## 3 Proposed Method
### Data Normalization  
The acquired data from all signatures is pre-processed by normalizing the alignment, size, and length before feature extraction. Because signatures have different values of starting point, size, and the number of samples, the data must be adjusted to equivalent conditions for more precise learning, modeling, and verification. To standardize starting points, each signature is aligned so that the starting point is set to (0, 0, 0, 0, 0). The processed data by alignment normalization \(Xa\) is calculated using Equation (2), where (\(Px_{0}\), \(Py_{0}\), \(Ax_{0}\), \(Ay_{0}\), \(Az_{0}\)) denotes the original starting point of the signature. The distance \(dist_{i}\) is omitted because it is unrelated to the starting points.  
\[Xa=(Px_{i}-Px_{0},Py_{i}-Py_{0},Ax_{i}-Ax_{0},Ay_{i}-Ay_{0},Az_{i}-Az_{0}), \quad i=0,...,n \tag{2}\]  
Even though the signatures are drawn on a restricted plane, i.e., smartphone screen, each signature is different in size. This size variation could lead to performance degradation of signature verification. Therefore, the proposed scheme makes the signatures into similar size by size normalization which ensures that all values of points range between 0 and 1. The processed data by size normalization \(Xs\) is calculated from \(Xa\) using Equation (3), where \(max(Px)\) is the maximum value among \(Px_{0}\), \(Px_{1}\),... \(Px_{n}\).  
\[\begin{split}Xs&=((\frac{Px_{i}-max(Px)}{min(Px)- max(Px)}),(\frac{Py_{i}-max(Py)}{min(Py)-max(Py)}),(\frac{Ax_{i}-max(Ax)}{min(Ax)- max(Ax)}),\\ &(\frac{Ay_{i}-max(Ay)}{min(Ay)-max(Ay)}),(\frac{Az_{i}-max(Az)} {min(Az)-max(Az)})),\quad i=0,...,n\end{split} \tag{3}\]  
Length normalization refers to the process of equalizing signature length, i.e., the number of samples. Each signature length is normalized to a mean length of the subject from \(Xs\) using Equations (4) and (5), where \(n\) is the original length of \(Xs\), \(m\) is the mean length of the subject, \(k\) is rate for equal  
Figure 1: Original signature.  
division, and \(t\) is a proper integer. \(x^{\prime}_{0}\) and \(x^{\prime}_{m-1}\) are, respectively, the first and last samples. They are kept constant to equalize the starting and ending points of all signatures. After length normalization, the sequence of \(x^{\prime}_{i}\), i.e., \(X^{\prime}\), is a signature normalized using three normalization processes.  
\[\begin{split}& k_{0}=K_{1}=n/m\\ & K_{j+2}=k_{j+1}+k_{j},\quad j=0,...,m-3\\ &\quad i=1\mid i\leq m-2\\ &\quad if\quad t\leq k_{i}<t+1\\ &\quad x^{\prime}_{i}=x_{t}+(x_{t+1}-x_{t})\times(k_{i}-t)\\ &\quad x^{\prime}_{0}=x_{0}\\ &\quad x^{\prime}_{m-1}=x_{n-1}\end{split} \tag{4}\]  
Figure 2 shows an example of data normalization. Figure 1(a) shows \(Xa\), with the starting point of (0, 0). Figure 1(b) shows \(Xs\), with all values existing between 0 and 1. Figure 1(c) shows \(X^{\prime}\). In addition, the number of points, i.e., the number of samples, are changed to the mean length of the subject.  
Figure 3 shows the sequence of \(x^{\prime}_{i}\), i.e., a normalized signature \(X^{\prime}\) according to the sequence of 150 samples.  
Figure 2: Data normalizations of: (**a**) alignment; (**b**) size; and (**c**) length.