# Improving Juvenile Age Estimation Based on Facial Landmark Points and Gravity Moment
## 4 The Proposed Method  
The overall illustration of our proposal is shown in Figure 2. As can be seen, facial images annotated with the 68 facial landmark points serve the system, which obtains a combination of shape and wrinkles information, and represents it as terms that can be used describe the age. In order to explore the distinguishing power of the terms, TF-IGM is used to assign weights to the term. Next, the computed weights are used to build a matrix of predictive probabilities, which is utilized for the detection of the age. Consequently, we introduced a scheme we name LaGMO, which is a combination of facial landmark points and TF-IGM for juvenile age detection.  
The following is an encapsulation of the key tasks that constitute LaGMO.  
Definition 1: Transferring Landmark points to landmark-term vectors (features):  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Case** & **F** & **Class/Sample** & **F/Frequency** & **Sorted Des. Order** & **IGM Value** & **GMO Value** \\ \hline Case 1 & \(f_{j1}\) & 6 / 100 & \{100, 100, 0, 0, 0, 0\} & \(f_{j1}\)\(>\)\(f_{j2}\)\(>\)\(f_{j3}\)\(>\)\(f_{j4}\)\(>\)\(f_{j5}\) & 0.333 & 0.3  
Given the 68 facial landmark points, we denote \(AAM\) of an image img as vector \(LM_{(img)}=(x_{1},x_{2},x_{3},x_{4},...,x_{135},x_{136})\), 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. For the vector \(LM_{(img)}\), we transferred the points into a string form such that the \(j^{th}\) member represents the \(j^{th}\) string of the form \(F-j-LOC\), where \(F\) represents the appearance information, \(j\) is the \(j^{th}\) element and \(LOC=round(\frac{x_{j}}{width})\). Width is hard coded. In line with the process described in Section 3.1, we obtained a compact term denoted by \(f_{j}\).  
**Definition 2**.: _Obtaining term dictionary: If for the whole space there are \(n\) different terms, then they constitute a dictionary denoted as \(TDIC=(f_{1},f_{2},...,f_{jn})\). Then for any arbitrary \(f_{j}\), we assign weight by Equation (9), resulting in a new vector denoted as \(LMM_{(img)}=w(f_{j1}),w(f_{j2}),...,w(f_{jn})\), where the \(n^{th}\) element is between 0 and 1. The bigger the weight the higher the importance of the weight in the class. Then we define a weighted term matrix as follows._  
\[wMtrx=w(f_{j})(TDIC_{size})\cdot|n| \tag{10}\]  
where \(w(f_{j})\) is the weight of the term, \(TDIC_{size}\) is the size of the dictionary and \(|n|\) is the total number of age classes.  
**Definition 3**.: _Establishing the relationship between the features and ages: To establish the relationship between the features and ages, we classified all \(LMM_{(img)}\) samples into different age classes according to the age denoted as \(c=\{c_{1},c_{2},...,c_{n}\}\) with \(n\) representing the \(n^{th}\) class. All samples in the class have the same ground-truth age \(c\)._  
**Definition 4**.: _Restricting the matrix table to correspond to juvenile age: Since our proposal considers the relative order of vector \(LMM_{(img)}\) as age labels, we treated \(LMM_{(img)}\) as labels of the order \(LMM_{(img)}\in\{1,2,...,n\}\), where \(n\) is the number of age classes based on the entire dataset. We reduced the dataset to correspond to juveniles using Equation (11)._  
\[X_{n}^{+} =\{(LMM_{(img)},c_{n})|c_{n}\ >n\},\] \[X_{n}^{-} =\{(LMM_{(img)},c_{n})|c_{n}\ \leqslant n\} \tag{11}\]  
Next, we resolved \(X_{n}^{-}\) into a new dataset denoted by _tdic_ and utilized Equation (10) to build the new search table we denote \(wMtrx_{new}\). By this strategy, we avoided the rather exhaustive search throughout the larger table, thereby reducing the overhead in both time and space.  
Figure 2: Overview of the proposed LaGMO. \(P(LLM_{(img)}|wMtrx_{new})\) is the predictive matrix.