*4.1. Stability to Image Noise*

Textural features of images can be disturbed by noise to a great extent. In this paper, Gaussian noise, speckle noise, and salt and pepper noise are added to the inertinite microscopic images to investigate the influence of noise on the multifractal spectrum. Figure 5 gives the images of pyrofusinite with the addition of Gaussian noise (0-mean and variance of 0.05), speckle noise (variance of 0.05), and salt and pepper noise (density of 2%), respectively.

**Figure 5.** Typical images of pyrofusinite with different noises. (**a**) Gaussian noise; (**b**) speckle noise; (**c**) salt and pepper noise.

The multifractal spectra *f*(*α*) of the pyrofusinite images with various noises were calculated and the comparisons with the original image were done. As shown in Figure 6, the spectrum of pyrofusinite image with speckle noise is almost identical to the original one, which illustrates that the speckle noise has a slight influence on pyrofusinite image. In addition, we extract and report the values of multifractal descriptors of eight groups of inertinite macerals with different noises in Table 1. For comparisons, the statistical features, such as angular second moment (ASM), entropy, moment of inertia (IM), and correlation based on GLCM, were calculated and listed in Table 2. From Tables 1 and 2, we can find that the multifractal descriptors are relatively stable, while the GLCM-based texture descriptors are sensitive to noise, and the value of IM fluctuates significantly with different noises.

**Figure 6.** Multifractal spectra of the microscopic images of pyrofusinite with different noises.


**Table 1.** Multifractal descriptors of typical microscopic images of inertinite with different noises. Sample labels (a)–(h) are consistent with the labels of typical images in Figure 1.

To further clarify that our multifractal descriptors of *α*min, *α*max and *f*max possess better anti-noise stability than that of GLCM-based texture parameters, the average relative errors of feature descriptors from typical microscopic images with different noises are calculated and displayed in Figure 7. We can see that three types of noises all have a great influence on GLCM-based texture parameters, especially for the moment of inertia, whose relative error is much higher than 100% for each maceral of inertinite. The parameter of IM of secretinite is highly sensitive to different noise with a relative error close to 400%. Furthermore, the relative errors of the three multifractal texture descriptors are particularly low among the seven parameters, none of which exceeds 15%, and that of *f*max even closes in on zero. From the results of noise immunity experiment, it is clear that our descriptors of *α*min, *α*max, and *f*max possess great stability to various noises.

**Figure 7.** Relative errors of texture descriptors of typical inertinite microscopic images with different noises. (**a**) Gaussian noise; (**b**) speckle noise; (**c**) salt and pepper noise.


**Table 2.** GLCM-based descriptors of typical microscopic images of inertinite with different noises. Sample labels (a)–(h) are consistent with the labels of typical
