# Improving Juvenile Age Estimation Based on Facial Landmark Points and Gravity Moment
## 3 Preliminaries
### Facial Landmark-Term by AAM  
AAM is a statistical shape and appearance model that presents a holistic view of the face [10; 16]. Given an annotated facial image, the shape information can be considered as the dominant points defining specific landmarks of the face, including the forehead, eye corner, forehead, face cheeks, etc., or an interpolated linkage of the points around the whole face [45]. However, when considered separately, we observe that the landmark points have no inherent ability to describe the face. Their independence appears similarly to the individual units of a natural language. As with the 26 letters of the English Language, the landmark points cannot by themselves describe the age of a face. However, when represented at a higher level, they can be considered to represent age. As with the quality of work by [14], we corroborate that the 68 landmark points can be transferred into meaningful terms (features) similar to words in natural language. Then, for different term vectors (features), the vectors can represent the age of the face. For the AAM of the face image, 68 landmark points can be expanded such that the same terms for different AAMs can be treated as the same features. We denote the AAM of an image \(img\) as vector \(LM_{(img)}\) and expand it to  
Figure 1: 68 facial landmark points. cover the 68 points as \(LM_{(img)}=(x_{1},x_{2},x_{3},x_{4},\,...,\,x_{135},x_{136})\), where \(x_{1},x_{2}\) is the first landmark point, \(x_{3},x_{4}\) the second landmark point and \(x_{135},x_{136}\) is the last landmark point, aiming to obtain a term of the form \(F-j-LOC\), where \(F\) represents the appearance information and \(j\) is the \(j^{th}\) element with \(LOC\) representing the location of the \(j^{th}\) point measured as \(LOC=round(\frac{x_{j}}{width})\). However, as shown in Figure 1, the 68 points of the AAM do not fully represent the holistic appearance of the face, as the distribution of the points falls within the areas marked off by the points. Additionally, certain regions of the face tend to show early signs of aging and must therefore be considered for a more holistic appearance representation. Therefore, we adapt the proposal in [46] to obtain wrinkle information from the corners of the eyes and the forehead region. We annotate the wrinkles in these regions with five points representing each wrinkle's shape. The wrinkle information is obtained using a bounding box placed around the annotations and only maintaining high-frequency information by difference-of-Gaussian. The wrinkle is warped in a mean shape and then transformed in pose parameters. By using the second derivative of the Lorentzian function to fit, the average of every parameter is obtained. Thus, the process leads to the transformation of the wrinkle into a vector of meaningful parameters representing the shape and appearance of the wrinkle. The parameters of the wrinkle vector include the following:  
* \(cx,cy\), the center of the wrinkle;
* \(d\), the geodesic distance between first and last points;
* \(a\), angle in degrees;
* \(C\), curvature computed as least-squares minimization using Equation (1);
* \(D\), depth of the wrinkle;
* \(W\), the width of wrinkle. \[min||Y-CX^{2}||_{2}^{2} \tag{1}\]  
with \(Y(resp.X)\) the coordinates (resp. abscissa) of the wrinkle centered at the origin, and with the horizontal alignment of the first and last points. Since there can be an unequal number of wrinkles in faces, a technique is used to obtain the wrinkle information from six equally sized non-overlapping segments of the target region. For every wrinkle, an estimate of the probability density is obtained. The strategy leads to a vector which incidentally is high dimensional. The problem is mitigated by introducing an approximation of an arbitrary joint probability of random variables by computing in pairs, every joint probability for every random variable. The computations are by means of Kernel Density Estimation (KDE) with a Gaussian and a standard deviation of 1.5.