# Application of a Novel Automatic Method for Determining the Bilateral Symmetry Midline of the Facial Skeleton Based on Invariant Moments

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Image Creation

#### 2.2. Feature Extractors

#### 2.2.1. Pseudo-Zernike Moments-PZMs

#### 2.2.2. Independent Component Analysis—ICA

**A**is an unknown mixture matrix

**A**${R}^{n\times n}$. The FastICA algorithm [31] was used to perform the task through the columns of its mixture matrix, which contain the main feature vectors.

#### 2.2.3. Principal Component Analysis—PCA

#### 2.3. Geometric Moments—Central Point

## 3. Results and Discussions

#### 3.1. Classification

**A**, described in Equation (13), was used as its feature vectors. For PCA, the eigenvectors based on eigenvalue order on the covariance matrix or its principal components were used. In total, feature vectors of size 35 were used to represent ICA and PCA.

#### 3.2. Midpoint Calculation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Disclosure Statement

## References

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**Figure 2.**An example of images in the IdentifyMe database that required pre-processing in which the image was (

**a**) cropped, (

**b**) rotated clockwise, and (

**c**) rotated counter clockwise.

**Figure 3.**Pre-processing step to vertically align images. (

**a**) The six cephalometric landmarks are identified. (

**b**) A grid is then added to the image and rotated so that the landmarks can be horizontally and vertically aligned.

**Figure 4.**The unrotated image (original) was labeled as “0” and subsequent images were labelled based on the angle of rotation.

**Figure 6.**Classification accuracy for different feature descriptors, using k-NN and Euclidean distance for images rotated from −14${}^{\xb0}$ to 15${}^{\xb0}$ with a resolution of 1${}^{\xb0}$.

**Figure 7.**Classification accuracy for different feature descriptors, using k-NN and Euclidean distance for images rotated from −14${}^{\xb0}$ to 15${}^{\xb0}$ with a resolution of 0.5${}^{\xb0}$.

**Figure 8.**Center of the images calculated using the moment technique. The symmetrical line is constructed by connecting the midpoint (red star) to the PZMs results (black dots).

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**MDPI and ACS Style**

Dalvit Carvalho da Silva, R.; Richard Jenkyn, T.; Alexander Carranza, V.
Application of a Novel Automatic Method for Determining the Bilateral Symmetry Midline of the Facial Skeleton Based on Invariant Moments. *Symmetry* **2020**, *12*, 1448.
https://doi.org/10.3390/sym12091448

**AMA Style**

Dalvit Carvalho da Silva R, Richard Jenkyn T, Alexander Carranza V.
Application of a Novel Automatic Method for Determining the Bilateral Symmetry Midline of the Facial Skeleton Based on Invariant Moments. *Symmetry*. 2020; 12(9):1448.
https://doi.org/10.3390/sym12091448

**Chicago/Turabian Style**

Dalvit Carvalho da Silva, Rodrigo, Thomas Richard Jenkyn, and Victor Alexander Carranza.
2020. "Application of a Novel Automatic Method for Determining the Bilateral Symmetry Midline of the Facial Skeleton Based on Invariant Moments" *Symmetry* 12, no. 9: 1448.
https://doi.org/10.3390/sym12091448