Digital Image Processing Method for Characterization of Fractures, Fragments, and Particles of Soil/Rock-Like Materials

mathematics

Article Digital Image Processing Method for Characterization of Fractures, Fragments, and Particles of Soil/Rock-Like Materials

Zizi Pi, Zilong Zhou, Xibing Li and Shaofeng Wang *

School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [email protected] (Z.P.); [email protected] (Z.Z.); [email protected] (X.L.) * Correspondence: [email protected]

Abstract: Natural soil and rock materials and the associated artiﬁcial materials have cracks, fractures, or contacts and possibly produce rock fragments or particles during geological, environmental, and stress conditions. Based on color gradient distribution, a digital image processing method was proposed to automatically recognize the outlines of fractures, fragments, and particles. Then, the fracture network, block size distribution, and particle size distribution were quantitatively characterized by calculating the fractal dimension and equivalent diameter distribution curve. The proposed approach includes the following steps: production of an image matrix; calculation of the gradient magnitude matrix; recognition of the outlines of fractures, fragments, or particles; and characterization of the distribution of fractures, fragments, or particles. Case studies show that the fractal dimensions of cracks in the dry mud layer, ceramic panel, and natural rock mass are 1.4332, 1.3642, and 1.5991, respectively. The equivalent diameters of fragments of red sandstone, granite, and marble produced in quasi-static compression failures are mainly distributed in the ranges of

20–40 mm, 25–65 mm, and 10–35 mm, respectively. The fractal dimension of contacts between mineral particles and the distribution of the equivalent diameters of particles in rock are 1.6381 and

Citation: Pi, Z.; Zhou, Z.; Li, X.; 0.8–3.6 mm, respectively. The proposed approach provides a computerized method to characterize Wang, S. Digital Image Processing quantitatively and automatically the structure characteristics of soil/rock or soil/rock-like materials. Method for Characterization of By this approach, the remote sensing for characterization can be achieved. Fractures, Fragments, and Particles of Soil/Rock-Like Materials. Mathematics Keywords: soil/rock-like materials; digital image processing; color gradient; block size distribution; 2021, 9, 815. https://doi.org/10.3390/ fractal dimension; particle size distribution math9080815

Academic Editor: Javier Martínez 1. Introduction Received: 17 February 2021 Natural geomaterials (soil and rock) and the associated artiﬁcial materials have many Accepted: 6 April 2021 Published: 9 April 2021 discontinuous structures such as cracks, fractures, and contacts and possibly produce fragments or particles during geological, environmental, and stress conditions [1–7]. The distri-

Publisher’s Note: MDPI stays neutral butions of discontinuous structures and particles can determine the physical-mechanical with regard to jurisdictional claims in characteristics and failure modes of these materials. The size distribution of fragments published maps and institutional affil- reflects the failure behavior or breakage performance. Characterizations of discontinu- iations. ous structures, particles, and fragments are important for many operations in mining and geotechnical fields, such as geo-mechanical assessment and stability analysis of rock mass, performance analysis of rock breakage, and geometrical characterization of the internal structure of rock-like material. Digital image processing is a digital-based and computer-based approach applying various mathematical algorithms to extract significant Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. information from the images. Digital image processing has been widely used in a range This article is an open access article of engineering topics in recent years to determine the characteristics of microscopic and distributed under the terms and macroscopic structures in the natural and human-made materials [8]. conditions of the Creative Commons Many efforts have been made to collect and characterize discontinuous structures in Attribution (CC BY) license (https:// rock exposures. Customarily, handheld equipment such as a geological compass-clinometer, creativecommons.org/licenses/by/ a measuring tape, and a roughness profile gauge were used to make the in situ measurement 4.0/). of discontinuity geometry of rock mass exposures. This method was undesirable for several

Mathematics 2021, 9, 815. https://doi.org/10.3390/math9080815 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 815 2 of 13

reasons: high operation risk, limited measurement scale, high time consumption, and no uninterpreted record of the rock face. As an alternative approach to in situ measurement, photogrammetry allows for discontinuity geometry to be measured from remotely sensed photographs of rock mass exposures [9,10]. Afterwards, digital images were used instead, which promoted the production of automated methods to firstly detect discontinuities and to then analyze discontinuity geometry as the data comprising digital images can be mathematically processed [11]. Some digital image processing techniques, such as the greyscale threshold, greyscale smoothing, segmentation, and edge detection, have been used to identify discontinuity traces in digital images of rock mass exposures to characterize the fracture length, orientation, and position [11–13]. In these efforts, it was difficult to fully reflect the information of the natural rock mass exposure with gray- level images. The complication of the recognition of discontinuity traces defined by edges resulted in the binary image from segmentation containing a large number of error pixels. The optimal threshold value for each image impeded the creation of an automatic mapping tool [14]. The manual, automatic, and semiautomatic methods had been used to characterize discontinuity traces using photographs of rock exposures, and the case studies on actual rock images were taken to conduct comparative analyses to quantify the accuracy of fracture detection and characterization [11,14] Fragmentation is an important operation in order to control and minimize the loading, hauling, crushing, classification, and processing costs in mining and minerals engineering [15–18]. Many methods based on photo-analysis were proposed to estimate the size distribution of rock fragments [19–22]. The aforementioned methods were performed by hand or through image processing techniques by computer, which must delineate the individual rock fragments in an image. A computer program using statistical procedures was developed to automatically determine the size distribution of rock fragments based on high-resolution images [23]. Subsequently, digital image software was developed to evaluate fragmentation, which gradually became a worldwide accepted tool in the mining and mineral processing industries with the emergence of the continuous fragmentation monitoring system [24]. Some scholars compared fragmentation measurements using photographic and image analysis techniques on the actual images of fragments. They found that the photographic method was time-consuming, and manual editing was required to improve the accuracy of the image processing method [25]. The image analysis approaches could reach a satisfactory accuracy via comparisons with calculations from manual outlining and measurement [23–25]. The aforementioned methods based on image analyses provide useful tools for estimating the fracture and block size distributions. However, there is still a lack of a comprehensive and consolidated method to automatically recognize discontinuous structures and to characterize the distributions of discontinuous structures and block sizes with multiple scales. In addition, the graying process decreases the dimensionality of multicolor images and reduces recognition accuracy, although it is a common process in demarcations based on image processing techniques. This study proposed a digital image processing method based on the color gradient distribution of a multicolor image. This approach was used to automatically recognize the outlines of fractures/cracks/contacts, fragments, and particles. Then, the fracture network, block size distribution, and particle size distribution were quantitatively characterized. Case studies were conducted to confirm the proposed method.

2. Methodology 2.1. Image Processing Method Based on Color Gradient The colors of fractures in rock and the outlines of rock fragments or particles are different from the colors of other zones in a digital image. There are high color gradient values at rock fractures and the outlines of rock fragments or particles. Therefore, a color gradient-based image processing method was proposed to automatically recognize the outlines of rock fractures, rock fragments, and rock particles. Then, the fractal dimension Mathematics 2021, 9, x FOR PEER REVIEW 4 of 13

1 ln(NDC ( )) ln( ) (6) F  where C is a regressed constant.

Step 5—Calculate the area, SAO , of the connected zone enclosed by each outline of

the rock fragment or rock particle, and convert SAO into the equivalent diameter d of

a circle for which the area S A is equal to the area enclosed by outline, which can be ex- Mathematics 2021, 9, 815 pressed as 3 of 13 d 2 SS (7) AO4 A of the fracture network in rocks and the particle size distribution of rock fragments or particlesThen, can the be characterized.frequency histograms The detailed and steps distribution are shown curves in Figure of 1equivalent and described diameters of asthe follows: rock fragments or rock particles can be drawn from the size statistics.

Original color image consisting of three primary colors: red, green and blue (RGB)

Image read

Image matrix Cm×n×3 including pixel location and associated tricolor values cijk

c c c c g = (ijk, 1, ijk , , )22 ( ijk 1, , ijk , , ) ijk xy

Gradient magnitude matrix G(m-1)×(n-1)×3 including gradient magnitudes of tricolor values gijk Boundary of Noise reduction fracture, Boundaryfragment or a

Fracture distribution Size statistics

Fractal Fractal Void rate Frequency Size dimension distributiondimension histogram distribution of fracture of boundaryfractures of size curve networks line

FigureFigure 1. 1.Flowchart Flowchart of of the the image-processing image-processing method method for analyzing for analyzing the properties the properties of rock fracturing of rock fractur- anding block/particleand block/particle size distribution. size distribution. Step 1—Read the original color image consisting of three primary colors (red, green, and2.2. blue)Description to produce of the an Proposed image matrix Image expressed Processing as Method Equation (1) including pixel locations and theIn associatedorder to clearly tricolor describe values. the proposed image processing method, the calculation processes of the fractal dimension of the world coastline and the size distribution of the C = [c ] (1) connected land were taken as the examplem×n×3 shownijk in Figure 2. The calculation process was

wheresimilarC mto× nthe×3 is analyses the image of matrix the fractal and c ijkdimensionis the tricolor and value the size of a distribution pixel. of rock fractures, rockStep fragments, 2—Calculate or rock the particles. color gradient First, the based world on Equation coastline (2) map to achieve(Figure a2a) gradient was simplified magnitudeinto a grap matrixh (Figure expressed 2b), with as Equationthe different (3), includingcolors denoting gradient seawater magnitudes and of land, tricolor by marking values.seawater In addition,and land the with isolated blue and pixel white, with a respectively, peak value of using color gradienta drawing will software. be deleted Secondly, forthe simple mean noiseof the reduction. color gradient The box was with calculated3 × 3 pixels and was drawn used in to the ﬁnd color the insolated gradient pixel distribution with a peak value of color gradient by one pixel as the set size in the search. If the color gradient in the center position presents a sudden rise, this pixel will be considered noise. s 2 2 ci,j+1,k − ci,j,k ci+1,j,k − ci,j,k g = ( ) + ( ) (2) ijk ∆x ∆y

G(m−1)×(n−1)×3 = [gijk] (3) Mathematics 2021, 9, 815 4 of 13

where gijk is the gradient magnitude, G(m−1)×(n−1)×3 is the gradient magnitude matrix, and ∆x and ∆y are the differential steps in the horizontal and vertical directions for calculating the gradient magnitude, in which the steps are set to one pixel in the horizontal and vertical directions. Step 3—Outline the rock fractures or the boundaries of rock fragments or particles, and use binarization to store the locations of outlines in a binary digital image according to the color gradient magnitude and the following criteria:

gij > a (4)

gij1 + gij2 + gij3 g = (5) ij 3

where gij is the mean of the gradient magnitude values of the tricolor values and a is the threshold value to recognize the outline. Step 4—Calculate the fractal dimension of the fracture network in rock to quantitatively characterize the developmental degree of fractures in rock. The box-counting dimension is used to determine the fractal property of the fracture network. The binarized image is covered by a sequence of square grids with ascending sizes through a measuring box scale along the horizontal and vertical measuring directions. For each grid, the number of boxes intersected by the fracture network, N(δ), and the lengths of the sides of the boxes, δ, are recorded. Then, the natural logarithm of N(δ) is plotted against the natural logarithm of 1/δ and the straight line is regressed by Equation (6). The slope of the regressed straight line is the fractal dimension, DF. 1 ln(N(δ)) = D ln( ) + C (6) F δ where C is a regressed constant. Step 5—Calculate the area, SAO, of the connected zone enclosed by each outline of the rock fragment or rock particle, and convert SAO into the equivalent diameter d of a circle for which the area SA is equal to the area enclosed by outline, which can be expressed as

d2 S = π = S (7) AO 4 A Then, the frequency histograms and distribution curves of equivalent diameters of the rock fragments or rock particles can be drawn from the size statistics.

2.2. Description of the Proposed Image Processing Method In order to clearly describe the proposed image processing method, the calculation processes of the fractal dimension of the world coastline and the size distribution of the connected land were taken as the example shown in Figure2. The calculation process was similar to the analyses of the fractal dimension and the size distribution of rock fractures, rock fragments, or rock particles. First, the world coastline map (Figure2a) was simpliﬁed into a graph (Figure2b), with the different colors denoting seawater and land, by marking seawater and land with blue and white, respectively, using a drawing software. Secondly, the mean of the color gradient was calculated and drawn in the color gradient distribution graph shown in Figure2c. Then, the coastline was recognized and located in a binary digital image shown in Figure2d. Finally, the fractal dimension of the coastline was obtained using the regression analysis (Figure2f) of the box-counting dimension. The schematic diagram of the measuring process for calculating the box-counting dimension is shown in Figure2e. Meanwhile, the connected domain of land (Figure2g) was distinguished. The areas of connected domains were calculated and stored in a vector to obtain the frequency histogram and cumulative distribution curve (Figure2i) of land sizes by calculating equivalent diameter, as shown in Figure2h. The fractal dimension of world coastline is 1.3625, Mathematics 2021, 9, x FOR PEER REVIEW 5 of 13

graph shown in Figure 2c. Then, the coastline was recognized and located in a binary digital image shown in Figure 2d. Finally, the fractal dimension of the coastline was obtained using the regression analysis (Figure 2f) of the box-counting dimension. The schematic diagram of the measuring process for calculating the box-counting dimension is shown in Figure 2e. Meanwhile, the connected domain of land (Figure 2g) was distinguished. The areas of connected domains were calculated and stored in a vector to obtain Mathematics 2021, 9, 815 5 of 13 the frequency histogram and cumulative distribution curve (Figure 2i) of land sizes by calculating equivalent diameter, as shown in Figure 2h. The fractal dimension of world coastlineand theis 1.3625, land sizes and are the mostly land distributed sizes are below mostly 2000 distributed km, as determined below by 2000 analyses km, fromas determined by analysesthe proposed from digitalthe proposed image processing digital method. image processing method.

(a) World coastline map (b) Simplified world coastline map (c) Color gradient distribution

Average color gradient Average Vertical direction/pixel Vertical

Horizontal direction/pixel

(f) Fractal dimension of coastline (e) Calculating fractal dimension (d) Recognized coastline

)

(

Measuring Measuring direction ln n 4 pixels 3 Goodness of fit: R2=0.9995 2 δ

Measuring direction ln(1/δ) Measuring box (δ×δ pixels) (i) Equivalent diameter distribution of land (h) Calculating equivalent diameter (g) Connected domain of land

Equivalent A’ d A

SA’=π =SA Figure 2. DescriptionFigure of the image-processing2. Description of method the image for determining-processing the fractalmethod dimension for determining of the world the coastline fractal and dimension of the the size distributionworld of land, coastline including (anda) a worldthe size coastline distribution map, (b) aof simpliﬁed land, including world coastline (a) a map,world (c) coastline color gradient map, (b) a simpli- distribution, (d) colorfied gradient-based world coastline recognition map, of the(c) coastline,color gradient (e) calculation distribution, process of (d the) color fractal gradient dimension,-based (f) result recognition of g h of the fractal dimensionthe ofcoastline, the coastline, (e) (calculation) connected domainprocess of of land, the ( fractal) calculation dimension, process of (f equivalent) result of diameter, the fractal and dimension of (i) the result of equivalent diameter distribution of land. the coastline, (g) connected domain of land, (h) calculation process of equivalent diameter, and (i) the result3. Application of equivalent of the diameter Proposed distribution Image Processing of land. Method 3.1. Fractal Dimension of Fracture Network 3. ApplicationRock and of soilthe materialsProposed often Image produce Processing a crack or fracture Method network under loads from 3.1. Fractalgeological Dimension and environment of Fracture effects. Network It is an important task to characterize the fracture network to quantitatively determine the development and distribution of fractures. For Rockthe soil and material soil materials of a mud layer often under produce water, a many crack cracks or fracture (Figure3a) network will emerge under after loads from geologicalthe water and recedes, environment induced effects. by tensile It stress is an generated important from task water to characterize evaporation, thermal the fracture network expansion,to quantitatively and cold shrinkagedetermine under the highdevelopment temperatures and and distribution drought conditions. of fractures. The For the proposed image-processing method was used to determine the fractal dimension of cracks soil materialin the dry of mud a mud layer. layer The calculation under water, process many is shown cracks in Figure (Figure3. The result3a) will shows emerge that after the waterthe recedes, fractal dimension induced ofby cracks tensile in the stress dry mud generated layer is 1.4332. from water evaporation, thermal expansion, andFor acold ceramic shrinkage panel that under is a brittle high material temperatures made by artiﬁcial and drought sintering conditions. of clay, the The pro- posedcrack image network-processing (Figure 4methoda) will be was produced used byto thermaldetermine expansion the fractal and cold dimension shrinkage of cracks in during high-temperature sintering and long-term service conditions. As shown in Figure4, the dry mud layer. The calculation process is shown in Figure 3. The result shows that the fractal dimension of cracks in the dry mud layer is 1.4332.

Mathematics 2021, 9, x FOR PEER REVIEW 6 of 13

(a) Cracks of mud layer (b) Color gradient distribution (c) Recognized cracks (d) Fractal dimension of cracks

Mathematics 2021, 9, 815 6 of 13

)

δ (

Mathematics 2021, 9, x FOR PEER REVIEW N 6 of 13 the proposed image processing method was used to determineln the fractal dimensions of the crack network in the ceramic panel. The calculated fractal dimension is 1.3642.

Goodness of fit: R2=0.9984 (a) Cracks of mud layer (b) Color gradient distribution (c) Recognized cracks (d) Fractal dimension of cracks ln(1/δ)

Figure 3. Fractal dimension of the crack network of the mud layer, including (a) cracks of the mud layer, (b) color gradient

) δ

distribution, (c) recognized cracks, and (d) fractal dimension of the crack network. (

N ln For a ceramic panel that is a brittle material made by artificial sintering of clay, the

crack network (Figure 4a) will be produced by thermal expansionGoodness ofand fit: R2=0.9984 cold shrinkage dur-

ing high-temperature sintering and long-term service conditions.ln(1/δ) As shown in Figure 4, the proposed image processing method was used to determine the fractal dimensions of Figure 3. Fractal dimension of the crack network of the mud layer, including (a) cracks of the mud layer, (b) color gradient Figure 3. Fractal dimensionthe of the crack crack network network in of thethe mudceramic layer, panel. including The ( acalculated) cracks of thefractal mud dimension layer, (b) color is 1.3642. gradient distribution,distribution, (c) recognized (c) recognized cracks, cracks, and and(d) (fractald) fractal dimension dimension of of the the crack crack network. network.

(a) Ceramic panel (b) Color gradient distribution (c) Recognized cracks (d) Fractal dimension of cracks For a ceramic panel that is a brittle material made by artificial sintering of clay, the crack network (Figure 4a) will be produced by thermal expansion and cold shrinkage during high-temperature sintering and long-term service conditions. As shown in Figure 4, the proposed image processing method was used to determine the fractal dimensions of

the crack network in the ceramic panel. The calculated fractal dimension is 1.3642.

)

( N

(a) Ceramic panel (b) Color gradient distribution (c) Recognized cracks ln (d) Fractal dimension of cracks

Goodness of fit: R2=0.9981

)

δ (

N ln(1/δ)

ln Figure 4. FractalFigure 4. dimensionFractal dimension of the of crack the crack network network in in a a ceramic ceramic panel, panel, including including (a) cracks (a) cracks in the ceramicin the panel,ceramic (b) colorpanel, (b) color gradient disgradienttribution, distribution, (c) recognized (c) recognized cracks, cracks, and and (d (d) )fractal fractal dimensiondimension of theof the crack crack network. network. For the rock mass, there are many cracks or fractures (Figure5a) due to geological Forprocesses the rock including mass, tectonics, there weathering,are many andcracks diagenesis, or fractures and the effects(Figure of human-made5a) due to geological Goodness of fit: R2=0.9981 processesexcavation. including It is importanttectonics, for weathering quality assessments,, and diagenesis, strength prediction, and the effects and stability of human-made analysis of rock masses to characterize the fracture distribution. As shown in Figure5, the ln(1/δ) excavation.proposed It methodis important was used for to quality determine assessments, the fractal dimension strength of prediction, the fracture networkand stability analysis of rock masses to characterize the fracture distribution. As shown in Figure 5, the Figure 4. Fractal dimension of thein crack rock mass network based in on a the ceramic color gradient-basedpanel, including processing (a) cracks of ain digital the ceramic image of panel, a natural (b) color gradient distribution, (c) recognizedproposedrock cracks, mass. method Byand distinguishing was (d) fractal used dimensionto the determine color differencesof the the crack fractal of network rock dimension fractures. from of the color fractur gradiente network in rock distribution,mass based the on rock the fractures color gradient were recognized-based from processing the natural of rock a digital mass. The image fractal of a natural dimension of the fracture network regressed to 1.5991. rock mass. By distinguishing the color differences of rock fractures from color gradient For Thethe aforementionedrock mass, there application are many examples cracks show or that fractures the proposed (Figure image-processing 5a) due to geological distribution,processesmethod including based the rock on thetectonics, fractures color gradient weathering were distribution recognized, and of diagenesis, a digitalfrom imagethe and natural can the automatically effects rock mass.of human and The- madefractal dimensionexcavation.programmatically of It theis important fracture recognize network for cracks quality orregressed fractures.assessments, to Then, 1.5991. strength the associated prediction, fractal dimension and stability analysis Theof therock aforementioned crack masses or fracture to characterize network application in soil the examples and fracture rock materials show distribution. that can be the calculated. proposedAs shown The image resultsin Figure-processing 5, the methodcan based provide on quantitative the color information gradient distribution to determine theof a distribution digital image characteristics can automatically and and proposeddevelopment method degree was ofused cracks to or determine fractures in the soil fractal and rock dimension materials. of the fracture network in programmatrock massIn orderbasedically to on comparerecognize the color the proposedcracks gradient or methodology fractures.-based processing withThen, the the existing of associated a digital methods, fractalimage the afore- dimensionof a natural of therock crack mentionedmass. or By fracture actualdistinguishing imagenetwork of fracture inthe soil color network and differences rock shot materials from theof rock on-site can fractures be rock calculated. mass from were usedcolorThe results gradient can providedistribution,to produce quantitative the the rock calculation information fractures time were and to fractaldeterminerecognized dimensions the from distribution by the different natural imagecharacteristics rock processing mass. The and fractal devel- opmentdimension degree of the of fracture cracks or network fractures regressed in soil and to 1.5991. rock materials. The aforementioned application examples show that the proposed image-processing method based on the color gradient distribution of a digital image can automatically and programmatically recognize cracks or fractures. Then, the associated fractal dimension of the crack or fracture network in soil and rock materials can be calculated. The results can provide quantitative information to determine the distribution characteristics and development degree of cracks or fractures in soil and rock materials.

Mathematics 2021, 9, 815 7 of 13

methods. The results were compared to the real situation by manual outlining, which can be regarded as a real result of fractal dimension. In the comparisons, the color gradient- based method proposed in this study was compared with the gray gradient-based, binary gradient-based, color image-based, gray image-based, and binary image-based methods. Finally, the proposed method was confirmed by manual outlining and the color-based method. The gradient-based methods were based on the gradient distribution of color, gray, or binary values for outlining the fracture network. The image-based methods were based on the tricolor, gray, or binary values for outlining the fracture network. The used computer was installed with Windows 10 × 64 system, Intel(R) Core(TM) i5-6200U CPU 2.30 GHz and RAM 8 GB. The comparative results are shown in Figure6 and summarized in Table1. Manual outlining can provide the real value of fractal dimension for conducting comparisons. However, the work outlining the fracture consumed four hours, which was a time-consuming and laborious task. Therefore, the image-processing methods were created to solve such a problem. The color gradient-based method proposed in this study consumed the longest calculation time for fracture identification and fractal dimension because it was required to determine the tricolor and the associated gradient. Nevertheless, it only took 1490.733 ms to produce the results. The results of fracture network and its fractal dimension achieved by the proposed color gradient-based method were closest to Mathematics 2021, 9, x FOR PEER REVIEW the results obtained by manual outlining. The difference rate was only 0.33%. Therefore, 7 of 13 the calculation accuracy and calculation efficiency of the proposed method were confirmed by the comparisons.

(a) Rock fractures (b) Color gradient distribution

(d) Fractal dimension of fractures (c) Recognized fractures

)

(

N ln

Goodness of fit: R2=0.9895

ln(1/δ)

Figure 5. FractalFigure 5.dimensionFractal dimension of the offracture the fracture network network in ina arock rock mass, including including (a) ( fracturesa) fractures in the in rock the mass, rock (b mass,) color (b) color gradient distribution, (c) recognized fractures, and (d) fractal dimension of the fracture network. gradient distribution, (c) recognized fractures, and (d) fractal dimension of the fracture network.

In order to compare the proposed methodology with the existing methods, the aforementioned actual image of fracture network shot from the on-site rock mass were used to produce the calculation time and fractal dimensions by different image processing methods. The results were compared to the real situation by manual outlining, which can be regarded as a real result of fractal dimension. In the comparisons, the color gradient-based method proposed in this study was compared with the gray gradient-based, binary gradient-based, color image-based, gray image-based, and binary image-based methods. Fi- nally, the proposed method was confirmed by manual outlining and the color-based method. The gradient-based methods were based on the gradient distribution of color, gray, or binary values for outlining the fracture network. The image-based methods were based on the tricolor, gray, or binary values for outlining the fracture network. The used computer was installed with Windows 10 × 64 system, Intel(R) Core(TM) i5-6200U CPU 2.30 GHz and RAM 8 GB. The comparative results are shown in Figure 6 and summarized in Table 1. Manual outlining can provide the real value of fractal dimension for conducting comparisons. However, the work outlining the fracture consumed four hours, which was a time-consuming and laborious task. Therefore, the image-processing methods were created to solve such a problem. The color gradient-based method proposed in this study consumed the longest calculation time for fracture identification and fractal dimension because it was required to determine the tricolor and the associated gradient. Neverthe- less, it only took 1490.733 ms to produce the results. The results of fracture network and its fractal dimension achieved by the proposed color gradient-based method were closest to the results obtained by manual outlining. The difference rate was only 0.33%. Therefore, the calculation accuracy and calculation efficiency of the proposed method were confirmed by the comparisons.

Mathematics 2021, 9, 815 8 of 13 Mathematics 2021, 9, x FOR PEER REVIEW 8 of 13

Color image of rock fractures Gradient distribution Recognized fractures Fractal dimension of fractures

)

( N

(a) ln

Goodness of fit: R2=0.9895

Gray image of rock fractures ln(1/δ)

)

δ (

(b) N ln

Goodness of fit: R2=0.9891

Binary image of rock fractures ln(1/δ)

)

( N

Goodness of fit: R2=0.9939 ln(1/δ)

Color image of rock fractures Recognized fractures Fractal dimension of fractures

)

( N

(d) ln

Goodness of fit: R2=0.9916

Gray image of rock fractures ln(1/δ)

)

δ (

(e) N ln

Goodness of fit: R2=0.9926

Binary image of rock fractures ln(1/δ)

)

( N

(f) ln

Goodness of fit: R2=0.9924

Outline image of rock fractures ln(1/δ)

)

( N

(g) ln

Goodness of fit: R2=0.9911

ln(1/δ) FigureFigure 6. 6. CalculationCalculation results results of of the the fractal fractal dimension dimension of of a a fracture fracture network network by by different different methods, methods, including including ( (aa)) the the color color gradientgradient-based-based method method proposed proposed in this study, (b)) a gray gradient-basedgradient-based method,method, ((cc)) aa binarybinary gradient-basedgradient-based method, method, ( d(d)) a acolor color image-based image-based method, method, ((ee)) aa graygray image-basedimage-based method,method, ((ff)) a binary image image-based-based method, and and ( (gg)) a a manual manual outlining outlining andand color color-based-based method. method.

Mathematics 2021, 9, 815 9 of 13

Table 1. Comparation of calculation methods.

Difference Rate of Fractal Calculation Time for Fractal Dimension Relative to Method Fracture Identiﬁcation and Dimension the Result by Manual Fractal Dimension/ms Outlining/% Color gradient-based method proposed in this study 1490.733 1.5991 0.33 Gray gradient-based method [11] 1003.207 1.5669 1.69 Binary gradient-based method [11] 1187.949 1.5299 4.01 Color image-based method (designed in this study 196.739 1.4181 11.02 for comparison) Gray image-based method 143.722 1.4911 6.44 Binary image-based method [14] 155.167 1.4775 7.3 157.846 Manual outlining and color-based method (designed (Before it, the outlining time 1.5938 – in this study for comparison) has 4 h)

3.2. Block Size Distribution of Rock Fragments A number of rock fragments with multiple shapes and sizes will be produced when rock failure occurs in rock mechanical experiments and field excavations. The block size distribution of rock fragments is an important parameter for estimating the energy consumption characteristic of a rock after peak strength. The digital image processing approach was proposed to calculate the block size distribution of rock fragments. Firstly, the boundaries of rock fragments will be drawn and stored in a binary color matrix using the recognition criterion based on the color gradient distribution of the original image of rock fragments. Then, the areas of the connected domains surrounded by fragment boundaries were calculated, and these areas were converted into circles with the same area values and equivalent diameters. Finally, the frequency histogram and cumulative distribution curve of the equivalent diameters of rock fragments can be achieved. Granite, marble, and sandstone, which are classified as magmatic, metamorphic, and sedimentary rocks, respectively, are commonly used rock specimens for investigating the mechanical properties of brittle materials. Therefore, rock fragments of red sandstone, granite, and marble were used as application examples to show the specific processes of the proposed image processing approach to characterize the block size distribution, as shown in Figure7 . The analysis results show that the equivalent diameters of fragments are mainly distributed in the ranges of 20–40 mm, 25–65 mm, and 10–35 mm for red sandstone, granite, and marble, respectively, under the same experimental conditions. Although the main distribution range of fragment diameter can be reflected, the distribution modality of fragment diameters did not present a special strict type with unimodality, bimodality, or multimodality because the number of fragments was few and did not have the statistical law. In addition, the randomness of rock fracturing also contributed to that result. Based on the proposed image processing approach, the boundaries of the rock fragments can be clearly recognized and the connected domains can be accurately demarcated. Then, an accurate frequency histogram and cumulative distribution curve of equivalent diameters can be automatically produced to conduct visualization of the block size distribution of rock fragments. Therefore, the proposed method can be used to characterize the block size distributions of rock fragments produced during opencast blasting, underground mining, tunneling, ore-rock crushing, etc.

3.3. Fractal Dimension and Particle Size Distribution of Microparticles in Rock Rocks are formed by cementation of many microparticles, inﬂuenced by long-term geological processes. The types, shapes, sizes, and distribution and cementation characteristics of microparticles determine the physical-mechanical properties of rock material. The fractal dimension of contacts between microparticles can reﬂect the shapes and the associated distribution of microparticles in rock. The equivalent diameter of particles and the associated distribution can characterize the particle size distribution in rock. The pro- Mathematics 2021, 9, 815 10 of 13

posed digital image processing method can be simultaneously used to calculate the fractal dimension of contacts and the particle size distribution of microparticles in rock, as shown in Figure8. Based on the color gradient distribution, the contacts between microparticles can be recognized, and then the connected domains of microparticles can be demarcated. Then, the fractal dimension of contacts and the equivalent diameter distribution of particles Mathematics 2021, 9, x FOR PEER REVIEW can be produced by using the calculation processes of the box-dimension and equivalent 10 of 13 diameter. The analysis results show that the fractal dimension of the contacts is 1.6381. The equivalent diameters of the microparticles are mainly distributed from 0.8–3.6 mm with a bimodality distribution, which means the sizes of microparticles are concentrated near distribution1 mm and of 2–3rock mm fragments. for the mentioned Therefore, rock material.the proposed The reason method for that can bimodality be used may to character- be determined by the mineral particle composition, bonding form, and special geological ize theconditions block size ofdiagenesis. distributions That resultof rock indicated fragments that the produced particle elements during with opencast sizes of 1 mm blasting, un- dergroundand 2–3 mining, mm be produced tunneling, and ore reserved-rock during crushing, the long etc. process of diagenesis.

(a) 1. Rock fragments of red sandstone (b) 1. Rock fragments of granite (c) 1. Rock fragments of marble

2. Color gradient distribution 2. Color gradient distribution 2. Color gradient distribution

3. Recognized boundaries of fragments 3. Recognized boundaries of fragments 3. Recognized boundaries of fragments

4. Connected domain of fragments 4. Connected domain of fragments 4. Connected domain of fragments

5. Equivalent diameter distribution of fragments 5. Equivalent diameter distribution of fragments 5. Equivalent diameter distribution of fragments

Figure 7. BlockFigure size 7. Blockdistribution size distribution of rock of fragments rock fragments produced produced in inrock rock mechanics mechanics tests,tests, which which include include (a) red (a) sandstone, red sandstone, (b) granite, and(b ()c granite,) marble. and (c) marble.

3.3. Fractal Dimension and Particle Size Distribution of Microparticles in Rock Rocks are formed by cementation of many microparticles, influenced by long-term geological processes. The types, shapes, sizes, and distribution and cementation characteristics of microparticles determine the physical-mechanical properties of rock material. The fractal dimension of contacts between microparticles can reflect the shapes and the associated distribution of microparticles in rock. The equivalent diameter of particles and the associated distribution can characterize the particle size distribution in rock. The proposed digital image processing method can be simultaneously used to calculate the fractal dimension of contacts and the particle size distribution of microparticles in rock, as shown in Figure 8. Based on the color gradient distribution, the contacts between microparticles can be recognized, and then the connected domains of microparticles can be demarcated. Then, the fractal dimension of contacts and the equivalent diameter distribution of particles can be produced by using the calculation processes of the box-dimension and equivalent diameter. The analysis results show that the fractal dimension of the contacts is 1.6381. The equivalent diameters of the microparticles are mainly distributed from 0.8–3.6 mm with a bimodality distribution, which means the sizes of microparticles are concentrated near 1 mm and 2–3 mm for the mentioned rock material. The reason for that bimodality may be determined by the mineral particle composition, bonding form, and special geological conditions of diagenesis. That result indicated that the particle elements with sizes of 1 mm and 2–3 mm be produced and reserved during the long process of diagenesis.

Mathematics 2021, 9, x FOR PEER REVIEW 11 of 13 Mathematics 2021, 9, 815 11 of 13

(a) Contacts and particles in rock (c) Recognized contacts (d) Fractal dimension of contacts

)

(

N ln

Goodness of fit: R2=0.9928

ln(1/δ) (b) Color gradient distribution (e) Connected domains of particles (f) Equivalent diameter distribution of particles

Figure 8. FractalFigure 8. dimensionFractal dimension and particle and particle size sizedistribution distribution of of microparticles microparticles in in rock, rock, including including (a) contacts (a) contacts and particles and particles in rock, (b) colorin rock, gradient (b) color distribution, gradient distribution, (c) recognized (c) recognized fractures fractures between between microparticles microparticles in in rock, rock, (d)) fractalfractal dimension dimension of fractures, (e) connectedof fractures, domains (e) connected of particles, domains of and particles, (f) the and frequency (f) the frequency histogram histogram and and cumulative cumulative distribution distribution curvecurve of of particle sizes. particle sizes. 4. Conclusions 4. ConclusionsA digital image-processing method based on color gradient distribution was proposed Ato automaticallydigital image recognize-processing the outlines method of rock based fractures, on rockcolor fragments, gradient and distribution rock particles. was pro- posedBy to using automatically this method, recognize the fracture the network, outlines block of rock size distribution,fractures, rock and particlefragments size, and rock distribution of rock can be quantitatively characterized. The proposed approach can particles.automatically By using produce this method, the fractal the dimension fracture of network, a fracture orblock contact size network distribution, in rock and and particle size distributionthe size distribution of rock curves can of be rock quantitatively fragments and mineralcharacterized. particles. The These proposed approaches approach can can automaticallyquantitatively produce characterize the thefractal rock dimension structure characteristics of a fracture by means or contact of a computer. network Based in rock and the sizeon thisdistribution study, the following curves of conclusions rock fragments can be drawn: and mineral particles. These approaches can (a) The calculation examples show that the proposed image-processing method can quantitativelyautomatically characterize and programmatically the rock recognize structure cracks characteristics or fractures and by then means calculate of the a computer. Basedfractal on this dimensions study, the of crackfollowing or fracture conclusions networks. can The be fractal draw dimensionn: can quantita- (a)tively The reﬂect calculation the distribution examples characteristic show that and developmentthe proposed degree image of cracks-processing or fractures method can automaticallyin soil and and rock programmatically materials. The calculation recognize results cracks show that or fractures the fractal dimensionsand then calculate of the fractalcracks dimensions in the dry of mud crack layer, or fracture ceramic panel,networks. and natural The fractal rock mass dimension are 1.4332, can 1.3642, quantitatively and 1.5991, respectively. reflect the(b) distribution The outlines of characteristic rock fragments and produced development after rock failure degree can beof clearlycracks recognized or fractures in soil and rockby the materials. proposed image-processing The calculation method. results Then, show the frequencythat the histogramfractal dimensions and cumulative of cracks in the drydistribution mud layer, curves ceramic of equivalent panel, diameters and natural of rock fragments rock mass can beare automatically 1.4332, 1.3642, obtained and 1.5991, respectively.to characterize the block size distribution of rock fragments. The proposed method can be used to characterize the block size distributions of rock fragments produced in rock (b)fragmentations The outlines during of rock opencast fragments blasting, produced underground after mining, rock tunneling, failure can and ore-rockbe clearly recog- nizedcrushing. by the proposed The calculation image examples-processing show that method. the equivalent Then, diameters the frequency of fragments histogram of red and cu- mulativesandstone, distribution granite, and curves marble of produced equivalent in compression diameters failures of rock are fragments mainly distributed can be in automatically theobtained ranges ofto 20–40 characterize mm, 25–65 the mm, block and 10–35 size mm,distribution respectively. of rock fragments. The proposed method can be used to characterize the block size distributions of rock fragments produced in rock fragmentations during opencast blasting, underground mining, tunneling, and ore-rock crushing. The calculation examples show that the equivalent diameters of fragments of red sandstone, granite, and marble produced in compression failures are mainly distributed in the ranges of 20–40 mm, 25–65 mm, and 10–35 mm, respectively. (c) Based on the proposed method, the contacts between microparticles in rock can be accurately recognized, and the connected domains of microparticles can be demarcated subsequently. Then, the fractal dimension of contacts and the equivalent diameter distribution of particles can be calculated and are 1.6381 and 0.8–3.6 mm, respectively, for the mineral particles in the rock specimens.

Mathematics 2021, 9, 815 12 of 13

(c) Based on the proposed method, the contacts between microparticles in rock can be accurately recognized, and the connected domains of microparticles can be demarcated subsequently. Then, the fractal dimension of contacts and the equivalent diameter distribution of particles can be calculated and are 1.6381 and 0.8–3.6 mm, respectively, for the mineral particles in the rock specimens. (d) The comparative analyses show that the proposed color gradient-based method presents the best performance and is closest to the actual result assessed by manual outlining. Compared to the existing methods, the proposed method is a comprehensive and consolidated approach that can automatically recognize discontinuous structures and characterize the distributions with multiple scales. Meanwhile, this approach fully utilizes multicolor information in color images. The above improvement signiﬁcantly increases the recognition accuracy.

Author Contributions: Conceptualization, S.W.; methodology, Z.P.; validation, S.W.; formal analysis, Z.P.; writing—original draft preparation, Z.P.; writing—review and editing, S.W.; supervision, X.L. and Z.Z.; funding acquisition, S.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China, grant number 51904333. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Acknowledgments: The work described in this paper was supported by the National Natural Science Foundation of China (No. 51904333), for which the authors are very thankful. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Wang, S.; Li, X.; Wang, S. Separation and fracturing in overlying strata disturbed by longwall mining in a mineral deposit seam. Eng. Geol. 2017, 226, 257–266. [CrossRef] 2. Dai, B.; Chen, Y.; Zhao, G.; Liang, W.; Wu, H. A Numerical study on the crack development behavior of rock-like material containing two intersecting flaws. Mathematics 2019, 7, 1223. [CrossRef] 3. Ran, X.; Xue, L.; Zhang, Y.; Liu, Z.; Sang, X.; He, J. Rock classification from field image patches analyzed using a deep convolutional neural network. Mathematics 2019, 7, 755. [CrossRef] 4. Ye, X.; Wang, S.; Li, Q.; Zhang, S.; Sheng, D. The negative effect of installation on the performance of a compaction-grouted soil nail in poorly graded Stockton Beach Sand. J. Geotech. Geoenviron. Eng. 2020, 146, 04020061. [CrossRef] 5. Qiu, J.; Li, X.; Li, D.; Zhao, Y.; Hu, C.; Liang, L. Physical Model Test on the Deformation Behavior of an Underground Tunnel Under Blasting Disturbance. Rock Mech. Rock Eng. 2021, 54, 91–108. [CrossRef] 6. Gao, K.; Li, S.; Han, R.; Li, R.; Liu, Z. Study on the propagation law of gas explosion in the space based on the goaf characteristic of coal mine. Saf. Sci. 2020, 127, 104693. [CrossRef] 7. Cai, X.; Zhou, Z.; Zang, H.; Song, Z. Water saturation effects on dynamic behavior and microstructure damage of sandstone: Phenomena and mechanisms. Eng. Geol. 2020, 276, 105760. [CrossRef] 8. Yue, Z.; Chen, S.; Tham, L. Finite element modeling of geomaterials using digital image processing. Comput. Geotech. 2003, 30, 375–397. [CrossRef] 9. Brown, E.T. Rock Mass Characterisation Testing and Monitoring. ISRM Suggested Methods; Pergamon Press: Oxford, UK, 1981. 10. Harrison, J.P. Improved Analysis of Rock Mass Geometry Using Mathematical and Photogrammetric Methods. Ph.D. Thesis, Imperial College, London, UK, 1993. 11. Reid, T.R.; Harrison, J.P. A semi-automated methodology for discontinuity trace detection in digital images of rock mass exposures. Int. J. Rock Mech. Min. Sci. 2000, 37, 1073–1089. [CrossRef] 12. Hagan, T.O. A case of terrestrial photogrammetry in deep-mine rock structure studies. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 1980, 15, 191–198. [CrossRef] 13. Maerz, N. Photoanalysis of Rock Fabric. Ph.D. Thesis, University of Waterloo, Waterloo, ON, Canada, 1990. 14. Lemy, F.; Hadjigeorgiou, J. Discontinuity trace map construction using photographs of rock exposures. Int. J. Rock Mech. Min. Sci. 2003, 40, 903–917. [CrossRef] 15. Wang, S.; Sun, L.; Li, X.; Wang, S.; Du, K.; Li, X.; Feng, F. Experimental investigation of cuttability improvement for hard rock fragmentation using conical cutter. Int. J. Geomech. 2021, 21, 06020039. [CrossRef] Mathematics 2021, 9, 815 13 of 13

16. Jiang, R.; Dai, F.; Liu, Y.; Li, A. Fast marching method for microseismic source location in cavern-containing rockmass: Performance analysis and engineering application. Engineering 2021.[CrossRef] 17. Wang, S.; Li, X.; Yao, J.; Gong, F.; Li, X.; Du, K.; Tao, M.; Huang, L.; Du, S. Experimental investigation of rock breakage by a conical pick and its application to non-explosive mechanized mining in deep hard rock. Int. J. Rock Mech. Min. Sci. 2019, 122, 104063. [CrossRef] 18. Du, K.; Li, X.; Tao, M.; Wang, S. Experimental study on acoustic emission (AE) characteristics and crack classiﬁcation during rock fracture in several basic lab tests. Int. J. Rock Mech. Min. Sci. 2020, 133, 104411. [CrossRef] 19. Ghosh, A.; Daemen, J.; van Zyl, D. Fractal based approach to determine the effect of discontinuities on blast fragmentation. In Proceedings of the 31st U.S. Symposium on Rock Mechanics (USRMS), Golden, CO, USA, 18–20 June 1990. 20. Hunter, G.; McDermott, C.; Miles, N.; Singh, A.; Scoble, M. Review of image analysis techniques for measuring blast fragmentation. Min. Sci. Technol. 1990, 11, 19–36. [CrossRef] 21. Liu, M.; Liu, J.; Zhen, M.; Zhao, F.; Liu, Z. A comprehensive evaluation method of bench blast performance in open-pit mine. Appl. Sci. 2020, 10, 5398. [CrossRef] 22. Mojtabai, N.; Farmer, I.; Savely, J. Optimization of rock fragmentation in bench blasting. In Proceedings of the 31st U.S. Symposium on Rock Mechanics (USRMS), Golden, CO, USA, 18–20 June 1990; pp. 897–904. 23. Kemeny, J.; Devgan, A.; Hagaman, R.; Wu, X. Analysis of rock fragmentation using digital image processing. J. Geotech. Eng. 1993, 119, 1144–1160. [CrossRef] 24. Sanchidrian, J.; Segarra, P.; Lopez, L. A Practical Procedure for the Measurement of Fragmentation by Blasting by Image Analysis. Rock Mech. Rock Eng. 2006, 39, 359–382. [CrossRef] 25. Sudhakar, J.; Adhikari, G.; Gupta, R. Comparison of Fragmentation Measurements by Photographic and Image Analysis Techniques. Rock Mech. Rock Eng. 2006, 39, 159–168. [CrossRef]