## 4 Applying MFDFA to the Outline of Coal and Gangue  
The sampled images of coal and gangue are compressed and binarized to detect edges. As shown in Figure 4, the identified edges of coal and gangue are highlighted with magenta curves. Even though some irrelevant scattered points are detected, it is not difficult to identify the outline curve series from the detected edges. The identified edges are carried out and transformed into the polar coordinates from the center at the accuracy of 0.1 degrees. The random walk series of the outline curves of coal and gangue can be obtained accordingly with 3600 points in Figure 5.  
According to the MFDFA algorithm introduced in Section 3, the MFDFA fluctuation functions \(F_{q}(s)\) for the outline curves of coal and gangue are shown versus the scale \(s\) in a log-log plot with \(q=-3\), \(-1\), \(1\), \(3\) in Figure 6. The \(q\)-order generalized Hurst parameter can now be defined and viewed as the slopes \(H_{q}\) of regression lines for each \(q\)-order fluctuation function \(F_{q}(s)\). The contrasting \(q\) dependence of the gangue outline curve compared with that of coal can clearly be seen in Figure 6.  
Figure 4: Images of: coal (**a**); and gangue (**b**). Edge detections of: coal (**c**); and gangue (**d**). The identified edges of coal and gangue are highlighted with magenta curves. Figure 5: Outline curves of: coal (**a**); and gangue (**b**). The identified edges are carried out and transformed into the polar coordinates from the center at the accuracy of 0.1 degrees. The \(q\)-order generalized Hurst parameter can now be defined and viewed as the slopes \(H_{q}\) of regression lines for each \(q\)-order fluctuation function \(F_{q}(s)\). The contrasting \(q\) dependence of the gauge outline curve compared with that of coal can clearly be seen. Next, the multifractal spectrum of the outline curves of coal and gangue can be obtained (Figure 7). The origin of multifractality of a time series can be distinguished as two different types, i.e., the multifractality due to: (i) the different long-range correlations of the number fluctuations; and (ii) the broadness of probability density function (PDF) of the distributions [29]. The easiest way to eliminate the correlations for (i) is shuffling the original series into random order, since the multifractality is due to the probability density, which is not affected by the shuffling  
Figure 6: Fluctuation functions of the outline curves for: (**a**) coal; and (**b**) gangue. The \(q\)-order generalized Hurst parameter can now be defined and viewed as the slopes \(H_{q}\) of regression lines for each \(q\)-order fluctuation function \(F_{q}(s)\). The contrasting \(q\) dependence of the (**a**) coal outline curve compared with that of (**b**) gangue can clearly be seen in the above figure. Figure 7: Multifractal spectrum of: (**a**) coal and (**b**) gangue outline curve series. The width of the multifractal spectrum is defined as \(\Delta h\). procedure. For (ii), the surrogate process of data, defined as replacing the phase of discrete Fourier transform (DFT) coefficients of the original data with a set of pseudo independent and identically distributed quantities in \((-\pi,pi)\), can change the broad PDF of the original data into the Gaussian distribution and seldom destroys the intrinsic long-range correlations of the original data [29]. \[F_{q}(s)/F^{shuf}(s) \sim s^{h(q)-h_{shuf}(q)}=s^{h_{corr(q)}} \tag{16}\] \[F_{q}(s)/F^{surr}(s) \sim s^{h(q)-h_{surr}(q)}=s^{h_{PDF(q)}} \tag{17}\]  
where \(H_{shuf}(q)\) is the generalized Hurst parameter of the shuffled data, \(H_{corr}(q)\) is the generalized Hurst exponent of the only long-range correlation data, \(H_{surr}(q)\) denotes the generalized Hurst parameter of the surrogate data, and \(H_{PDF}(q)\) indicates the generalized Hurst exponent of the only broad PDF data. If only the fat PDF causes the multifractality of time series, then \(H(q)=H_{shuf}(q)\) and \(H_{corr}(q)=0\).