## 2 Related Works
### Support Vector Machine Algorithm  
The support vector machine (SVM) is a supervised learning algorithm based on the concept of hyperplanes that aims to separate a set of objects with maximum margin. First proposed by [PERSON] and [PERSON] [6], the SVM performs binary classification tasks based on statistical learning theory. Unlike the empirical risk minimization (ERM) method, used in conventional classifiers, the SVM learns by minimizing errors using the structural risk minimization (SRM) method. In this study, the SVM model used for image processing was employed [7,8].  
Let X [\(x_{1},x_{2},...,x_{N}\)] be a training set containing \(N\) feature vectors in d-dimensional feature space and associated with two class labels {-1, +1}. The linear SVM determines two classes that are linearly separable by defining a hyperplane that separates these feature vectors as follows.  
A hyperplane can be expressed by the following Equation (1):  
\[\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}=0 \tag{1}\]  
where \(\mathrm{w}^{\mathrm{t}}\) is the weight vector, \(\mathrm{x}\) is the input vector, and \(\mathrm{b}\) is the bias constant. A parallel hyperplane can be expressed by Equations (2) and (3):  
\[\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}=1 \tag{2}\]  
\[\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}=-1 \tag{3}\]  
If the learning data can be separated linearly, the distance between two hyperplanes is \(2/\|\mathrm{w}\|\).  
The algorithm searches for a hyperplane that minimizes \(\|\mathrm{w}\|\) while satisfying Equations (4) and (5).  
\[\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}\geq 1\ \forall t_{i}=1 \tag{4}\]  
\[\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}\leq-1\ \forall t_{i}=-1 \tag{5}\]  
Equations (4) and (5) can be combined as Equation (6):  
\[t_{i}\Big{(}\mathrm{w}^{\mathrm{t}}\cdot\mathrm{x}+\mathrm{b}\Big{)}\geq 1 \tag{6}\]  
To obtain a better separation among the classes, this distance should be maximized by minimizing \(\|w\|^{2}\) using the Lagrangian function with the most suitable optimal hyperplane obtained using Equation (7).  
\[f(x)=\sum_{i=1}^{N}\alpha_{i}w_{i}K(x_{i}\cdot x)+b \tag{7}\]  
where \(\alpha_{i}\) is the Lagrange multiplier of a dual optimization problem that describes the separating hyperplane \(w_{i}(x_{i}\cdot x)\) with kernel function \(K(x_{i}\cdot x)\), and \(b\) is the threshold parameter of the hyperplane. If \(f(x)\geq 0\), then \(x\) is classified as a member of the class +1; else, it is classified as a member of the second class (\(-1\)). If the data space is linearly separable, the linear kernel function \(K(x_{i}\cdot x_{i})=x_{i}^{T}x_{i}\) is sufficient to solve this optimization problem. Otherwise, other nonlinear kernel functions, such as the radial basis function (RBF) and sigmoid function, can be applied. However, nonlinear kernel function might require the complex tuning of hyperparameters which has values that can affect the performance of the SVM classifiers significantly.  
Thus, the SVM applied in this study uses the linear kernel function. The SVM algorithm is essentially a binary classifier which classifies objects into the positive and negative classes. The one-against-all (OAA) SVM [9], which can handle multiple classes, was employed to classify the three classes (i.e., B1, B2, and B3) in this study. The OAA SVM classifies one class as +1 (positive), while the other classes are classified as -1 (negative). Unlike the B1 type, any microvessel with the morphologies shown in Figure 1 is classified as either the B2 or B3 type. Therefore, when the OAA SVM learning method is applied, the B1 type is classified as positive, while the B2 and B3 types, which have mixed types of microvessels, are classified as negative.