# Improving Juvenile Age Estimation Based on Facial Landmark Points and Gravity Moment
## 3 Preliminaries
### Measuring Weight by TF-IGM
#### 3.5.1 Limitations of IGM  
However, empirical tests revealed some cases of redundancies in the computation of the weight based on the IGM factor. For instance, in the case of five features \(f_{j1}\), \(f_{j2}\), \(f_{j3}\), \(f_{j4}\) and \(f_{j5}\) having individual feature frequencies (TF) in six age classes corresponding to {100, 100, 0, 0, 0, 0}, {40, 40, 0, 0, 0, 0, 0}, {23, 23, 0, 0, 0, 0}, {11, 11, 0, 0, 0, 0} and {2, 2, 0, 0, 0, 0} respectively. If there are 100 samples in each age class, then standard IGM computes the weight of each feature as 0.333, regardless of the sorted distinguishing power being \(f_{j1}>f_{j2}>f_{j2}>f_{j3}>f_{j4}>f_{j5}\). We refer to this as case 1. Additionally, consider the case of five features \(f_{j6}\), \(f_{j7}\), \(f_{j8}\), \(f_{j9}\) and \(f_{j10}\) and a dataset of five age classes each with 10 samples and corresponding feature frequencies (TF) of {10, 0, 0, 0, 0}, {8, 0, 0, 0, 0}, {5, 0, 0, 0}, {3, 0, 0, 0, 0} and {1, 0, 0, 0, 0} respectively. Intuitively, the order of the class distinguishing power should be \(f_{j6}>f_{j7}>f_{j8}>f_{j9}>f_{j10}\). However, per Equation (5), the IGM values are all the same as 1.0. This can be considered as case 2. Obviously, computing the weight based on the standard IGM does not fully depict the distinguishing power, since in both cases, features with different frequencies are assigned the same values. In addressing these limitations, we introduced a common logarithm denoted by \(K\) to the standard IGM equation such that \(K=log_{10}[S_{total}(f_{j(max)})/S_{f_{j(max)}}]\), where \(S_{total}(f_{j(max)})\) isthe total face samples in the age class, in which \(f_{j}\) occurs the most and \(S_{f_{j(max)}}\) is the number of the face samples in the age class in which \(f_{j}\) occurs the most. For clarity, we denote the intervention as \(GMO\) expressed as follows.  
\[GMO(f_{j})=\frac{f_{j1}}{\sum_{r=1}^{n}f_{jr}\cdot R+K} \tag{7}\]  
As shown in Table 1, the weights computed by \(GMO\), using Equation (7), show unique values for features \(f_{j1}\) through \(f_{j10}\). This shows that, whereas in case 1, the IGM values are the same for the different features, the values as computed by \(GMO\) are unique. This is indicative of the distinguishing abilities of \(GMO\) over IGM.  
Consequently, Equation (8) is proposed to compute the weight as follows.  
\[W(f_{j})=TF(f_{j},s_{k})\cdot(1+\lambda\cdot GMO(f_{j})) \tag{8}\]  
where \(TF\) represents the local weighting factor, \(GMO\) is the global weighting factor and \(\lambda\) is a co-efficient that maintains the balance between the two factors. \(W\) is the weight and \(s_{k}\) is the sample image.  
Intuitively, the local weighting factor should be lowest since a feature occurring 68 times in an age class is generally less than 68 times as important as a feature that rarely occurs [48]. Similar to the method in [49], the TF was reduced by the introduction of square root to Equation (8), resulting in new weight expressed by Equation (9).  
\[w(f_{j})=[TF(f_{j},s_{k})]^{\frac{1}{2}}\cdot(1+\lambda)\cdot[GMO(f_{j})] \tag{9}\]  
We utilize the new GMO-based weight for our proposed scheme.