# A Robust and Automatic Method for the Best Symmetry Plane Detection of Craniofacial Skeletons

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

- -
- Cephalometric methods
- -
- Morphometric methods
- -
- ICP-based methods

#### 2.1. Cephalometric Methods

- -
- -
- of best fit passing through a selection of midline landmarks [31].

#### 2.2. Morphometric Methods

#### 2.3. ICP-Based Methods

- -
- -

- -
- the final solution is strictly affected by the initial one;
- -
- the function to be minimized likewise considers all the points of the model, including local defectiveness, such as holes or bony prominences, that should not be considered in the evaluation of symmetry plane of the skull.

## 3. The Mirroring and Weighted Approach

#### 3.1. The Published Method

_{0}), carried out with a PCA algorithm, which is then refined iteratively until its final estimation.

_{f}of the MSP is obtained by the Levenberg-Marquardt algorithm [37] that iteratively minimizes the following objective function (OF

_{i}), whose expression at the i-th step is:

- -
- n is the number of points of the source point cloud (PC);
- -
- p
_{j}is the j-th point belonging to PC; - -
- TS(PC
_{m,i}) is the tessellated surface of the mirrored data at i-th iteration (PC_{m,i}); - -
- d
_{Hauss}(p_{j},TS(PC_{m}_{,i})) is the Hausdorff distance between p_{j}and TS(PC_{m}_{,i}); - -
- w
_{i,j}is the weight associated to p_{j}at i-th iteration.

_{j}on the source PC and the tessellated surface TS(PC

_{m,i}), instead of point-point, in order to avoid the asymmetry in surface sampling and to make the distance as independent as possible on points density.

_{i,j}play an important role in the functionalities of the method being presented. It is expressed as the product of two specific weights:

_{s,i}

_{,j}and w

_{r,i}

_{,j}are both expressed according to the Leclerc function [38], which has its maximum in correspondence with the symmetry plane:

- -
- d
_{i,j}is the distance between p_{j}and the symmetry plane Π_{i}(at the i-th iteration). - -
- σ
_{s}and σ_{r}define, respectively, the distance and the radius values for which the weight is the 36.79% of its maximum value [38].

_{s,i}

_{,j}works to reduce the effects of the asymmetries in the acquisition process which are mainly located far from the symmetry plane. The weight w

_{r,i}

_{,j}works as a filter which excludes from the registration process any local asymmetries, whether they are near or far from the symmetry plane.

#### 3.2. Main Limitations and Proposed Improvements

_{i,j}(Equation (2)) and the objective function’s (Equation (1)) minimization strategy.

- -
- the acquisition quality is not affected by the distance from the symmetry plane;
- -
- the most symmetrical areas may be those that are farthest from the symmetry plane;
- -
- the weight w
_{s,i}_{,j}proves to be not only useless but also negatively affecting the results.

_{s,i}

_{,j}has been neglected and the weight w

_{i,j}works only as a filter which excludes from the registration process any local asymmetries:

_{r}guarantee a robust registration, whereas small values afford an extremely accurate registration of the symmetric parts, excluding all asymmetries. In the experiments described hereafter, since we analyze real skulls with large asymmetries, the value of σ

_{r}is assumed to be the 50 percent the maximal width of the skull.

## 4. Experiments and Results

- -
- TC#1 and TC#2 are two healthy real skulls (Figure 6a,b), to demonstrate the reliability of each method in real cases. For TC#2 an incomplete skull model has been chosen because, commonly, the TC images acquisition addresses only the region of interest to reduce the irradiation risks.
- -
- TC#3, TC#4 and TC#5 are real skulls, each with a large defect (so, large asymmetry) (Figure 6c–e), to demonstrate the reliability and the robustness of the new method compared with the other approaches. These three case studies include one unilateral defect, one bilateral defect and one defect crossing the MSP.

_{point}that is the Euclidean distances between each point of the source point cloud (p

_{j}) with respect to its closer tessellated surface triangle (TS(PC

_{m,i})) of the mirrored configuration. The mirrored configuration is obtained reflecting the PC upon the estimated symmetry plane. The AV is calculated as the median instead of the mean in order to reduce the weight of the little strongly asymmetrical regions in the calculation. The distance point-triangle makes it possible to avoid the asymmetries due to the surface model sampling. Preliminarily, it has been verified that for the previously shown synthetic mesh (Figure 1a), the AV value is zero. For real skulls, based on the previously mentioned considerations, the AV value cannot be zero; however, the method for the smaller AV performs the best localization of the symmetry plane.

_{point}maps for the TC#5 and the four methods here compared are shown. It is evident that the estimation of the symmetry plane carried out with the proposed method determines a better localization of the asymmetries of the skull.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**The MSP resulting from: (

**a**) the MaWR method as presented in [28]; (

**b**) the MaWR improved method.

**Figure 6.**The real test cases here analyzed. (

**a**) and (

**b**) are two healthy skulls; (

**c**) skull with large unilateral defect; (

**d**) skull with large bilateral defect; (

**e**) skull with large defect crossing the MSP.

**Figure 7.**The AV maps for the TC#5 and the four methods here compared. (

**a**) MWR-method; (

**b**) Mpm-method; (

**c**) Cplm1-method; (

**d**) Cplm2-method.

**Figure 8.**Skull with craniofacial dysmorphism due to hypertelorism combined with a severe form of plagiocephaly. (

**a**) MSP resulting from the proposed method; (

**b**) MSP resulting from the proposed method applied to the user selected area highlighted in green.

Landmark | Abbr | Definition |
---|---|---|

Basion | Ba | The most anterior point on the margin of the foramen magnum in the mid-sagittal plane |

Nasion | N | Most anterior point of the frontonasal suture in the mid-sagittal plane |

Crista Galli | CG | Most superior point of the Crista Galli |

Incisive Foramen | IF | The midpoint of the Incisive Foramen |

Frontozygomatic Suture | FZS | The most medial and anterior point of left (FZSL) or right (FZSR) frontozygomatic suture at the level of the lateral orbital rim |

Supraorbital Foramen | SOF | The midpoint of supraorbital foramen |

Fontorbitomaxillare | FOM | Lateral point of the frontomaxillary suture on the medial margin of the orbit |

Frontonasomaxillare | FNM | The intersection of the nasomaxillary, frontomaxillary, and frontonasal sutures |

Sella | S | Center of the sella turcica |

Pogonion | Pog | Most anterior point of the bony chin in the median plane |

Anterior Nasal Spine | ANS | Most anterior point midpoint of the anterior nasal spine of the maxilla |

Test Cases | MWR-Method | Mpm-Method | Cplm1-Method | CPLM2-METHOD |
---|---|---|---|---|

TC#1 | 0.95 | 1.02 | 0.98 | 0.97 |

TC#2 | 0.66 | 1.09 | 0.72 | 0.73 |

TC#3 | 0.78 | 0.88 | 1.01 | 1.01 |

TC#4 | 0.76 | 0.88 | 0.80 | 1.48 |

TC#5 | 1.30 | 1.56 | 1.59 | -- |

Test Cases | MWR-Method | Mpm-Method | Cplm1-Method | Cplm2-Method |
---|---|---|---|---|

TC#1 | 0.95 | 1.02 | 0.98 | 0.97 |

TC#2 | 0.66 | 1.09 | 0.72 | 0.73 |

TC#3 | 0.78 | 0.88 | 1.01 | 1.01 |

TC#4 | 0.76 | 0.88 | 0.80 | 1.48 |

TC#5 | 1.30 | 1.56 | 1.59 | -- |

TC#6 | 1.11 | 1.33 | 1.43 | 1.21 |

TC#7 | 0.75 | 0.81 | 0.88 | 1.24 |

TC#8 | 0.63 | 0.72 | 0.76 | 0.75 |

TC#9 | 0.65 | 0.80 | 0.75 | 0.76 |

TC#10 | 1.02 | 1.13 | 1.18 | 1.27 |

TC#11 | 1.26 | 1.52 | 1.38 | 1.55 |

TC#12 | 0.87 | 1.24 | 1.25 | 1.20 |

TC#13 | 1.04 | 1.11 | 1.42 | 1.39 |

TC#14 | 0.76 | 0.84 | 0.86 | 1.10 |

TC#15 | 0.89 | 1.07 | 1.21 | 1.20 |

TC#16 | 0.69 | 0.92 | 0.90 | 0.98 |

TC#17 | 1.18 | 1.40 | 1.24 | 1.37 |

TC#18 | 0.70 | 0.85 | 0.97 | 0.92 |

TC#19 | 1.23 | 1.39 | 1.56 | 1.55 |

TC#20 | 0.84 | 1.09 | 0.93 | 0.92 |

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**MDPI and ACS Style**

Di Angelo, L.; Di Stefano, P.; Governi, L.; Marzola, A.; Volpe, Y.
A Robust and Automatic Method for the Best Symmetry Plane Detection of Craniofacial Skeletons. *Symmetry* **2019**, *11*, 245.
https://doi.org/10.3390/sym11020245

**AMA Style**

Di Angelo L, Di Stefano P, Governi L, Marzola A, Volpe Y.
A Robust and Automatic Method for the Best Symmetry Plane Detection of Craniofacial Skeletons. *Symmetry*. 2019; 11(2):245.
https://doi.org/10.3390/sym11020245

**Chicago/Turabian Style**

Di Angelo, Luca, Paolo Di Stefano, Lapo Governi, Antonio Marzola, and Yary Volpe.
2019. "A Robust and Automatic Method for the Best Symmetry Plane Detection of Craniofacial Skeletons" *Symmetry* 11, no. 2: 245.
https://doi.org/10.3390/sym11020245