Fractal Dimension Spectrum Research Articles

Blood Plasma Film Multifractal Scanning in COVID-19 Consequences Diagnostics.

A 3D phase scanning method was applied to study blood plasma facies, generating layered polarization maps of the object field. The most sensitive parameters to changes in birefringence distribution were identified. Multifractal analysis using wavelet transforms and fractal dimension spectra provided specific insights into the scale self-similarity of the polarization maps. The multifractal spectra of ellipticity distributions were algorithmically derived, revealing that the third- and fourth-order statistical moments were most sensitive to changes in the supramolecular networks of the facies. These findings were successfully applied to differentiate post-COVID-19 effects with high accuracy.

Characterization of rutting damage based on two-dimensional image analysis of changes in mesoscopic aggregate properties of asphalt mixtures

Accurately predicting the rutting development law of asphalt pavement is of great significance for preventing and controlling pavement diseases. The aggregate skeleton of the asphalt mixture is the main load-bearing part and plays an important role in its anti-rutting performance. However, quantitative and mesoscopic evaluation indicators and standards for aggregate skeletons have not been established. This study aims to reveal the relationship between the mesostructured characteristics of asphalt mixtures and rutting damage. This paper uses two-dimensional image technology to analyze the dynamic change process of internal skeleton information of two graded asphalt mixtures under cyclic wheel load. According to the fractal theory, the fractal dimension and multifractal spectrum of the full-depth aggregate were calculated. The results indicate that the linear regression model between the mesoscopic skeleton information of asphalt mixture and the average rutting depth during the rutting test can be used to quantify the rutting performance. There is an obvious correlation between the fractal dimension of the section aggregate and the loading times, and there is a certain linear relationship between the multifractal spectrum indexes and the loading times. It indicates that the fractal theory index of aggregate can reflect the change of the macro performance of loading times.

Nanoscale surface dynamics of spatial patterns of polymeric bilayered particles loaded with essential oil.

Gelatin/PCL bilayered particles loaded with Piper nigrum essential oil was synthesized aiming to access their morphological and surface dynamic patterns. Atomic force microscopy (AFM) was applied to investigate the 3D morphology and multifractal aspects of the particles surface. The AFM maps revealed spherical surfaces and well dispersed particles, besides a rougher surface on the loaded system. Minkowski functionals showed that shape of the rough peaks was similar in the unloaded and loaded systems; however, the presence of deep valleys on the loaded particles revealed their rougher pattern. Multifractal analysis revealed that unloaded and loaded particles presented multifractal behavior with different surface dynamics. The loaded surface presented a greater width of the multifractal spectrum and smaller difference of fractal dimensions, confirming their more vertically growing. These results can be useful in the development of novel polymeric-based particles loaded with essential oil. Their unique surface dynamics can provide enhanced physical properties and performance in emerging biotechnological applications.

Application of fractal and multifractal analysis on Blue Nile drainage patterns in the morphostructural analysis of the Ethiopian highlands, Ethiopia

Fractal geometry methods provide a quantitative description of complex landscape networks, such as drainage patterns. This article focuses on the determination of the fractal dimension ( F D) and Renyi multifractal spectrum ( F Dα) of drainage patterns of the Blue Nile Basin in the Ethiopian Highlands. The Ethiopian Highlands are one of the most tectonically influenced areas in the world. The spatial distribution of drainage patterns reflects a decreasing influence of tectonics with increasing distance from the Main Ethiopian Rift (MER) to the center of the Ethiopian Highlands. The novelty of the paper is given by the calculation of F D and F Dα of drainage patterns as a part of morphostructural analysis. We analyzed how the values of F D and F Dα in each of 145 local drainage patterns changed depending on the changes of tectonic influences on landscape formation. Our methodologies provided a way of distinguishing the morphologies of tectonic and non-tectonic drainage patterns using the fractal/multifractal technique. The results of the study show that old dendritic drainage patterns located in the western part with low tectonic activity and an absence of faults had the lowest F D and F Dαvalues. The opening up of the MER and the neotectonic uplift of the Ethiopian Highlands accelerated the development of drainage patterns by activating river erosion and river piracy. Dendritic drainage patterns of the volcanic platforms have changed to young deep cut trellis patterns, and their F D and F Dα values are increased. Accelerating Quaternary uplift created the recent high landscape in the eastern part of the Blue Nile Basin, where the tectonic activities correspond to the MER. This young Quaternary landscape is characterized by prevailing deep and narrow valleys, and the presence of morphodynamic phenomena (erosion, landslides, earthquakes), where the trellis drainage patterns have the highest F D and F Dα.values.

Fractal and multifractal characterization of stochastic fracture networks and real outcrops

The fractal dimension and multifractal spectrum are widely used to characterize the complexity of natural fractures. However, systematic investigations, considering impacts of different fracture geometrical properties (fracture lengths, orientations, center positions) and system sizes, on the fractal and multifractal characterization of complex fracture networks are insufficient. Here, we utilize an in-house developed DFN modeling software, hatchfrac, to construct stochastic fracture networks with prescribed distributions and systematically study the impact of three geometrical properties of fractures and system sizes on the fractal and multifractal characterization. We calculate the single fractal dimension and multifractal spectrum with the box-counting method. The single fractal dimension, D, and the difference of singularity exponent, Δα, are used to represent the fractal and multifractal patterns, respectively. We find that fracture lengths, orientations and system sizes positively correlate with D and Δα, while the system size has the most significant impact among the four parameters. D is uncorrelated with fracture positions (FD), which means that a single fractal dimension cannot capture the complexity caused by clustering effects. However, Δα has a strong negative correlation with FD, implying that clustering effects make fracture networks more complex, and Δα can capture the difference. We also digitize 80 outcrop maps with a novel fracture detection algorithm and calculate their fractal dimension and multifractal spectrum. We find wide variations of D and Δα on those outcrop maps, even for outcrops at similar scales, indicating that a universal indicator for characterizing fracture networks at different scales or the same scale is almost impossible. D and Δα have negligible correlations with scales, supporting the self-similarity patterns of natural fracture networks.

Studying the fractal properties of Ceres

Currently, the asteroid Ceres belongs to small celestial bodies with the most well-known physical parameters. The study of the structural and real properties of Ceres is an urgent and modern task, the solution of which will make it possible to develop the evolutionary theory of a minor planet. In this work, the fractal properties of the dwarf planet Ceres were analyzed using data from the Dawn space mission. Using the expansion in a harmonic series in spherical functions the height parameters of the structural model of Ceres, a 3D model of Ceres was constructed. The analysis showed that the resulting system has a complex multiparameter fractal configuration. The study of such objects requires the use of harmonic multiparameter methods. Multivariate fractal analysis allows to represent systems similar to the Ceres model in the form of a spectrum of fractal dimensions. The advantage of fractal analysis is the ability to explore local areas of the physical surface. In this work, the Minkowski algorithm was used for this purpose. At the final stage, an overdetermined system was solved for various local areas of topocentric information in order to postulate a model that takes into account external measures. Fractal dimensions D are determined for local regions and the entire model of the planet. Fractal dimensions vary from 1.37 to 1.92 depending on the longitude and latitude of Ceres. The main results are as follows: 1) the structure of the Ceres surface varies more strongly in longitude; 2) the structure of Ceres is smoother in latitude; 3) the coefficient of self-similarity changes rather quickly in longitude, which indicates that different local regions of the minor planet were formed under the influence of various physical processes. It is necessary to emphasize that the resulting fractal dimensions are significantly scattered both in longitude and latitude of Ceres. This fact confirms the presence of a complex structure in the spatial model of a minor planet. This also applies to the actual physical surface of Ceres. The results of the work allow us to conclude that fractal modeling can give independent values of the fractal dimension both for the entire model of Ceres and for its local macrostructural regions.

Fractal and Multifractal Analysis of PET-CT Images for Therapy Assessment of Metastatic Melanoma Patients under PD-1 Inhibitors: A Feasibility Study.

Simple SummaryThe fractal dimension (FD) and the multifractal spectrum (MFS) are nonlinear quantitative measures which express the heterogeneity in the distribution of the tracer, F-18-fluorodeoxyglucose, (18F-FDG), in the body of patients suffering from metastatic melanoma. Given the well-documented, high accumulation of the tracer in tumor/metastatic sites, the measures expressing the tracer distribution also express the extent of metastases in the body. As such, FD and MFS can be employed to detect the presence of melanoma and to monitor the therapeutic outcome using the PET-CT follow-up digitized scans of the patients. In the present study, the FD and MFS measures of patients are evaluated before and during treatment with PD-1 inhibitors and are compared with the corresponding values of healthy controls. The MFS predictions agree with the PET Response Evaluation Criteria for Immunotherapy (PERCIMT) in 81% of the cases, while the FD agrees in 77% of all cases. Therefore, the quantitative MFS is proposed as an additional, alternative biomarker for monitoring the immunotherapy outcome in melanoma patients, after treatment with PD-1 inhibitors.Longitudinal whole-body PET-CT scans with F-18-fluorodeoxyglucose (18F-FDG) in patients suffering from metastatic melanoma were analyzed and the tracer distribution in patients was compared with that of healthy controls. Nineteen patients with metastatic melanoma were scanned before, after two and after four cycles of treatment with PD-1 inhibitors (pembrolizumab, nivolumab) applied as monotherapy or as combination treatment with ipilimumab. For comparison eight healthy controls were analyzed. As quantitative measures for the comparison between controls and patients, the nonlinear fractal dimension (FD) and multifractal spectrum (MFS) were calculated from the digitized PET-CT scans. The FD and MFS measures, which capture the dispersion of the tracer in the body, decreased with disease progression, since the tracer particles tended to accumulate around metastatic sites in patients, while the measures increased when the patients’ clinical condition ameliorate. The MFS measure gave better predictions and were consistent with the PET Response Evaluation Criteria for Immunotherapy (PERCIMT) in 81% of the cases, while FD agreed in 77% of all cases. These results agree, qualitatively, with a previous study of our group when treatment with ipilimumab monotherapy was considered.

Adsorption behaviors of near-critical carbon dioxide on organic-rich shales: Modeling, multifractality, and kinetics

Studies on the near-critical carbon dioxide (NC-CO2) adsorption behavior of organic-rich shales are important for shallow shale gas exploration and hydrocarbon recovery enhancement of liquid CO2 fracturing, which further contributes to geological CO2 sequestration and global carbon neutrality. We performed CO2 isothermal adsorption and kinetic experiments on six organic-rich shale samples at a temperature of 303.15 K and pressures of up to 5 MPa to investigate the modeling, multifractality, and kinetics of NC-CO2 adsorption behavior on shales. The equilibrium data were fitted to the Langmuir, Freundlich, BET, modified BET (M−BET), DA, DR, and OK adsorption models; the multifractality of the isotherms was analyzed; and pseudo-first-order, pseudo-second-order, Elovich, and Avrami's fractional-order models were adopted to analyze the adsorption kinetics. Furthermore, an error analysis technique was employed to compare the accuracies of various models for fitting the experimental data. The results showed that various adsorption models presented different fitting effects; however, the M−BET, Freundlich, and DA adsorption models performed better over the entire studied pressure range. The fitting results of the M−BET model indicated that the number of adsorption layers ranged from 2.528 to 4.663 and had a multilayer adsorption mechanism. The multifractality of the adsorption behavior was described by the generalized fractal dimension and multifractal singularity spectra; we demonstrated that some multifractal parameters characterize the NC-CO2 adsorption behavior on shales well. Additionally, the TOC, clay, and quartz contents show close covariations with the isothermal adsorption and multifractal parameters. Moreover, the adsorption process was TOC-dependent and subject to Avrami's fractional-order model. Finally, we propose conceptual three-stage adsorption and geological storage models to understand the nature of the NC-CO2 adsorption behavior of shales.

Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China

Multifractal theory provides an effective method for the scientific quantification of the river network features. This method has been applied to estimate river network structure in previous research, but there are few temporal and spatial analysis studied for large river basins based on multifractals. In this study, the Yellow River Basin and its nine provinces were selected as the study area, and the generalized fractal dimension and multifractal spectrum of the study area during the period 2000–2020 were calculated and analyzed. We analyzed the response relationship between the river network changes and the urbanization process. Results indicated that the river network of the study area exhibited obvious multifractal properties, which were mainly affected by dense river networks. The complexity of the studied river network has shown a tendency to decrease over time. The rate of change in the river network structure has a high positive correlation with urbanization, that is, the higher the rate of urbanization, the greater the impact on the river network structure. Additionally, the changes of the river network structure are more affected by urbanization during the rapid urbanization stage. We applied multifractal analysis to study the river network structure changes, which is of great significance for scientifically quantifying fluvial characteristics and studying the development and evolution of river networks.

Development of Novel Fractal Method for Characterizing the Distribution of Blood Flow in Multi-Scale Vascular Tree.

Blood perfusion is an important index for the function of the cardiovascular system and it can be indicated by the blood flow distribution in the vascular tree. As the blood flow in a vascular tree varies in a large range of scales and fractal analysis owns the ability to describe multi-scale properties, it is reasonable to apply fractal analysis to depict the blood flow distribution. The objective of this study is to establish fractal methods for analyzing the blood flow distribution which can be applied to real vascular trees. For this purpose, the modified methods in fractal geometry were applied and a special strategy was raised to make sure that these methods are applicable to an arbitrary vascular tree. The validation of the proposed methods on real arterial trees verified the ability of the produced parameters (fractal dimension and multifractal spectrum) in distinguishing the blood flow distribution under different physiological states. Furthermore, the physiological significance of the fractal parameters was investigated in two situations. For the first situation, the vascular tree was set as a perfect binary tree and the blood flow distribution was adjusted by the split ratio. As the split ratio of the vascular tree decreases, the fractal dimension decreases and the multifractal spectrum expands. The results indicate that both fractal parameters can quantify the degree of blood flow heterogeneity. While for the second situation, artificial vascular trees with different structures were constructed and the hemodynamics in these vascular trees was simulated. The results suggest that both the vascular structure and the blood flow distribution affect the fractal parameters for blood flow. The fractal dimension declares the integrated information about the heterogeneity of vascular structure and blood flow distribution. In contrast, the multifractal spectrum identifies the heterogeneity features in blood flow distribution or vascular structure by its width and height. The results verified that the proposed methods are capable of depicting the multi-scale features of the blood flow distribution in the vascular tree and further are potential for investigating vascular physiology.

Multifractal Characteristics Analysis Based on Slope Distribution Probability in the Yellow River Basin, China

Multifractal theory provides a reliable method for the scientific quantification of the geomorphological features of basins. However, most of the existing research has investigated small and medium-sized basins rather than complex and large basins. In this study, the Yellow River Basin and its sub-basins were selected as the research areas, and the generalized fractal dimension and multifractal spectrum were computed and analyzed with a multifractal technique based on the slope distribution probability. The results showed that the Yellow River Basin and its sub-basins exhibit clear multifractal characteristics, which indicates that the multifractal theory can be applied well to the analysis of large-scale basin geomorphological features. We also concluded that the region with the most uneven terrain is the Yellow River Downstream Basin with the “overhanging river”, followed by the Weihe River Basin, the Yellow River Mainstream Basin, and the Fenhe River Basin. Multifractal analysis can reflect the geomorphological feature information of the basins comprehensively with the generalized fractal dimension and the multifractal spectrum. There is a strong correlation between some common topographic parameters and multifractal parameters, and the correlation coefficients between them are greater than 0.8. The results provide a scientific basis for analyzing the geomorphic characteristics of large-scale basins and for the further research of the morphogenesis of the forms.

Experimental Study on Monitoring of Dike Piping Process Based on Acoustic Emission Technology

Acoustic emission (AE) is employed in this study to monitor piping failure process in the dike subjected to varied head action. Piping is the main reason for the seepage failure of dikes and dams, and the AE characterization of piping process is investigated. The laboratory experiments and data analyses were carried out using specially designed physical model and data acquisition systems. The monitored data were analyzed using four methods, namely, AE ringing count, fractal dimension, frequency domain analyses and b value. The analysis results showed that the piping failure process can be divided into 3 phases: seepage erosion, piping generation, and piping development. In different stage, the activity of the AE signals is related to seepage flow and seepage pressure. Based on the fractal dimension and waveform energy spectrum distribution analysis in the entire process of piping failure, the measured acoustic signals in the test can reflect both a seepage-induced and a soil-seepage induced AE. The AE ringing count fractal dimension and waveform energy spectrum can be used as new index monitoring the occurrence and development of piping. The abrupt change of AE fractal dimension and AE energy spectrums curve with time can be considered as the precursor to piping occurrence and piping development. What’s more the b value of the AE signal can be used to describe the piping state.

Beautiful chaotic patterns generated using simple untrained recurrent neural networks under harmonic excitation

This paper focuses on the intrinsic chaotic behavior of recurrent neural networks under harmonic excitation. The chaotic behaviors of some untrained two-neuron RNN with strange attractors are studied in detail. Taking the input as an additional dimension, the intrinsic manifold shaped by an RNN is visualized using 3-D phase portraits. A series of isomorphic topological structures are generated to provide different aspects of the intrinsic manifold. Local fractal dimension is introduced to visualize the repetitive folding of attractors, which exhibits a spectrum of fractal dimensions concentrating around 1, 1.5 and 2. By using maximal Lyapunov exponent (MLE), our results show that when tuning the amplitude or frequency of the harmonic input, periodic and chaotic motions in the phase space emerge alternatingly. For a certain RNN, bifurcation with the topological transformation of the manifold can be observed when MLEs approach zero. Algorithm 1 is designed to search for various topological structures with potential chaotic patterns from massive randomly generated RNNs. Beautiful chaotic patterns are generated and presented in the form of a gallery appended in the end. This paper provides a novel prospect to generate topological structures with beautiful chaotic patterns using RNNs.

Multifractal analysis of embryonic eye structures from female mice with dietary folic acid deficiency. Part I: Fractal dimension, lacunarity, divergence, and multifractal spectrum

Abstract The purpose of this study was to investigate effects of low maternal folic acid (FA) status on many eye structures in embryonic mice using multifractal analysis, a method for describing complex structural patterns. To do this, we used the FracLac plugin for ImageJ to analyse binary digital images of cornea, lens, retina, and vitreous after immunohistochemical analysis using anti-collagen IV or anti-laminin-1 antibodies. The box-counting and gliding box methods were used to generate metrics including the fractal dimension, lacunarity, divergence, and multifractal spectrum distance metrics for each ocular structure. Results were analysed using bootstrap resampling techniques and the Kruskal-Wallis significance test. For all structures and staining methods studied, differences between embryos from control vs FA-deficient females were detectable in lacunarity plots and multifractal spectra, and statistically significant in cornea, lens, retina and vitreous. Multifractal analysis detected even slight differences in tissue textures presumably attributable to altered expression of extracellular matrix molecules.

Density and tracer statistics in compressible turbulence: Phase transition to multifractality.

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state. This is demonstrated to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers; however, at finite Mach numbers, the assumption of mixing is needed, which we demonstrate to be not self-evident because of the possibility of self-organization.

Using the Binary Transformation for the Characterization of Structure-Texture Boundaries of Mineral Multifractal Surfaces

The possibilities of using fractal analysis for detecting atypical areas on digital images of feldspar (cracks, kinks, adhesions and structural changes) are investigated. The selection of texture heterogeneities and morphological features is carried out using the field of fractal dimension D, the self-similarity coefficient field K, and the measure field SRGB (triangle areas in the coordinate system of properties). The influences of the displacement step as well as the size of the scanning “window” are estimated. The proposed segmentation of digital images of minerals allows one to eliminate the regions with different ranges of fractal measure values and to describe complex multifractal surfaces of minerals taking into account the spectrum of fractal dimensions, because such fractals, along with geometric features, possess certain statistical properties. The application of the binary transformation of the obtained fractal measure fields to the selected threshold value revealed a sensitivity to a quantitative and qualitative description of the complexity of structural and texture boundaries of the surface. This fact which can be used to predict the results of the ores crushing.

Radar target detection method based on cross‐correlation singularity power spectrum

Recently, radar target detection based on fractal theory has received great attention. Especially in the sea clutter background, fractal modelling, fractal dimension, and multifractal spectrum are widely used in radar signal processing. Here, a novel target detection method based on cross-correlation singularity power spectrum (CSPS) is proposed. Firstly, the CSPS theory and algorithm is derived based on SPS; secondly, based on the SPS and CSPS analysis of radar sea clutter, radar target detection parameters are designed, and the detection flow and algorithm based on the CSPS method is presented in detail. The proposed methodology is tested on sea clutter, both with and without targets, from the Ice Multi-parameter Imaging X -Band radar data set. The simulation results indicate that the proposed method performs better than the conventional multifractal-based methods and traditional CFAR methods, and the detection probability of detecting low-observable targets within sea clutters is close to 100%.

RUMOR SITUATION RECOGNITION BASED ON MULTIFRACTALS

In order to effectively identify rumors and further block their propagation, this present study established a rumor situation identification model based on multifractals and support vector machine (SVM), which considered the fractal theory and the situation of rumor propagation. First, the representative rumors and nonrumors about food safety were selected. The multifractal analysis was used to calculate the generalized fractal dimension spectrum of the two target states. After this, the generalized fractal dimension values were used as the input target eigenvector for the SVM, which was subsequently trained to obtain the best classification results. The experimental results show that the model can quickly and effectively identify rumors from the communication situation, thus providing further help for governance.

Fractal Dimension and Power Spectrum of Electroencephalography Signals of Sleep Inertia State

Human brain functions and behaviors during the transition state between sleep and wakefulness are not similar to these at alert wakefulness state. The transition state, which is called sleep inertia, has many unpleasant and dangerous effects on many situations that require full attention and fast response, such as driving. Within 30 minutes after waking up from sleep, the driver’s performance might be impaired due to the sleep inertia effects. Groups of drivers that may drive within a short period after waking up are: workers who travel early in the morning; secondary drivers of long distance bus who sleep in the bus before taking over the job from the primary drivers; night travelers; and long haul truck drivers who stop at the rest area to sleep for a while and continue driving. Previous research used subjective self-report measurement, eye tracker, and a driving simulator to analyze the driver’s performance during sleep inertia state. The physiological measures of the drivers, such as their brain signals have also been studied. However, the brain signals which are recorded in Electroencephalography (EEG) are typically analyzed in perspective of the power spectrum. This study proposes a hybrid of EEG features, which are fractal dimension and power spectrum, supported by behavioral data which is the driver’s reaction time. This study finds the features that significantly differentiate between normal and sleep inertia drivers based on the classification accuracy and p-value of the statistical ANOVA. This study compares the results with other features (power spectrum, variance, sample entropy), and between EEG channel. This study record the EEG from the Fz, T7, Cz, Pz, and O1 channels. This study uses subjective and behavioral measurements to support the results. The results show that the hybrid of fractal dimension estimated by Katz’s algorithm at the O1 channel and delta power from the Fz channel, and alpha power from the O1 channel, have better classifications than the power spectrum alone. Furthermore, the reaction time recorded from the LED reaction time task shows a significant difference between drivers with sleep inertia and normal (alert) drivers.

The study of coal tectonic disturbance using multifractal analysis of coal specimen images obtained by means of scanning electron microscopy

The possibility of using a multifractal approach to study the tectonic disturbance of coals has been investigated. The relationship between the coal disturbance and the asymmetry of fractal spectra of coal images obtained by means of scanning electron microscopy (SEM) is revealed: it has been established that undisturbed coals are characterized, as a rule, by a symmetric fractal dimension spectrum, and the disturbed coals are described by a fractal spectrum with some degree of asymmetry. It is shown that if fractal spectra of images have a symmetric appearance, then brightness distributions of these images are well fitted by a lognormal curves and parameters of these fittings can be estimated through characteristics of the fractal spectra. By using multifractal analysis of images for more than 140 test coal specimens from the quiet zone of a seam and the outburst zone, differences in the brightness distributions for images of coals with various degrees of disturbance were revealed. The basis of the research is the assumption that differences in the structure of disturbed and undisturbed coals are reflected in histograms of the brightness distributions for images of coal specimens. According to the results of multifractal analysis of images for the test coal specimens, it was established that the brightness distributions for images of the surface of undisturbed coal specimens are lognormal, while the brightness distributions for images of the surface of highly disturbed coal specimens, in most cases, deviate from the lognormal one. The conducted studies allow us to conclude about the applicability of the multifractal approach for assessing the degree of coal disturbance using digital images of coal specimens.