Research status and prospects of the fractal analysis of metal material surfaces and interfaces

As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions.Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces.However,fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials.From the perspective of two-dimensional planes and three-dimensional curved surfaces,this paper reviews the current research status of the fractal analysis of metal precipitate interfaces,metal grain boundary interfaces,metal-deposited film surfaces,metal fracture surfaces,metal machined surfaces,and metal wear surfaces.The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized.Starting from three perspectives of fractal analysis,namely,research scope,image acquisition methods,and calculation methods,this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed.It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.

A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

New seismic attribute:Fractal scaling exponent based on gray detrended fluctuation analysis

Seismic attributes have been widely used in oil and gas exploration and development. However, owing to the complexity of seismic wave propagation in subsurface media, the limitations of the seismic data acquisition system, and noise interference, seismic attributes for seismic data interpretation have uncertainties. Especially, the antinoise ability of seismic attributes directly affects the reliability of seismic interpretations. Gray system theory is used in time series to minimize data randomness and increase data regularity. Detrended fluctuation analysis （DFA） can effectively reduce extrinsic data tendencies. In this study, by combining gray system theory and DFA, we propose a new method called gray detrended fluctuation analysis （GDFA） for calculating the fractal scaling exponent. We consider nonlinear time series generated by the Weierstrass function and add random noise to actual seismic data. Moreover, we discuss the antinoise ability of the fractal scaling exponent based on GDFA. The results suggest that the fractal scaling exponent calculated using the proposed method has good antinoise ability. We apply the proposed method to 3D poststack migration seismic data from southern China and compare fractal scaling exponents calculated using DFA and GDFA. The results suggest that the use of the GDFA-calculated fractal scaling exponent as a seismic attribute can match the known distribution of sedimentary facies.

Fractal analysis of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces

The fractal dimensions in different topological spaces of polyferric chloride-humic acid （PFC-HA） flocs, formed in flocculating different kinds of humic acids （HA） water at different initial pH （9.0, 7.0, 5.0） and PFC dosages, were calculated by effective densitymaximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions （De） of PFC-HA floes were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7,0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respecively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 （logA vs. logdL） and D3 （logVsphere vs. logdL） of PFC-HA floes decreased with the increase of PFC dosages, and PFC-HA floes showed a gradually looser structure. At the optimum dosage of PFC, the D2 （logA vs. logdL） values of the flocs show 14%-43% difference with their corresponding Dr, and they even had different tendency with the change of initial pH values. However, the D2 values of the floes formed at three different initial pH in HA solution had a same tendency with the corresponding Df. Based on fractal Frenkel-Halsey-HiU （FHH） adsorption and desorption equations, the pore surface fractal dimensions （Ds） for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA floes dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

Identification of Geochemical Anomaly by Multifractal Analysis

The separation of anomalies from geochemical background is an important part of data analysis because lack of such identifications might have profound influence on or even distort the final analysis results. In this article, 1 672 geochemical analytical data of 11 elements, including Cu, Mo, Ag, Sn, and others, from a region within Tibet, South China, are used as one example. Together with the traditional anomaly recognition method of using the iterative mean ±2σ local multifractality theory has been utilized to delineate the ranges of geochemical anomalies of the elements. To different degrees, on the basis of original data mapping, C-A fractal analysis and singularity exponents, Sn differs from the other 10 elements. Moreover, geochemical mapping results based on values of the multifractal asymmetry index for all elements delineate the highly anomalous area. Similar to other 10 elements, the anomalous areas of Sn delineated by the asymmetry index distribute along the main structure orientations. According to the asymmetry indexes, the 11 elements could be classified into 3 groups： （1） Ag and Au, （2） As-Sb-Cu-Pb-Zn-Mo, and （3） Sn-Bi-W. This parageneflc association of elements can be used to interpret possible origins of mineralization, which is in agreement with petrological analysis and field survey results.

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis

· AIM: To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method.·METHODS: This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal(24 images) and pathological(148 images)states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software Image J. Statistical analyses were performed for these groups using Microsoft Office Excel2003 and Graph Pad In Stat software.·RESULTS: It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy(DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images(segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR(NPDR) images(segmented and skeletonized versions).The average of fractal dimensions D for the normal images(segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images(segmented and skeletonized versions). The lowestvalues were found for the corresponding values of severe NPDR images(segmented and skeletonized versions).· CONCLUSION: The fractal analysis of fundus photographs may be used for a more complete understanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension.Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from healthy individuals.

Use of digital image analysis combined with fractal theory to determine particle morphology and surface texture of quartz sands

The particle morphology and surface texture play a major role in influencing mechanical and hydraulic behaviors of sandy soils. This paper presents the use of digital image analysis combined with fractal theory as a tool to quantify the particle morphology and surface texture of two types of quartz sands widely used in the region of Vitória, Espírito Santo, southeast of Brazil. The two investigated sands are sampled from different locations. The purpose of this paper is to present a simple, straightforward,reliable and reproducible methodology that can identify representative sandy soil texture parameters.The test results of the soil samples of the two sands separated by sieving into six size fractions are presented and discussed. The main advantages of the adopted methodology are its simplicity, reliability of the results, and relatively low cost. The results show that sands from the coastal spit(BS) have a greater degree of roundness and a smoother surface texture than river sands(RS). The values obtained in the test are statistically analyzed, and again it is confirmed that the BS sand has a slightly greater degree of sphericity than that of the RS sand. Moreover, the RS sand with rough surface texture has larger specific surface area values than the similar BS sand, which agree with the obtained roughness fractal dimensions. The consistent experimental results demonstrate that image analysis combined with fractal theory is an accurate and efficient method to quantify the differences in particle morphology and surface texture of quartz sands.

Fractal Method for Statistical Analysis Geological Data

This paper establishes the phase space in the light of spacial series data , discusses the fractal structure of geological data in terms of correlated functions and studies the chaos of these data . In addition , it introduces the R/S analysis for time series analysis into spacial series to calculate the structural fractal dimensions of ranges and standard deviation for spacial series data -and to establish the fractal dimension matrix and the procedures in plotting the fractal dimension anomaly diagram with vector distances of fractal dimension . At last , it has examples of its application .

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.

Mineralization-related geochemical anomalies derived from stream sediment geochemical data using multifractal analysis in Pangxidong area of Qinzhou-Hangzhou tectonic joint belt, Guangdong Province, China

Distinguishing geochemical anomalies from background is a basic task in exploratory geochemistry. The derivation of geochemical anomalies from stream sediment geochemical data and the decomposition of these anomalies into their component patterns were described. A set of stream sediment geochemical data was obtained for 1 880 km 2 of the Pangxidong area, which is in the southern part of the recently recognized Qinzhou-Hangzhou joint tectonic belt. This belt crosses southern China and tends to the northwest (NE) direction. The total number of collected samples was 7 236, and the concentrations of Ag, Au, Cu, As, Pb and Zn were measured for each sample. The spatial combination distribution law of geochemical elements and principal component analysis (PCA) were used to construct combination models for the identification of combinations of geochemical anomalies. Spectrum-area (S-A) fractal modeling was used to strengthen weak anomalies and separate them from the background. Composite anomaly modeling was combined with fractal filtering techniques to process and analyze the geochemical data. The raster maps of Au, Ag, Cu, As, Pb and Zn were obtained by the multifractal inverse distance weighted (MIDW) method. PCA was used to combine the Au, Ag, Cu, As, Pb, and Zn concentration values. The S-A fractal method was used to decompose the first component pattern achieved by the PCA. The results show that combination anomalies from a combination of variables coincide with the known mineralization of the study area. Although the combination anomalies cannot reflect local anomalies closely enough, high-anomaly areas indicate good sites for further exploration for unknown deposits. On this basis, anomaly and background separation from combination anomalies using fractal filtering techniques can provide guidance for later work.

Implication of Surface Fractal Analysis to Evaluate the Relative Sensitivity of Topography to Active Tectonics,Zagros Mountains, Iran

Fractal geometry is increasingly becoming a useful tool for modeling and quantifying the complex patterns of natural phenomena. The Earth's topography is one of these phenomena that have fractal characteristics. This paper investigates the relative sensitivity of topography to active tectonics using ASTER Global Digital Elevation Model. The covering divider method was used for direct extraction of surface fractal dimension(D surf) to estimate the roughness-surface of topography with aid of geographic information system(GIS)techniques. This evaluation let us highlight the role of the geomorphic and tectonic processes on the spatial variability of fractal properties of natural landforms.Geomorphic zones can be delineated using fractal dimension mapping in which variability of surface fractal dimension reflects the roughness of the landform surface and is a measure of topography texture. Obtained results showed this method can be a quick and easy way to assess the distribution of land surface deformation in different tectonic settings. The loose alluvial deposits and irregularities derived by tectonic activity have high fractal dimensions whereas the competent formations and higher wavelength folded surfaces have lower fractal dimensions.According to the obtained results, the Kazerun Fault Zone has a crucial role in the separation of the Zagros Mountain Ranges into the different lithological,geomorphological and structural zones.

Integrating image processing and deep learning for effective analysis and classification of dust pollution in mining processes

A comprehensive evaluation method is proposed to analyze dust pollution generated in the production process of mines.The method employs an optimized image-processing and deep learning framework to characterize the gray and fractal features in dust images.The research reveals both linear and logarithmic correlations between the gray features,fractal dimension,and dust mass,while employing Chauvenel criteria and arithmetic averaging to minimize data discreteness.An integrated hazardous index is developed,including a logarithmic correlation between the index and dust mass,and a four-category dataset is subsequently prepared for the deep learning framework.Based on the range of the hazardous index,the dust images are divided into four categories.Subsequently,a dust risk classifcation system is established using the deep learning model,which exhibits a high degree of performance after the training process.Notably,the model achieves a testing accuracy of 95.3%,indicating its efectiveness in classifying diferent levels of dust pollution,and the precision,recall,and F1-score of the system confrm its reliability in analyzing dust pollution.Overall,the proposed method provides a reliable and efcient way to monitor and analyze dust pollution in mines.

Fractal analysis of retinal vasculature in normal subjects on ultra-wide field fluorescein angiography

AIM: To evaluate the fractal feature of the retinal vasculature of normal eyes on a stereographic projected and montaged ultra-wide field(UWF) fluorescein angiography(FA).METHODS: Prospective, observational, cross-sectional study. Totally 59 eyes of 31 normal subjects were imaged using the Optos 200 Tx. Images obtained at different gaze angles stereographically projected and montaged. The early-phase UWF FA frames were processed to segment the retinal vasculature and the results were exported as binary masks. The fractal dimension(FD) was calculated using the box-counting method.RESULTS: The global FD for the entire retina was 1.6±0.04, with no difference between males and females(1.59±0.04 vs 1.61±0.04, P=0.084) or between right and left eyes(1.6±0.04 vs 1.6±0.05, P=0.61). FD was nonuniformly distributed among four quadrants(P<0.001) and decreased as the distance from the fovea increased(P<0.001). A negative association was observed between FD and age(R=-0.37, P=0.006), and this relationship was observed in the posterior and mid-peripheral retina(P<0.05) but absent in far-periphery(P>0.05).CONCLUSION: Fractal geometry is non-uniformly distributed across the retina in normal eyes and decreases from the fovea to the far-periphery. Subjects with an older age tend to have a smaller FD, however, the FD in the farperiphery does not appear to be influenced by age.

Ti(C, N) BASED CERAMETS AND ITS FRACTAL ANALYSIS OF FRACTURE SURFACES

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Chaos, Fractal and Recurrence Quantification Analysis of Surface Electromyography in Muscular Dystrophy

We analyze muscular dystrophy recorded by sEMG and use standard methodologies and nonlinear chaotic methods here including the RQA. We reach sufficient evidence that the sEMG signal contains a large chaotic component. We have estimated the correlation dimension (fractal measure), the largest Lyapunov exponent, the LZ complexity and the %Rec and %Det of the RQA demonstrating that such indexes are able to detect the presence of repetitive hidden patterns in sEMG which, in turn, senses the level of MU synchronization within the muscle. The results give also an interesting methodological indication in the sense that it evidences the manner in which nonlinear methods and RQA must be arranged and applied in clinical routine in order to obtain results of clinical interest. We have studied the muscular dystrophy and evidence that the continuous regime of chaotic transitions that we have in muscular mechanisms may benefit in this pathology by the use of the NPT treatment that we have considered in detail in our previous publications.

Visual Evaluation of the Morphologic Structure of Nonwovens Using Image Analysis and Fractal Geometry

Nonwovens are fiber materials which are based on nonwoven technologies. For the complexity and randomness of nonwovens morphologic structures, it is difficult to express them effectively using classical method. Fractal geometry gives us a new idea and a powerful tool to study on irregularity of geometric objects. Therefore, we studied on the pore size, pore shape, pore size distribution and fiber orientation distribution of real nonwovens using fractal geometry combined with computer image analysis to evaluate nonwovens’ morphologic structures.

Fractal description and quantitative analysis of normal brain development in infants

We examined the fractal pattern of cerebral computerized tomography images in 158 normal infants aged 0 3 years, based on the quantitative analysis of chaotic theory. Results showed that the fractal dimension of cerebral computerized tomography images in normal infants remained stable from 1.86-1.91. The normal distribution range in the neonatal period, 1-2 months old infants, 1-2 year old infants, and of 2-3 year old infants was 1.88 1.90 （mean： 1.891 3 ± 0.006 4）, 1.89-1.90 （mean： 1.892 7 ±0.004 5）, 1.86-1.90 （mean： 1.886 3 ± 0.008 5）, and 1.88-1.91 （mean： 1.895 8±0.008 3）, respectively. The spectrum width of the multifractal spectrum （△α） in normal infants was 1.4618. These data suggest that the spectral width parameters of the multifractal spectrum and the fractal dimension criteria in normal children may be useful as a practical specific parameter for assessing the fractal mode of brain development in normal infants.

GENERALIZED FLOW ANALYSIS OF NON-NEWTONIAN VISCO-ELASTICFLUID FLOW THROUGH FRACTAL RESERVOIR

In this paper, fractal geometry theory is used to combine with the seepage flow mechanics to establish the relaxation models of non_Newtonian visco_elastic fluid flow in fractal reservoirs. A method to scale the fractal properties of a fractal reservoir by the double parameters (d f ,d s ) and to describe the generalized flow characteristics of visco_elastic fluid by four parameters (d f ,d s ,λ v,λ p) are presented. Exact solutions and asymptotic solutions have been obtained by using the Laplace_Weber and Laplace_orthogonal transforms with both infinite and finite reservoirs. The pressure transient behavior of non_Newtonian visco_elastic fluid flow through a fractal reservoir are studied by using the numerical Laplace transform inversion and asymptotic solutions. The law of pressure change for various fractal parameters is obtained.

Fractal Analysis of Mobile Social Networks

Fractal and self similarity of complex networks have attracted much attention in recent years. The fractal dimension is a useful method to describe the fractal property of networks. However, the fractal features of mobile social networks （MSNs） are inadequately investigated. In this work, a box-covering method based on the ratio of excluded mass to closeness centrality is presented to investigate the fractal feature of MSNs. Using this method, we find that some MSNs are fractal at different time intervals. Our simulation results indicate that the proposed method is available for analyzing the fractal property of MSNs.