# A Vertex-Based 3D Authentication Algorithm Based on Spatial Subdivision

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Vertex-Based 3D Authentication Algorithms

#### 2.2. Tsai et al.’s Tree-Based Information Hiding Algorithm

## 3. The Proposed Algorithm

#### 3.1. Preprocessing Process

_{N}and the number of polygons be ${P}_{N}$. After reading the information of vertex ${V}_{i}$ and polygon ${P}_{K}$, a bounding volume of the 3D model is established on the basis of Cartesian coordinates. The magnitude of the bounding volume, $D{L}_{BV}$, is determined using the distance between the boundary points $B{P}_{M}$ and $B{P}_{m}$ as follows:

#### 3.2. Vertex Encoding Process

#### 3.3. Authentication Code Generation Process

_{2}= (1$\underset{\_}{6}$1)

_{10}, generating the decimals 0.1, 0.6, and 0.1; and the encoding result of ${P}_{8}$ is converted to (111$\underset{\_}{001}$00)

_{2}= (7$\underset{\_}{1}$0)

_{10}, producing the decimals 0.7, 0.1, and 0.0.

#### 3.4. Authentication Code Embedding Process

## 4. Experimental Evaluations

## 5. Conclusions and Future Studies

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Examples of Tsai et al.’s algorithm: (

**a**) the BSP spatial subdivision result; (

**b**) the BSP tree structure.

**Figure 4.**Visual effects of our test models: (

**a**) Bunny Model; (

**b**) Cow Model; (

**c**) Golf ball Model; (

**d**) Lucy Model.

– | Length | |||||
---|---|---|---|---|---|---|

Embedding Threshold | – | 0.005 | 0.003 | 0.001 | 0.0005 | 0.0003 |

Model | Bunny | 24 | 26 | 30 | 33 | 36 |

Cow | 24 | 27 | 30 | 33 | 36 | |

Golf ball | 24 | 26 | 30 | 33 | 36 | |

Lucy | 23 | 26 | 30 | 33 | 35 |

– | Model Distortion | |||||
---|---|---|---|---|---|---|

Embedding Threshold | – | 0.005 | 0.003 | 0.001 | 0.0005 | 0.0003 |

Model | Bunny | 0.148% | 0.097% | 0.037% | 0.018% | 0.009% |

Cow | 0.148% | 0.075% | 0.037% | 0.019% | 0.009% | |

Golf ball | 0.150% | 0.106% | 0.037% | 0.019% | 0.009% | |

Lucy | 0.166% | 0.083% | 0.037% | 0.019% | 0.010% |

– | Number | |||||
---|---|---|---|---|---|---|

Embedding Threshold | – | 0.005 | 0.003 | 0.001 | 0.0005 | 0.0003 |

Model | Bunny | 57.96 | 63.81 | 68.87 | 78.81 | 105.64 |

Cow | 57.97 | 65.39 | 70.18 | 89.15 | 105.89 | |

Golf ball | 68.66 | 73.64 | 79.87 | 90.97 | 115.35 | |

Lucy | 60.63 | 65.31 | 73.56 | 89.68 | 105.30 |

Algorithm | [17] | [18] | [19] | [20] | [21] | [22] | [23] | [24] | Proposed |
---|---|---|---|---|---|---|---|---|---|

Blind extraction | Semi-blind | Blind | Semi-blind | Semi-blind | Blind | Blind | Blind | Blind | Blind |

Embedded ratio | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}$ | ${N}_{V}-6$ |

Embedding capacity | 1 | 48 | 1 | 1 | 3 | 1 | 1 | 32 | 36 |

Distortion control | Yes | Yes | No | Yes | No | No | No | No | Yes |

Robustness * | VR | R, S, T | R, S, T | VR | VR | None | T | None | T, S, VR |

Embedding method | QIM | QIM | CM | QIM | HC | LSBs | LSBs | LSBs | MC |

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**MDPI and ACS Style**

Tsai, Y.-Y.; Tsai, Y.-S.; Chi, I.-T.; Chan, C.-S.
A Vertex-Based 3D Authentication Algorithm Based on Spatial Subdivision. *Symmetry* **2018**, *10*, 422.
https://doi.org/10.3390/sym10100422

**AMA Style**

Tsai Y-Y, Tsai Y-S, Chi I-T, Chan C-S.
A Vertex-Based 3D Authentication Algorithm Based on Spatial Subdivision. *Symmetry*. 2018; 10(10):422.
https://doi.org/10.3390/sym10100422

**Chicago/Turabian Style**

Tsai, Yuan-Yu, Yu-Shiou Tsai, I-Ting Chi, and Chi-Shiang Chan.
2018. "A Vertex-Based 3D Authentication Algorithm Based on Spatial Subdivision" *Symmetry* 10, no. 10: 422.
https://doi.org/10.3390/sym10100422