# The Classification of Inertinite Macerals in Coal Based on the Multifractal Spectrum Method
## 3 Methods
### Multifractal Analysis and Feature Extraction  
We express grayscale images of inertinite macroals as two-dimensional matrices and analyze them in accordance with the multifractal detrended fluctuation analysis introduced previously. It is worth mentioning that in the partitioning process, the upper-right and bottom areas are ignored since the image sizes of \(M\) and \(N\) are not particular multiples of the small square \(s\). Hence, we can repeat the partitioning process in the other three directions. Taking the typical microscopic images in Figure 1 as examples, we calculate the scaling exponent \(h(q)\) with different values of \(q\) in the range from \(-6\) to \(6\); then, the corresponding function \(\tau(q)\) can be obtained according to Equation (8). The result \(\tau(q)\) is given in Figure 2, and the inset displays the scaling exponent \(h(q)\). We can find that the function \(\tau(q)\) is nonlinear with respect to \(q\), which indicates that the exponent \(\tau(q)\) is dependent on \(q\). Nonlinearity also confirms that the microscopic images of inertinite do possess multifractal nature.  
According to Equations (9) and (10), we calculate the multifractal spectra of the macroals of inertinite, which are displayed in Figure 3. Their graphs are typically barbed, indicating that different parts with different singularities have different fractal dimensions, confirming the multifractal properties of our microscopic images. The multifractal singularity spectrum is a single-peak map normally, and several important multifractal feature parameters can be extracted as the texture descriptors of the corresponding image, such as the minimum value of the local singularity \(\alpha_{\text{min}}\), the maximum value of the local singularity \(\alpha_{\text{max}}\), and the maximum value of the spectrum \(f_{\text{max}}\).  
Figure 3: Multifractal spectra of microscopic images of the typical inertinite macroals. (**a**) Pyrofusinite; (**b**) oxyfusinite; (**c**) semifusinite; (**d**) secretinite; (**e**) funginite; (**f**) macroinite; (**g**) inertodetrinite; (**h**) micirinite.  
Figure 2: Dependence of \(\tau(q)\) and \(h(q)\) on \(q\) for the typical microscopic images of inertinite macroals. (**a**) Pyrofusinite; (**b**) oxyfusinite; (**c**) semifusinit; (**d**) secretinite; (**e**) funginite; (**f**) macroinite; (**g**) inertodetrinite; (**h**) micirinite.  
Additionally, the multifractal descriptors of \(\alpha_{\min}\), \(\alpha_{\max}\), and \(f_{\max}\) are used to build a three-dimensional space to test the distinguishing ability of each of the two groups. We calculated the multifractal spectra of 480 grayscale images in the inertinite data set, and their corresponding multifractal descriptors are plotted in Figure 4, respectively. We can find that it is not difficult to distinguish different groups due to the fact that the same components are clustered together and different macerals are separated in the space. It is worth mentioning that a small number of combinations of macerals have a certain degree of overlap due to a high similarity between their textures. However, the majority of combinations are separable in our three-dimensional space.