## 5 Methodology Used For Classification  
In this work, we have chosen three of the most popular classifiers algorithms frequently used in different classification scenarios. Figure 7 illustrates the general overview of the methodology used for classification. As we can see, in phase 1, the extraction of the handcrafted features is performed. The texture operators used are those already described in Section 4. Next, in phase 2, the classification is carried out using one of the three classifiers described in this section. In phase 3, the results are evaluated considering each possible combination \(feature\times classifiers\) in isolation. Finally, in phase 4, the fusions combining the outputs of the classifiers with the best individual performances are evaluated, using late fusion strategies (i.e., max rule, sum rule and product rule) proposed by [PERSON] et al. [15]. Equations (17)-(19) describe the mathematical details behind the max, product and sum combinations rules, respectively. In these equations, \(x\) is the pattern to be classified, \(c\) is the number of classes involved in the problem, \(n\) is the number of classifiers involved in the combination, \(\omega_{k}\) represents a class, with \(k\in 1..c\), and \(P(\omega_{k}|l_{i}(x))\) is the probability that \(x\) belongs to the class \(\omega_{k}\) according to the classifier \(i\). \[\text{Max Rule }(x)=arg\max_{k=1}^{c}\max_{i=1}^{n}P(\omega_{k}|l_{i}(x)) \tag{17}\]  
Figure 6: Binary size distributions from an area closing granulometry, \(\lambda\in[1,\cdots,50]\). One size distribution was computed for each image from the dataset introduced in Section 3. Figure 7: General overview of the methodology used for classification. \[\text{Product Rule}\left(x\right)=arg\max_{k=1}^{c}\prod_{i=1}^{n}P(\omega_{k}|l_{i} (x)) \tag{18}\] \[\text{Sum Rule}\left(x\right)=arg\max_{k=1}^{c}\sum_{i=1}^{n}P( \omega_{k}|l_{i}(x)) \tag{19}\]  
Three classifiers' algorithms were applied in this work: Support Vector Machines (SVM), K-Nearest Neighbor and Logistic Regression. **SVM:** Support Vector Machine (SVM) was first proposed by [PERSON] [PERSON] (1966). The SVM algorithm is able to perform the classification by determining a hyperplane that best separates the classes in the training data [PERSON] (1966). In this work, we used the Gaussian kernel, and cost and gamma parameters were tuned using a grid search. **k-NN:** k-NN is an instance-based algorithm widely used for classification. The K-Nearest Neighbor algorithm for binary classifications is considered simple when compared to other machine learning algorithms [PERSON] (1966). Despite its simplicity, k-NN is still one of the top 10 classification algorithms in machine learning [PERSON] (1966). This simplicity lies in the fact that it assumes all instances as points in the \(\mathbb{R}^{n}\) dimensional space and uses a distance metric (e.g., the Euclidean distance is frequently used in this case) to decide whether the element belongs to class A or class B [PERSON] (1966); [PERSON] (1966). In the experiments, various numbers of neighbors were tested, and k = 5 was chosen as it performed better than the other odd values. **Logistic Regression:** Logistic Regression is a special case of Regression [PERSON] (1966). Logistic Regression uses the following equation:  
\[p(X)=\frac{e^{\beta_{0}+\beta_{1}}X}{1+e^{\beta_{0}+\beta_{1}}X}=\frac{1}{1+ e^{-}\beta_{0}+\beta_{1}X},\]  
in which \(\beta_{0}\) and \(\beta_{1}\) are associated with every independent variable and are calculated by the likelihood method based on the dataset. Reglog is a statistical technique that establishes a relationship between the variable of interest and the probability of the outcome occurring; this probability has the value of success (1) and failure (0) [PERSON] (1966). The values \(\beta_{0}\) and \(\beta_{1}\) assume the value that maximizes the probability of the observed sample [PERSON] (1966).