*4.2. Stability to Image Blurring*

Due to the limitations of objective conditions and the interference from human factors, some inevitable phenomena may occur in the process of microscopic image acquisition, such as motion blur caused by lens jitter and defocusing blur caused by inaccurate focusing, while stable texture parameters should have good immunity to these kinds of fuzzy degradation [31].

Taking the image of pyrofusinite as an example, motion blur and defocusing blur degradation are processed as shown in Figure 8. We plot their multifractal spectra in Figure 9 and compare the spectra with the original one. As can be seen from the multifractal spectra, after image blurring, the value of *f*max fluctuates slightly between 2 and 2.05, indicating the extracted texture parameter *f*max is not sensitive to blurring.

**Figure 8.** Typical images of pyrofusinite with different blurred processing methods. (**a**) Motion blurred image; (**b**) defocus blurred image.

**Figure 9.** Multifractal spectra of the microscopic images of pyrofusinite with different blurring.

In order to demonstrate the robustness of the multifractal descriptors to image blurring more convincingly, we perform motion blur and defocus blur on all microscopic images labeled (a)–(h) form Figure 1. Then the parameters of *α*min, *α*max, and *f*max are calculated; besides, the GLCM-based texture features are also calculated for comparison. Figure 10 shows the average error of these texture features. For eight types of inertinite macerals, the relative error of *f*max is close to 0, *α*min and *α*max are between 0% and 15%, which indicates that the multifractal features have excellent robustness and are insensitive to blurring. However, GlCM-based features are susceptible to image blurring. For example, the relative error of the second-order moment of the semifusinite with defocus blurring is even higher than 300%; the parameter of energy is relatively stable in the microscopic images of inertinite macerals, all of which are less than 40%. The above analysis depicts that the three multifractal descriptors possess great stability to different kinds of blurring.

**Figure 10.** The relative errors of textural descriptors of typical inertinite microscopic images with different blurring types. (**a**) Motion blurred image; (**b**) defocus blurred image.

### **5. Classification Experiment**

#### *5.1. Experiment Design*

Considering small samples, SVM is employed to build a classifier for the classification of inertinite macerals [32]. To address the non-linear and the multi-classes problem in this paper, the input data are mapped into high-dimensional space with a non-linear mapping, and the relevant classification function can be expressed as

$$F(\mathbf{x}) = \text{sgn}[\sum\_{i=1}^{n} a\_i y\_i \mathcal{K} \left(\mathbf{x}\_i, \mathbf{x}\right) + \mathbf{b}\_0],\tag{11}$$

where *ai*, *i* = 1, ..., *n* are Lagrange multipliers, the class to which a sample is assigned is labeled *yi*, and *K* (*xi*, *x*) represents a kernel function, which is the radial basis function (RBF) kernel function here.

The classification model for inertinite macerals with the SVM-based classifiers is illustrated in Figure 11. To implement the multi-classification, we construct a classifier group with 28 RBF-SVM classifiers to distinguish eight groups of inertinite macerals based on the one-against-one (1A1) technique and optimize the error parameter (usually designated *c*) and parameter *γ* in RBF kernel function by a grid search [33,34]. Besides, 40 of the microscopic samples per group are used for training, and the remaining 20 samples for testing and each classifier is used to distinguish two different classes of inertinite macerals. Then, the remaining testing samples per group are input into the trained classifiers. The specific testing process is as follows.

Step 1. Calculate the texture descriptors of *α*min, *α*max and *f*max for each image in the testing set.

Step 2. Input the texture data obtained in the previous step into the classifier group in turn.

Step 3. Count the votes in eight groups; the testing image is classified into the group with the best poll numbers.

Step 4. Repeat the above steps for the remaining images, and finally, get the category for each training images.

**Figure 11.** Classification model for inertinite macerals with the SVM-based classifiers.

#### *5.2. Evaluation Measures*

The results of the automatic classification method are quantitatively evaluated by ensemble of popular measures. The measures used in our work comprise precision, recall, and F-measure.

The class agreement of the predicted labels with the positive labels given by the classifier is estimated by precision, and the validity of the positive label recognition is measured by recall. The F-measure is defined as a scaled harmonic mean of precision and recall.

$$\text{precision}\_{i} = \frac{tp\_i}{tp\_i + fp\_i} \tag{12}$$

$$\text{recall}\_{i} = \frac{tp\_{i}}{tp\_{i} + fn\_{i}} \tag{13}$$

$$\text{F-measure}\_{i} = \frac{2 \ast \text{precision}\_{i} \ast \text{recall}\_{i}}{\text{precision}\_{i} + \text{recall}\_{i}} \text{'} \tag{14}$$

where *tpi*, *f pi*, *tni*, and *f ni* denote the values of true positives, false positives, true negatives, and false negatives for class *i*, respectively. Using the above measurements, the performance of proposed classification model can be conducted for comparison purposes. Additionally, for the purpose of comprehensively evaluating the average performance of eight groups of inertinite macerals, we consider the average values of precision (macro-precision), the average values of recall (macro-recall), and macro-F, which is a scaled harmonic mean of macro-precision and macro-recall.
