# Three-Dimensional Tooth Model Reconstruction Using Statistical Randomization-Based Particle Swarm Optimization

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}and 7.41 micrometer

^{2}for both models, respectively. From the results compared with the particle swarm optimization (PSO) algorithm with the ICP method, it can be seen that the results from the proposed algorithm are much better than those from the PSO algorithm with the ICP method.

## 1. Introduction

## 2. Registration Method

**P**is the target point-cloud matrix ([

**p**

_{i}]

_{M}

_{×}

_{4}, M is the number of target point-cloud points),

**Q**is the source point-cloud matrix ([

**q**

_{j}]

_{N}

_{×}

_{4}, N is the number of source point-cloud points), and

**H**is the geometry transform. Finally, O(⋅) is an objective function. Since the transformation

**H**is estimated by finding the nearest neighbor [26] between a set of point-pairs (

**p**

_{j},

**q**

_{j}), the minimum error of the distance between two corresponding points can be considered [27]. Using the mean squared error (MSE), hence, the minimization problem, in this case, is calculated as

**H**is the 3-D transformation with 15 unknown parameters, i.e., 3 parameters from scaling (

**S**), 3 parameters from translation (

**T**), 3 parameters from rotation (

**R**), and 6 parameters from shearing (

**SH**) [28]. The matrix

**H**is computed as

**H**=

**T**×

**S**×

**R**×

**SH**,

_{x}, t

_{y}, and t

_{z}, respectively. Moreover, p, q, and r are set to 0 since they are perspective property values. Finally, s is always set to 1 because of the scaling factor.

**H**. Each individual in the swarm has 15 dimensions. The search space is defined as shown in Table 1.

#### Statistical Randomization-Based PSO (SR-PSO) Algorithm

**X**= $\left\{{\mathit{x}}_{j}|j=1\dots N\right\}$ be a set of N particles in the swarm in d-dimensional feature space. ${\mathit{x}}_{i}^{b}$ and

**x**

^{g}are the individual best of the i

^{th}particle and the global best of the swarm, respectively. The update equations for velocity and position of each particle are

_{1}and r

_{2}are randomly generated numbers from the uniform distribution within [0, 1]. c

_{1}and c

_{2}are the acceleration coefficients, and w is the inertia weight calculated by [29,30,31]

_{max}= χ, T is the number of iterations, α = 1 (normally 0 < α ≤ 2), and

_{min}, v

_{max}] where

^{th}dimension of intermediate particle $\left({x}_{j}^{tmi}\right)$ with K being the number of particles in the swarm,

- −
- Calculated from an average of all individual best $\left({x}_{ij}^{b}\right)$:$${x}_{j}^{tm1}=\frac{{\displaystyle {\sum}_{i=1}^{K}{x}_{ij}^{b}}}{K}$$
- −
- Calculated from a median of all individual best $\left({x}_{ij}^{b}\right)$:$${x}_{j}^{tm2}=\underset{1\le i\le K}{\mathrm{median}}({x}_{ij}^{b})$$
- −
- Calculated by random generate number from Gaussian distribution with mean and standard deviation computed from individual best positions:$${x}_{j}^{tm3}={\sigma}_{j}\times Z+{\mu}_{j},\mathrm{when}Z\sim N(0,1)$$
- −
- Calculated from the larger absolute value of the maximum and the minimum in that dimension:$${x}_{j}^{tm4}=\{\begin{array}{l}\underset{1\le i\le K}{\mathrm{min}}({x}_{ij}^{b})\mathrm{if}\left|\underset{1\le i\le K}{\mathrm{min}}({x}_{ij}^{b})\right|\left|\underset{1\le i\le K}{\mathrm{max}}({x}_{ij}^{b})\right|\\ \underset{1\le i\le K}{\mathrm{max}}({x}_{ij}^{b})\mathrm{else}\end{array}$$
- −
- Calculated from the smaller absolute value of the maximum and the minimum in that dimension:$${x}_{j}^{tm5}=\{\begin{array}{l}\underset{1\le i\le K}{\mathrm{min}}({x}_{ij}^{b})\mathrm{if}\left|\underset{1\le i\le K}{\mathrm{min}}({x}_{ij}^{b})\right|\left|\underset{1\le i\le K}{\mathrm{max}}({x}_{ij}^{b})\right|\\ \underset{1\le i\le K}{\mathrm{max}}({x}_{ij}^{b})\mathrm{else}\end{array}$$

^{th}intermediate particle (

**x**

^{tml}) is the best particle among the intermediate particles and it is better than the global particle, then the global best particle will be replaced by

**x**

^{tml}.

**x**

^{g}, we randomly select the individual best (${\mathit{x}}_{k}^{b}$) from all particles (instead of randomly selecting a particle except the best one as in [23]) using

_{i}, ub

_{i}], where lb

_{i}and ub

_{i}are the lower and upper bounds in the search space in the i

^{th}dimension. That is

**x**

^{g}, it is then updated using

## 3. Experimental Results

**H**

^{−1}(transform from source point-cloud to target point-cloud). The ICP method is used to fine-tune the resultant

**H**

^{−1}. Finally, the 3-D tooth models are reconstructed based on the registered source and target point-clouds.

^{−31}at the 2000th iteration. Figure 4 shows the registration progress image of the global best particle in the 1st, 50th, 500th, and 2000th iterations. It can be seen that the registration result in the last iteration is an almost perfect match.

^{−31}. The best final registration result is also shown in Figure 5.

_{min}, w

_{max}, c

_{1}, and c

_{2}were set to 0.4, 0.9, 2, and 2, respectively.

^{2}. When considering the final registration results of six consecutive views shown in Table 7, we find that the best result from the SR-PSO algorithm with the ICP method and α = 1.5 is 7.3666 micrometer

^{2}whereas that from the PSO algorithm with the ICP method is 17.1150 micrometer

^{2}. The result from the SR-PSO algorithm with the ICP method is better than that from the PSO algorithm with the ICP method. The final registration result of the regular-tooth model is shown in Figure 11. We can see that the final registration image provides a good visualization to human eyes.

^{2}whereas the comparable error of 7.4672 micrometer

^{2}is achieved by the PSO algorithm with the ICP method. We can also see in Figure 14 that the final 3-D registration image of the orthodontic-tooth model also provides a good visualization to human eyes.

## 4. Conclusions

^{2}and 7.4130 micrometer

^{2}, respectively. For the sake of comparison, the MSEs from the PSO algorithm with the ICP method for the two tooth models were 17.115 micrometer

^{2}and 7.4672 micrometer

^{2}, respectively. The SR-PSO algorithm with the ICP method provided better results than that of the PSO algorithm with the ICP method. Moreover, the resulting 3-D images from the SR-PSO algorithm with the ICP method are also viewable by human eyes and are useful for experts. In future work, we will implement this algorithm for the tooth models with defections in order to simulate dental caries in real situations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sivakumar, A.; Thangaswamy, V.; Ravi, V. Treatment planning in conservative dentistry. J. Pharm. Bioallied Sci.
**2012**, 4, S406–S409. [Google Scholar] [CrossRef] [PubMed] - Hugar, S.M.; Sogi, H.P.S.; Nalawade, T.M.; Sinha, A.; Hugar, S.; Mallikarjuna, R.M. Knowledge, attitude, and practices of oral health care in prevention of early childhood caries among parents of children in Belagavi city: A Questionnaire study. J. Fam. Med. Prim. Care
**2016**, 5, 286–290. [Google Scholar] [CrossRef] [PubMed] - Yanagisawa, R.; Sugaya, Y.; Kasahara, S.; Omachi, S. Tooth shape reconstruction from dental CT images with the region-growing method. Dentomaxillofacial Radiol.
**2014**, 43, 20140080. [Google Scholar] [CrossRef][Green Version] - Zhou, X.; Gan, Y.; Xiong, J.; Zhang, D.; Zhao, Q.; Xia, Z. A Method for Tooth Model Reconstruction Based on Integration of Multimodal Images. J. Heal. Eng.
**2018**, 2018, 1–8. [Google Scholar] [CrossRef] [PubMed] - Zhang, D.; Gan, Y.; Xiong, J.; Xia, Z. Three-dimensional tooth model reconstruction based on fusion of dental computed tomography images and laser-scanned images (Chineses Article). Shengwu Yixue Gongchengxue Zazhi
**2017**, 34, 7–14. (In Chinese) [Google Scholar] [PubMed] - Yau, H.-T.; Yang, T.-J.; Chen, Y.-C. Tooth model reconstruction based upon data fusion for orthodontic treatment simulation. Comput. Biol. Med.
**2014**, 48, 8–16. [Google Scholar] [CrossRef] [PubMed] - Srisilapanan, P.; Nirunsittirat, A.; Roseman, J. Trends over Time in Dental Caries status in Urban and Rural Thai Children. J. Clin. Exp. Dent.
**2017**, 9, e1201–e1206. [Google Scholar] [CrossRef] - Abdel-Basset, M.; Fakhry, A.E.; El-henawy, I.; Qiu, T.; Sangaiah, A.K. Feature and Intensity Based Medical Image Registra-tion Using Particle Swarm Optimization. J. Med. Syst.
**2017**, 41, 197. [Google Scholar] [CrossRef] - Sarvamangala, D.R.; Kulkarni, R.V. Swarm Intelligence Algorithms for Medical Image Registration: A Comparative Study. Commun. Comput. Inf. Sci.
**2017**, 776, 451–465. [Google Scholar] [CrossRef] - Khan, M.K.; Nystrom, I. A Modified Particle Swarm Optimization Applied in Image Registration. In Proceedings of the 2010 20th International Conference on Pattern Recognition, Istanbul, Turkey, 23–26 August 2010; IEEE: Piscataway Township, NJ, USA, 2010; pp. 2302–2305. [Google Scholar]
- Jara, R.I.; Imbachí; Buchelly, F.J.; Meschino, G.; Ballarin, V.L. Improved Particle Swarm Optimization algorithm applied to rigid registration in medical images. In VII Latin American Congress on Biomedical Engineering CLAIB, Proceedings of the VII Latin American Congress on Biomedical Engineering CLAIB 2016, Bucaramanga, Santander, Colombia, 26–28 October 2016; Torres, I., Bustamante, J., Sierra, D., Eds.; IFMBE Proceedings; Springer: Singapore, 2016; Volume 60, pp. 161–164. [Google Scholar]
- Yonghong, Y.; Jiying, L.; Qiang, W.; Tao, Z. Improved Particle Swarm Optimization Image Registration Based on Mutual Information. In Proceedings of the 2019 11th International Conference on Measuring Technology and Mechatronics Automation (ICMTMA), Qiqihar, China, 28–29 April 2019; IEEE: Piscataway Township, NJ, USA, 2019; pp. 450–453. [Google Scholar]
- Wachowiak, M.P.; Smolíková, R.; Zheng, Y.; Zurada, J.M.; Elmaghraby, A.S. An Approach to Multimodal Biomedical Im-age Registration Utilizing Particle Swarm Optimization. IEEE Trans. Evol. Comput.
**2004**, 8, 289–301. [Google Scholar] [CrossRef] - Chen, Y.-W.; Mimori, A. Hybrid Particle Swarm Optimization for Medical Image Registration. In Proceedings of the 2009 Fifth International Conference on Natural Computation, Tianjian, China, 14–16 August 2009; IEEE: Piscataway Township, NJ, USA, 2009; Volume 6, pp. 26–30. [Google Scholar]
- Zhan, X.; Cai, Y.; He, P. A three-dimensional point cloud registration based on entropy and particle swarm optimization. Adv. Mech. Eng.
**2018**, 10, 1–13. [Google Scholar] [CrossRef] - Zhan, X.; Cai, Y.; Li, H.; Li, Y.; He, P. A point-cloud registration algorithm based on normal vector and particle swarm optimization. Meas. Control.
**2019**, 53, 1–11. [Google Scholar] [CrossRef] - Ge, Y.; Wang, B.; Nie, J.; Sun, B. A point cloud registration method combining enhanced particle swarm optimization and iterative closest point method. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28–30 May 2016; IEEE: Piscataway Township, NJ, USA, 2016; pp. 2810–2815. [Google Scholar]
- Yousry, M.; Youssef, B.A.B.; El Aziz, M.A.; Sidky, F.I. 3D Point-cloud Registration Using Particle Swarm Optimization Based on Different Descriptors. Int. J. Sci. Eng. Res.
**2017**, 8, 558–564. [Google Scholar] - John, V.; Xu, Y.; Mita, S.; Long, Q.; Liu, Z.; Tan, Y.; Takagi, H.; Shi, Y. Registration of GPS and Stereo Vision for Point Cloud Localization in Intelligent Vehicles Using Particle Swarm Optimization. In Lecture Notes in Computer Science; Springer International Publishing: Berlin/Heidelberg, Germany, 2017; Volume 10385, pp. 209–217. [Google Scholar]
- Zhang, L.; Yang, B.; Wang, L.; Zhao, X.; Zhou, J.; Li, M.; Han, Y. Three-dimensional Cement Image Registration Based on Multi-layer PSO and Mutual Information. In Proceedings of the 2016 3rd International Conference on Informative and Cybernetics for Computational Social Systems (ICCSS), Jinzhou, China, 26–29 August 2016. [Google Scholar]
- Kennedy, J.; Eberhart, E. Particle swarm optimization. In International Conference on Neural Networks (ICNN); IEEE: Perth, WA, Australia, 1995; pp. 1942–1948. [Google Scholar]
- Sun, L.; Song, X.; Chen, T. An Improved Convergence Particle Swarm Optimization Algorithm with Random Sampling of Control Parameters. J. Control. Sci. Eng.
**2019**, 2019, 1–11. [Google Scholar] [CrossRef] - Fajr, R.; Bouroumi, A. An Improved Particle Swarm Optimization Algorithm for Global Multidimensional Optimization. J. Intell. Syst.
**2017**, 29, 127–142. [Google Scholar] [CrossRef] - Besl, P.; McKay, N.D. A method for registration of 3-D shapes. IEEE Trans. Pattern Anal. Mach. Intell.
**1992**, 14, 239–256. [Google Scholar] [CrossRef] - Wang, F.; Zhao, Z. A survey of iterative closest point algorithm. In Proceedings of the 2017 Chinese Automation Congress (CAC), Jinan, China, 20–22 October 2017; IEEE: Piscataway Township, NJ, USA, 2017; pp. 4395–4399. [Google Scholar] [CrossRef]
- Duda, R.O.; Hart, P.E.; Stork, D.G. Pattern Classification, 2nd ed.; John Willey & Sons: Hoboken, NJ, USA, 2001. [Google Scholar]
- Chen, Y.; Medioni, G. Object modelling by registration of multiple range images. Image Vision Comput.
**1992**, 10, 145–155. [Google Scholar] [CrossRef] - FitzGibbon, A.W. Robust registration of 2D and 3D point sets. Image Vis. Comput.
**2003**, 21, 1145–1153. [Google Scholar] [CrossRef] - Eberhart, R.C.; Shi, Y. Comparing inertia weights and constriction factors in particle swarm optimization. In Proceedings of the 2000 Congress on Evolutionary Computation, CEC00 (Cat. No.00TH8512), La Jolla, CA, USA, 16–19 July 2000; pp. 84–88. [Google Scholar]
- Bansal, J.C.; Singh, P.K.; Saraswat, M.; Verma, A.; Jadon, S.S.; Abraham, A. Inertia Weight strategies in Particle Swarm Optimization. In Proceedings of the Third World Congress on Nature and Biologically Inspired Computing, Salamanca, Spain, 19–21 October 2011; pp. 633–640. [Google Scholar]
- Rathore, A.; Sharma, H. Review on Inertia Weight Strategies for Particle Swarm Optimization. In Proceedings of the Advances in Intelligent Systems and Computing; Springer International Publishing: Berlin/Heidelberg, Germany, 2017; Volume 547, pp. 76–86. [Google Scholar]
- Lagarias, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E. Convergence Properties of the Nelder—Mead Simplex Method in Low Dimensions. SIAM J. Optim.
**1998**, 9, 112–147. [Google Scholar] [CrossRef][Green Version] - Glira, P.; Pfeifer, N.; Briese, C.; Ressl, C. A Correspondence Framework for ALS Strip Adjustments based on Variants of the ICP Algorithm. Photogramm. Fernerkund. Geoinformation
**2015**, 2015, 275–289. [Google Scholar] [CrossRef] - Glira, P.; Pfeifer, N.; Ressl, C.; Briese, C. Rigorous Strip Adjustment of Airborne Laserscanning Data Based on the ICP Algorithm. ISPRS Ann. Photogramm. Remote Sens. Spat. Inf. Sci.
**2015**, 2, 73–80. [Google Scholar] [CrossRef][Green Version] - Point-Cloud Tools for Matlab. Available online: www.geo.tuwien.ac.at/downloads/pg/pctools/pctools.html (accessed on 10 December 2020).

**Figure 2.**(

**a**) The original shape (Target point-cloud) and (

**b**) the 3-D transformation (Source point-cloud).

**Figure 4.**Registration results from the global best particle at the (

**a**) 1st, (

**b**) 50th, (

**c**) 500th, and (

**d**) 2000th iterations.

**Figure 6.**(

**a**) The superimposed image before registration, the final registration with α is equal to (

**b**) 1.0, and (

**c**) 2.0.

**Figure 9.**The best SR-PSO registration results for the pairs (

**a**) 1 and 2; (

**b**) 2 and 3; (

**c**) 3 and 4; (

**d**) 4 and 5; (

**e**) 5 and 6 of the regular-tooth model.

**Figure 10.**The best SR-PSO with the ICP registration results for the pairs (

**a**) 1 and 2; (

**b**) 2 and 3; (

**c**) 3 and 4; (

**d**) 4 and 5; (

**e**) 5 and 6 of the regular-tooth model.

**Figure 11.**The final registration result of six consecutive views of the regular-tooth model from the SR-PSO with the ICP.

**Figure 12.**The best SR-PSO registration results for the pairs (

**a**) 1 and 2; (

**b**) 2 and 3; (

**c**) 3 and 4; (

**d**) 4 and 5; (

**e**) 5 and 6 of the orthodontic-tooth model.

**Figure 13.**The best SR-PSO with ICP registration results for the pairs (

**a**) 1 and 2; (

**b**) 2 and 3; (

**c**) 3 and 4; (

**d**) 4 and 5; (

**e**) 5 and 6 of the orthodontic-tooth model.

**Figure 14.**The final registration result of six consecutive views of the orthodontic-tooth model from the SR-PSO with the ICP.

Parameters | Lower Bound | Upper Bound |
---|---|---|

t_{x}, t_{y}, t_{z} | –1.5 (cm) | 1.5 (cm) |

ɵ_{x}, ɵ_{y}, ɵ_{z} | –45 (deg) | 45 (deg) |

s_{x}, s_{y}, s_{z} | 0.8 (20% downscaling) | 1.2 (20% upscaling) |

sh_{1}, sh_{2}, sh_{3}, sh_{4}, sh_{5}, sh_{6} | –0.5 (cm) | 0.5 (cm) |

**Table 2.**Statistical randomization-based particle swarm optimization (SR-PSO) algorithm’s parameters settings in the experiment.

Parameters | Values | |
---|---|---|

Number of particles | K | 100 |

Number of iterations | T | 2000 |

Constriction coefficient | χ | 0.7298 |

α | 0.5, 1.0, 1.5, 2.0 | |

φ | 4.1 | |

Personal learning coefficient | c_{1} | 1.4962 |

Global learning coefficient | c_{2} | 1.4962 |

α | ||||
---|---|---|---|---|

0.5 | 1.0 | 1.5 | 2.0 | |

SR-PSO without ICP | 7.68 × 10^{−2} | 9.23 × 10^{−2} | 3.22 × 10^{−31} | 9.68 × 10^{−2} |

SR-PSO with ICP | 7.68 × 10^{−2} | 9.23 × 10^{−2} | 3.22 × 10^{−31} | 9.68 × 10^{−2} |

Model | Object View | Object Name | Points Number |
---|---|---|---|

Regular tooth model | 1 | Scan_0.asc | 28,807 |

2 | Scan_1.asc | 28,970 | |

3 | Scan_2.asc | 28,983 | |

4 | Scan_3.asc | 25,809 | |

5 | Scan_4.asc | 17,303 | |

6 | Scan_5.asc | 21,739 | |

Total | Six views | 151,592 | |

Orthodontic tooth model | 1 | Scan_0.asc | 25,301 |

2 | Scan_1.asc | 25,772 | |

3 | Scan_2.asc | 22,432 | |

4 | Scan_3.asc | 17,167 | |

5 | Scan_4.asc | 22,537 | |

6 | Scan_5.asc | 24,148 | |

Total | Six views | 137,357 |

**Table 5.**MSE from SRPSO and PSO for the regular-tooth model (the best value is in bold and the worst value is underlined).

MSE in Micrometer^{2} | |||||
---|---|---|---|---|---|

SR-PSO | PSO | ||||

View pairs | α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

1 vs. 2 | 5.9300 | 6.1910 | 6.3274 | 6.3470 | 7.0328 |

2 vs. 3 | 5.0026 | 4.9070 | 4.8937 | 4.9507 | 5.3218 |

3 vs. 4 | 5.8824 | 5.4310 | 6.0370 | 5.5086 | 8.1904 |

4 vs. 5 | 5.2666 | 5.3131 | 5.2807 | 32.312 | 6.7879 |

5 vs. 6 | 6.0431 | 5.8163 | 5.9118 | 5.8166 | 6.5468 |

**Table 6.**MSE from SR−PSO with iterative closet point (ICP) and PSO with ICP for the regular-tooth model (the best value is in bold).

MSE in Micrometer^{2} | |||||
---|---|---|---|---|---|

SR-PSO with ICP | PSO with ICP | ||||

View pairs | α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

1 vs. 2 | 5.8628 | 5.8636 | 5.8632 | 5.8809 | 5.8156 |

2 vs. 3 | 4.8906 | 4.8912 | 4.8860 | 4.8884 | 4.8844 |

3 vs. 4 | 5.4030 | 5.4024 | 5.4017 | 5.4031 | 5.4055 |

4 vs. 5 | 5.1326 | 5.1253 | 5.1747 | 5.1261 | 5.1261 |

5 vs. 6 | 5.6829 | 5.6880 | 5.6828 | 5.6882 | 5.6956 |

**Table 7.**MSE of the final registration of six consecutive views (micrometer

^{2}) for the regular-tooth model (the best value is in bold).

SR-PSO with ICP | PSO with ICP | |||
---|---|---|---|---|

α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

7.3670 | 7.3668 | 7.3666 | 7.3667 | 17.1150 |

MSE in Micrometer^{2} | |||||
---|---|---|---|---|---|

SR-PSO | PSO | ||||

View pairs | α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

1 vs. 2 | 5.5553 | 5.7212 | 5.6724 | 5.7199 | 28.973 |

2 vs. 3 | 6.2493 | 6.2868 | 6.1613 | 6.2224 | 35.490 |

3 vs. 4 | 179.95 | 12.719 | 5.4687 | 12.779 | 6.7116 |

4 vs. 5 | 7.0814 | 6.7577 | 6.5847 | 8.6855 | 14.036 |

5 vs. 6 | 5.4288 | 5.3262 | 5.3638 | 5.4472 | 35.000 |

**Table 9.**MSE from SR-PSO with ICP and PSO with ICP for the orthodontic-tooth model (the best value is in bold).

MSE in Micrometer^{2} | |||||
---|---|---|---|---|---|

SR-PSO with ICP | PSO with ICP | ||||

View pairs | α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

1 vs. 2 | 5.5093 | 5.5101 | 5.5093 | 5.5127 | 5.5130 |

2 vs. 3 | 6.1441 | 6.1440 | 6.1440 | 6.1444 | 6.1570 |

3 vs. 4 | 5.2854 | 12.511 | 5.2706 | 12.514 | 5.2847 |

4 vs. 5 | 6.3945 | 6.3948 | 6.3946 | 6.3999 | 6.3933 |

5 vs. 6 | 5.2859 | 5.2801 | 5.2810 | 5.2814 | 5.2815 |

**Table 10.**MSE of the final registration of six consecutive views (micrometer

^{2}) for the orthodontic-tooth model (the best value is in bold).

SR-PSO with ICP | PSO with ICP | |||
---|---|---|---|---|

α = 0.5 | α = 1.0 | α = 1.5 | α = 2.0 | |

7.4130 | 7.4141 | 7.4141 | 7.4131 | 7.4672 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wongkhuenkaew, R.; Auephanwiriyakul, S.; Chaiworawitkul, M.; Theera-Umpon, N. Three-Dimensional Tooth Model Reconstruction Using Statistical Randomization-Based Particle Swarm Optimization. *Appl. Sci.* **2021**, *11*, 2363.
https://doi.org/10.3390/app11052363

**AMA Style**

Wongkhuenkaew R, Auephanwiriyakul S, Chaiworawitkul M, Theera-Umpon N. Three-Dimensional Tooth Model Reconstruction Using Statistical Randomization-Based Particle Swarm Optimization. *Applied Sciences*. 2021; 11(5):2363.
https://doi.org/10.3390/app11052363

**Chicago/Turabian Style**

Wongkhuenkaew, Ritipong, Sansanee Auephanwiriyakul, Marasri Chaiworawitkul, and Nipon Theera-Umpon. 2021. "Three-Dimensional Tooth Model Reconstruction Using Statistical Randomization-Based Particle Swarm Optimization" *Applied Sciences* 11, no. 5: 2363.
https://doi.org/10.3390/app11052363