# Quantifying Perceived Facial Asymmetry to Enhance Physician-Patient Communications
## 2 Materials and Methods
### Asymmetry Classifier  
The asymmetry classifier is an artificial neural network using back-propagation to adjust its inter-layer weights. The model has 14 nodes on its input layer, 10 nodes on the hidden layer, and 2 target nodes. The 14 facial features fed into the input layer nodes are the 14 asymmetry indices (AI's) computed from the 20 landmarks as defined in Equation (1). The number of hidden layer nodes being 10 is determined by cross-validation and by employing the rule of thumb: (a) The number of hidden neurons is between the size of the input layer and the size of the output layer; (b) The number of hidden neurons is approximately two-thirds of the size of the input layer, plus the size of the output layer. The two target nodes correspond to the binary coding of the three possible asymmetry classes: PN, PAN, and PA.  
An asymmetry index of a feature is its distance off the mid-sagittal plane if it is a medial landmark. On the other hand, the asymmetry index for a pair of bilateral landmarks is defined as the root sum squared (RSS) of the disparities of both landmarks on x-, y-, and z-directions. Equation (1) is calculated as follows:  
\[\text{AI}_{i}=\begin{cases}\delta_{x}(\textbf{L}(i))\\ \left(\left(\frac{L_{x}^{l}+L_{x}^{r}-2M_{x}}{2}\right)^{2}+\left(\frac{L_{y}^ {l}-L_{y}^{r}}{2}\right)^{2}+\left(\frac{L_{z}^{l}-L_{z}^{r}}{2}\right)^{2} \right)^{1/2}\end{cases},\text{ if }\textbf{L}(i)\in\{Ex,En,Al,Ch,Zy, Go\} \tag{1}\]  
where \(L=\textbf{L}(i)\) represents the landmark with ID being \(i\), as shown in Table 1. \(\text{AI}_{i}\) denotes the asymmetry index of landmark \(\textbf{L}(i)\in\{G,N,Prn,Sn,Ls,Li,Sto,Me\}\cup\{Ex,En,Al,Ch,Zy,Go\}\); \(\delta_{x}(L)\) is the distance of landmark \(L\) off the mid-sagittal plane; \(L_{x}^{l}\) denotes the \(x\)-coordinate of landmark \(L\)(left); \(L_{x}^{r}\) is the \(x\)-coordinate of landmark \(L\)(right); \(L_{y}^{r}\) denotes the \(y\)-coordinate of landmark \(L\)(left); \(L_{y}^{r}\) is the \(y\)-coordinate of landmark \(L\)(right); \(L_{z}^{l}\) denotes the \(z\)-coordinate of landmark \(L\)(left); \(L_{z}^{r}\) is the \(z\)-coordinate of landmark \(L\)(right); \(M_{x}\) denotes the \(x\)-coordinate of the mid-sagittal plane.  
Specifically, in this study, we define mid-sagittal plane as the yz-plane, i.e., \(M_{x}=0\); therefore, the definition of (1) can be simplified to Equation (2) as follows:  
\[\text{AI}_{i}=\begin{cases}\delta_{x}(\textbf{L}(i))\\ \left(\left(\frac{L_{x}^{l}+L_{x}^{r}}{2}\right)^{2}+\left(\frac{L_{y}^{l}-L_ {y}^{r}}{2}\right)^{2}+\left(\frac{L_{z}^{l}-L_{z}^{r}}{2}\right)^{2}\right) ^{1/2}\end{cases},\text{ if }\textbf{L}(i)\in\{Ex,En,Al,Ch,Zy, Go\} \tag{2}\]  
During the training process, each face's 14 AIs are fed as the input to the under-constructed classifier, which adjusts its inter-layer weight matrices of the input and hidden and hidden and output layers to generate the desired target that is encoded from the corresponding questionnaire results. In combination, the 14 AIs of a face form an asymmetry index vector, \(\overset{\cdot}{\text{AI}}=[\text{AI}_{1},\,\,\text{AI}_{2},\cdots,\text{AI}_ {14}]^{T}\). The \(\overset{\cdot}{\text{AI}}\)'s of the 64 faces are iteratively applied until the classifier reaches convergence and its mean square error from the desired target is below a pre-defined threshold.