# Extraction of Coal and Gangue Geometric Features with Multifractal Detrending Fluctuation Analysis

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## Abstract

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## 1. Introduction

- Read digital image obtained from the camera by computers.
- Determine the identification characteristics of coal and gangue.
- Find gangue materials based on the identification algorithms.
- Determine the size and the location of gangue, and then convert these information to the control signals of high pressure air nozzle.

## 2. Preliminaries

#### 2.1. Hurst Parameter, ACF, LRD, FD

#### 2.2. fBm

## 3. MFDFA Algorithm

- Transform original data into mean-reduced cumulative sums,$${Y}_{j}=\sum _{i=1}^{j}({X}_{i}-\overline{X}),\phantom{\rule{3.33333pt}{0ex}}j=1,\cdots ,N,$$
- Divide time series ${Y}_{j}$ into ${N}_{s}=\mathrm{int}(N/s)$ non-overlapping segments of equal length s, starting from the beginning. Since the length N of the series is often not a multiple of the considered time scale s, to not miss any data, another set of segments starting from the end of data is made. As a result, $2{N}_{s}$ segments are obtained covering the whole dataset.
- Calculate the local trend for each of the segments $k=1,\cdots ,2{N}_{s}$ by a least-square fit of the series.
- Calculate the mean square error ${F}^{2}(k,s)$ for the estimate of each segment k of length s.$${F}^{2}(k,s)=\frac{1}{s}\sum _{i=1}^{s}{\left(E[(k-1)s+i]-{p}_{k}\left[i\right]\right)}^{2},$$$${F}^{2}(k,s)=\frac{1}{s}\sum _{i=1}^{s}{\left(E[N-(k-{N}_{s})s+i]-{p}_{k}\left[i\right]\right)}^{2},$$
- Average all segments to obtain the qth order variance (or fluctuation) function ${F}_{q}\left(s\right)$ for each size s:$${F}_{q}\left(s\right)={\left(\frac{1}{2{N}_{s}}\sum _{k=1}^{2{N}_{s}}{\left[{F}^{2}(k,s)\right]}^{q/2}\right)}^{1/q}.$$$${F}_{0}\left(s\right)=\mathrm{exp}\left\{\frac{1}{4{N}_{s}}\sum _{k=1}^{2{N}_{s}}\mathrm{ln}\left({F}^{2}(k,s)\right)\right\},$$
- Repeat Steps (2)–(5) for different s evaluating new sets of variances ${F}_{q}\left(s\right)$.
- Plot ${F}_{q}\left(s\right)$ for each q in log–log scale and estimate the linear fit with least squares. If slope $h\left(q\right)$ varies with q, multifractality is suspected. Single slope shows monofractal scaling.
- Calculate multifractal exponent $t\left(q\right)$ as$$t\left(q\right)=qh\left(q\right)-1.$$
- Use Legendre transform to evaluate the q-order singularity–Hölder exponent $h\left(q\right)$ and corresponding dimension $D\left(q\right)$:$$\left\{\begin{array}{c}h\left(q\right)=dt\left(q\right)/dq\\ D\left(q\right)=qh\left(q\right)-t\left(q\right)\end{array}\right.$$

## 4. Applying MFDFA to the Outline of Coal and Gangue

## 5. Pattern Recognition Methods and Discussions

#### 5.1. Grayscale and Texture Features of the Image

#### 5.2. Discussions

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The automated coal–gangue separation system with computer vision. PLC: Programmable Logic Controller, LED: Light Emitting Diode.

**Figure 3.**Fractional Brownian motion (fBm) and fractional Gaussian noise (fGn). In the left panel, from top to bottom, are fBm with H = 0.5, 0.6, 0.7, and 0.8. In the right panel, from top to bottom, are fGn with H = 0.5, 0.6, 0.7, and 0.8. With the increase of Hurst exponent H, the coupling effects of fBm and fGn are strengthened.

**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.

**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 ${F}_{q}\left(s\right)$. 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 $\Delta h$.

**Figure 8.**Multifractal spectrum of the shuffled (

**a**) coal and (

**b**) gangue outline curve series. The multifractal spectrums become narrower after the removal of the long-range correlations.

**Figure 9.**Spectrum widths $\Delta \left({h}_{q}\right)$ of: (

**a**) the original coal and gangue curve series; and (

**b**) the shuffled series. A clear threshold value at 0.4 is marked with dot dash line in (

**b**) while no such confident line can be drawn in (

**a**).

**Figure 10.**Images of: (

**a**) coal; and (

**b**) gangue; and grayscale histograms of: (

**c**) coal; and (

**d**) gangue respectively.

**Figure 11.**(

**a**) Grayscale features of coal and gangue; (

**b**) Texture features of coal and gangue; (

**c**) Geometric features of coal and gangue.

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**MDPI and ACS Style**

Liu, K.; Zhang, X.; Chen, Y.
Extraction of Coal and Gangue Geometric Features with Multifractal Detrending Fluctuation Analysis. *Appl. Sci.* **2018**, *8*, 463.
https://doi.org/10.3390/app8030463

**AMA Style**

Liu K, Zhang X, Chen Y.
Extraction of Coal and Gangue Geometric Features with Multifractal Detrending Fluctuation Analysis. *Applied Sciences*. 2018; 8(3):463.
https://doi.org/10.3390/app8030463

**Chicago/Turabian Style**

Liu, Kai, Xi Zhang, and YangQuan Chen.
2018. "Extraction of Coal and Gangue Geometric Features with Multifractal Detrending Fluctuation Analysis" *Applied Sciences* 8, no. 3: 463.
https://doi.org/10.3390/app8030463