^{\prime}$ as described in Equation (10).

## 4. Experimental Results and Analysis

**M**| as shown in Equation (11).

**M**|. For example, if $\left|M\right|=2146$, then $S=4$. To evaluate the proposed method, we evaluate the invisibility, robustness and performance of the proposed method. Section 4.1 shows the invisibility evaluation of the proposed method. The robustness of the proposed method is described in Section 4.2, and the performance of the proposed method is shown in Section 4.3.

#### 4.1. Invisibility Evaluation

^{−6}to 4.113 × 10

^{−6}. This proves that the difference between the watermarked 3D triangle mesh and the original 3D triangle mesh is very small. Therefore, it proves that the invisibility of the proposed method is very high. Based on Equation (12), we concluded that the mean distance error is dependent on the number of watermarked vertices and the number of vertices. The number of watermarked vertices is dependent on the number of groups and the mean Menger curvature of each group. Therefore, we concluded that the mean distance error (the invisibility of the proposed method) is dependent on the number of groups. From Table 1, we concluded that the mean distance error is decreased according to the number of groups. Figure 6 shows the mean distance error according to the number of groups.

#### 4.2. Robustness Evaluation and Analysis

#### 4.3. Performance Evaluation

^{−1}). In the second part, he experimented on three 3D triangle meshes: bunny, casting and hand with a 3D printer. He discussed that the percentage that is precise for a casting object is smaller than 10

^{−5}. Therefore, we concluded that the accuracy of Yamazaki’s method is approximate 40%. In addition, the risk of Yamazaki’s method is the length of the watermark bits is fixed and equal to 256 bits for all test models. As a result, there is a limitation of watermark bits in this method, and attackers can remove the embedded watermark more easily. In Suzuki’s method [14], the watermark data are embedded in the printed objects in the 3D printing process by a complex system of laser and halogen lights. This requires a complex hardware system, but it could not embed all expected watermark bits inside 3D-printed objects. Suzuki experimented on his method with two 3D triangle meshes. Following the experimental results of Suzuki, the maximum length of the embedded watermark bits was 64 bits. In addition, Suzuki did not describe how to extract the embedded watermark data from the 3D-printed objects. Therefore, we considered the accuracy of Suzuki’s method to be approximately 0%. In our method, the length of watermark bits is flexible and can be changed by the user based on the number of facets in each model (see Table 1 and refer to Section 4, Paragraph 1, and Equation (1)). This helps the users change the content of watermark bits according to their purpose. As we explained in Section 4.2, the accuracy of our method is dependent on the quality of the 3D printer and 3D scanner. The maximum accuracy of our method is 74.42% with the test models shown in Table 2. Table 3 describes the comparison between the proposed method 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 Yamazaki and Suzuki.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Name Model | of Facets | of Groups | Mean Distance Error |
---|---|---|---|

Orient Tube | 464 | 19 | 4.0952 × 10^{−6} |

Orient Holder | 520 | 21 | 4.0452 × 10^{−6} |

Number 7 | 526 | 21 | 4.0042 × 10^{−6} |

Fidget | 750 | 31 | 4.1330 × 10^{−6} |

Valve Tube | 2062 | 64 | 3.1083 × 10^{−6} |

Pitco | 7442 | 232 | 3.1329 × 10^{−6} |

Diamond Grip | 8870 | 277 | 3.1274 × 10^{−6} |

Holder | 12,392 | 309 | 2.4931 × 10^{−6} |

Lion | 15,366 | 384 | 2.4991 × 10^{−6} |

3D Printer | 35,482 | 887 | 2.4993 × 10^{−6} |

Name Model | of Facets | of Groups | Accuracy (%) |
---|---|---|---|

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 Pitco | 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 |

Method No. | Watermarking Domain | Number of Test Models | Length of Watermark Bit | Accuracy (%) |
---|---|---|---|---|

Yamazaki’ method | Frequency | 3 | 256 | 40 |

Suzuki’s method | Spatial | 2 | 64 | ~0 |

Proposed method | Spatial | 10 | Flexible | 74.42 |

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**MDPI and ACS Style**

Pham, G.N.; Lee, S.-H.; Kwon, O.-H.; Kwon, K.-R.
A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering. *Symmetry* **2018**, *10*, 97.
https://doi.org/10.3390/sym10040097

**AMA Style**

Pham GN, Lee S-H, Kwon O-H, Kwon K-R.
A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering. *Symmetry*. 2018; 10(4):97.
https://doi.org/10.3390/sym10040097

**Chicago/Turabian Style**

Pham, Giao N., Suk-Hwan Lee, Oh-Heum Kwon, and Ki-Ryong Kwon.
2018. "A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering" *Symmetry* 10, no. 4: 97.
https://doi.org/10.3390/sym10040097