# Optimized Score Level Fusion for Multi-Instance Finger Vein Recognition
## 4 Proposed Method
### Fusion and Matching with BSSF and BSBF  
Next, the extracted HOG features are fused and matched using SVM based on the implementation of BSSF and BSBF. SVM is a classifier that classifies data by constructing a hyperplane to separate the training data of a class to the nearest training data of other classes with the largest distance. Another characteristic of SVM is the use of kernel function, which allow SVM to classify nonlinear data by creating a similarity matrix between the data for classification purpose.  
Given a sample of \(N\) independent and identically distributed training instances \(\{(x_{i},\ y_{i})\}_{i=1}^{N}\) where \(x_{i}\) is the \(D\) dimensional input vector (HOG feature) and \(y_{i}\in\{-1,1\}\) is its class label, and the kernel function \(k\) (\(x_{i},\ x_{i}\)) where \(x_{i}\) is the test instances. The SVM is trained by optimizing the following:  
maximize  
\[\sum_{i=1}^{N}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\alpha_{i} \alpha_{j}y_{i}y_{j}k\big{(}x_{i},x_{j}\big{)}\]  
with respect to  
\[\alpha\in\mathbb{R}_{+}^{N}\]  
subject to  
\[\sum_{i=1}^{N}\alpha_{i}y_{i}=0\ \ \ C\geq\alpha_{i}\geq\forall i \tag{2}\]  
where \(\alpha\) is the dual coefficients and \(C\) is the regularization hyperparameter. The hyperparameter \(C\) will be optimized by Bayesian optimizer. After training, the resulting discriminant function of the SVM is:  
\[s=score(x)=\sum_{i=1}^{N}\alpha_{i}y_{i}\ k\left(x_{i},x_{i}\right)+b \tag{3}\]  
The proposed system is tested with two kernel functions: linear kernel and radial basis function (RBF) kernel. The linear kernel formulation can be seen in the following:  
\[k_{linear}\big{(}x,x^{\prime}\big{)}=\big{\langle}x,x^{\prime}\big{\rangle} \tag{4}\]  
In summary, the hyperparameter \(\lambda\) of linear SVM is  
\[\lambda_{LinearSVM}=\{C\} \tag{5}\]  
The RBF kernel can be formulated with the following:  
\[k_{RBF}\big{(}x,x^{\prime}\big{)}=\exp\Bigl{(}-\gamma\parallel x-x^{\prime} \parallel^{2}\Bigr{)} \tag{6}\]  
where \(x\) and \(x^{\prime}\) are the HOG vector and \(\gamma\) is the width of the RBF kernel. The hyperparameter \(\gamma\) is optimized by Bayesian optimizer. In summary, the hyperparameters \(\lambda\) of RBF SVM are:  
\[\lambda_{RBFSVM}=\{C,\gamma\} \tag{7}\]  
BSSF works by multiplying the weight with the resulting score from SVM prediction and summing them. The summed score is then used to determine a match. The weights are optimized with Bayesian optimization. The BSSF function is given by:  
\[F_{s}=\sum_{i=1}^{n_{frags}}w_{i}\ score\Big{(}x_{i}^{i}\Big{)} \tag{8}\]  
where w is the weight of the score and \(F_{s}\) is the fused score.  
BSBF works by computing the score between two samples of a finger vein instance with Euclidean distance matcher. The Euclidean distance is defined as:  
\[d\big{(}x,x^{\prime}\big{)}=\sqrt{\sum_{i=1}^{D}\left(x_{i}-x_{i^{\prime}} \right)^{2}} \tag{9}\]The resulting scores from each finger vein instance are then concatenated as a score vector and normalized with z-score normalization as follows:  
\[s^{\prime}=\frac{s-mean(S)}{std(S)} \tag{10}\]  
where \(S\) is the score vector from the matcher, \(s\in S\), \(s^{\prime}\) is the normalized score, mean(\(S\)) is the arithmetic mean of \(S\), and std (\(S\)) is the standard deviation of \(S\). The normalized score vector is then fed into SVM for classification.