## 2 Materials and Methods
### Application of the TSK Inference System to Classify Objects  
Each of the object descriptors (area \(A\), fractal dimension \(D\), and eccentricity \(E\)) obtained in the first stage of the image processing pipeline (Figure 3) was considered a crisp input variable of the fuzzy system. That is, the input vector with \(n=3\) variables in Figure 4 is given by \((x_{1},x_{2},x_{3})=(A,D,E)\). Each of these variables was fuzzified using the C-means [44] fuzzy method with 3 classes (\(K=3\)) representing \(k=1\) high (\(H\)), \(k=2\) medium (\(M\)), and \(k=3\) low (\(L\)). Membership functions obtained in this way for the descriptor \(D\) (fractal dimension) are shown in Figure 5. The 3 membership functions of the other two descriptors were obtained in a similar fashion. A rule base was built considering the three descriptors with the 27 rules shown in Table 3. The rules were defined according to previous knowledge on the morphology and granulocyte of the additive materials (EVA and fiber) and the pores in the analyzed specimens. The outputs of the inference rules \(y_{r},\ r=1,2,\ldots,27\) are constant functions with 3 discrete values: \(1,2\), and \(3\), representing the classification of the object as pore, EVA, and fiber, respectively. The processing described in this section up to this point is illustrated in Figure 6.  
Figure 5: Membership functions of the fuzzy subsets LOW (\(L\)), MEDIUM (\(M\)), and HIGH (\(H\)) of the variable fractal dimension \(D\). With the rule base defined, the \(w_{r}\) weights of each rule \(r\) for each object using Formula (6) were calculated, where the sub-indexes \(k(r,i)\) are the fuzzy sets \(L,M,\) or \(H\) of the attribute \(i=A,D,E\) (column) in the rule \(r=1,2,\ldots,27\) (row) of Table 3.  
Once the weights of each rule are calculated, the output of the system \(Y\) is calculated according to Equation (5) using the output functions \(y_{r}\) in the last column of Table 3. As the weights vary in the interval \([0,1],\) the output \(Y\) is a real number that varies between 1 and 3 and is necessary to transform the output into a discrete variable with value \(1,2,\) or 3 representing the type of component, pore, EVA, or fiber, respectively. In this study, the closest class is assigned, i.e., the output class, will be \(z\in[1,3]\) if \(|Y-z|=\min_{j\in[1,3]}|Y-j|\). However, it is possible that the output is the exact mean value between two classes. In this case, \(z\) is assigned to the smaller class, which is the more frequent of the two in concrete due to their ordering. In other words, the most frequent class in the concrete studied is pore (1), followed by EVA (2) and fiber (3). Therefore, if \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(R_{r}\) & \(A\) & \(E\) & \(D\) & \(y_{r}\) & \(R_{r}\) & \(A\) & \(E\) & \(D\) & \(y_{r}\) \\ \hline \(R_{1}\) & L & L & L & 1 & R15 & M & M & H & 3 \\ \(R_{2}\) & L & L & M & 2 & R16 & M & H & L & 1 \\ \(R_{3}\) & L & L & H & 1 & R17 & M & H & M & 3 \\ \(R_{4}\) & L & M & L & 1 & R18 & M & H & H & 1 \\ \(R_{5}\) & L & M & M & 2 & R19 & H & L & L & 3 \\ \(R_{6}\) & L & M & H & 1 & R20 & H & L & M & 2 \\ \(R_{7}\) & L & H & L & 1 & R21 & H & L & H & 3 \\ \(R_{8}\) & L & H & M & 1 & R22 & H & M & L & 3 \\ \(R_{9}\) & L & H & H & 1 & R23 & H & M & M & 2 \\ \(R_{10}\) & M & L & L & 1 & R24 & H & M & H & 3 \\ \(R_{11}\) & M & L & M & 3 & R25 & H & H & L & 1 \\ \(R_{12}\) & M & L & H & 3 & R26 & H & H & M & 2 \\ \(R_{13}\) & M & M & L & 1 & R27 & H & H & H & 2 \\ \(R_{14}\) & M & M & M & 3 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rule base considering the descriptors of area \(A\), eccentricity \(E\), and fractal dimension \(D\). Each rule \(R_{r},\ r=1,2,\ldots,27\) defines a non-repeated classification of the 3 descriptors as (\(L\)) low, (\(M\)) medium, and (\(H\)) high. Column \(y_{r}\) its outputs 1, 2, and 3 corresponding to pore, EVA, and fiber, respectively. Figure 6: Fuzzification pipeline.