Research on Coal-Rock Fracture Image Edge Detection Based on Tikhonov Regularization and Fractional Order Differential Operator

Aiming at the conventional image edge detection algorithm, the first-order differential edge detection method is easy to lose the image details and the second-order differential edge detection method is more sensitive to noise. To deal with the problem, the Tikhonov regularization method is adopted to reconstruct the input coal-rock infrared images, so as to reduce the noise interference, and then, the reconstructed image is transformed by gray level. Finally, we consider the frequency characteristics and long memory properties of fractional differential, the classical first-order Sobel and second-order Laplacian edge detection algorithms are extended to fractional order pattern, and a new pattern of fractional order differential image edge detection is constructed to realize the coal-rock fracture edge features identification. &e results show that, compared with integer order differential, the error rate and omission rate of fractional order differential algorithm are smaller, the quality factor is larger, and the execution time and memory footprint are smaller. From the point of view of location criteria and location accuracy, the fractional order differential algorithm is better than the integer order. In addition, the proposed method is compared with Canny algorithm, B-spline wavelet transform, and multidirection fuzzy morphological edge detection method, can detect more coal-rock fracture infrared image edge details, and is more robust to noise.


Introduction
e coal-rock fracture detection is an effective means to help coal seam gas development, once coal-rock image fractures are accurately detected, which will play an important role in further exploiting coal seam gas [1,2].In the final analysis, the coal-rock image fracture detection is edge detection, the method can be basically divided into two major categories, one of two methods is based on the firstorder derivative method, which detects the boundary by finding the maximum and minimum values of the image first-order derivative, and the boundary is usually positioned in the direction of the maximum gradient [3,4].Another kind is the second-order derivative method; when the image first-order derivative is taken as the maximum, the second-order derivative is zero, so we can find the boundary by looking for the zero crossing point of the image second-order derivative [5].ese two methods are implemented by convoluting the specific template with the image to achieve the image boundary point detection, but there are differences in the detection effect.e first-order derivative method is easy to produce thicker edge, which results in the loss of part of the image information details.And the second-order derivative method has strong image detail detection ability, but it is very sensitive to noise [6].
Fractional order differential theory is a generalization of integral order differential theory, and it has been widely used in many research fields such as applied mathematics, medicine, and information science [7].In recent years, image processing is a new research hotspot with fractional differential theory and has been successfully applied to solve many problems such as image processing, image enhancement, image denoising, and edge detection [8].Bai and Feng [9] proposed an image denoising pattern based on fractional anisotropic diffusion equation and used the fractional Fourier transformation to solve the pattern, which effectively suppresses the "ladder effect" phenomenon produced by the traditional denoising method.Józwik [10] proposed a fractional order robust contour edge detector, which can selectively detect edges when the order of fractional differential is properly chosen.Chen et al. [11] constructed two different fractional edge detection operators using fractional differential instead of traditional firstorder differential.He et al. [12] proposed an edge detection operator based on compound derivative, which uses the combination of fractional differential and integral differential.Goddeke and Strzodka [13] introduced fractional calculus in the traditional Kalman filter, and a discrete fractional order Kalman filter algorithm is proposed in the linear space and nonlinear space to estimate the parameters and fractional order.In [14], fractional differential can be used to accurately detect the medical image edges and to effectively suppress noise, which will improve the speed and accuracy of medical diagnosis.
e above analysis shows that the edge detection method based on fractional order differential can not only effectively extract the image edge information but also preserve the image texture details, so fractional order differential edge detection is better than the integer order differential.Although the first-order Sobel and the second-order Laplacian operators are not ideal for edge detection, there is no denying that they still have a wide range of applications due to stronger universality and faster computing speed.erefore, if the fractional order differential is introduced into the firstorder and the second-order differential operators, it will inherit the advantages of the first-order and second-order differential operators and improve the edge detection effect, so which can provide a new effective method to detect image edge.
In order to effectively and accurately detect the image edge, we were inspired by the fractional differential theory, the first-order Sobel edge detection operator and the secondorder Laplacian edge detection operator is extended to fractional order pattern, which is used to extract the edge feature of coal-rock fracture infrared image.
e results show that, compared with integer order differential, fractional order differential can detect more image edge detail features and is more robust to noise.

Image Reconstruction
e image reconstruction model is uniformly described as follows [15][16][17][18][19]: where A represents a matrix of dimension m × n, u represents an n × 1 dimensional vector denoting the gray level values, and s represents an m × 1 dimensional vector denoting the normalized capacitance values.Image reconstruction is an ill-posed problem, and it is generally known that Tikhonov regularization is an efficient way to solve ill-posed problems.Its basic idea is to transform equation ( 1) into an optimization problem [20][21][22][23][24]: where λ is the regularization parameter and ‖•‖ denotes 1 norm.en, minimizing equation (2) yields the solution: In the MATLAB 7.0 environment, the Tikhonov regularization method is used to reconstruct the coal-rock infrared image.
Figure 1(a) shows the coal-rock fracture infrared image [25][26][27], Figure 1(b) represents the reconstructed image, and Figure 1(c) represents a gray-level transformation image.In Figure 1(a), we can see that the infrared image contains a lot of noise, and the reconstructed image noise is obviously reduced from Figure 1(b) using the Tikhonov regularization method.
erefore, our conclusion is that the Tikhonov regularization method can reduce the noise interference, which lays the foundation for further image edge detection.
e edge detection mentioned below uses gray-level transformation images as input images.

Integer Order Edge Detection Operator
3.1.1.First-Order Sobel Operator.Sobel edge detection method is a typical first-order gradient, it uses a pair of 3 × 3 mobile template in the image, and as a traditional method, the corresponding gradient based on each pixel as the center point can be calculated, finally produces the horizontal gradient map and vertical gradient map of image, respectively.For a continuous image u(x, y), the gradient of position (x, y) can be defined as the following form [28]: where G x , G y represents the gradient along the X-axis and the Y-axis, respectively.In an image, if Δx and Δy are computed according to the number of pixels between two pixels, let Δx � Δy � 2, and the differential form of the gradient component is represented as follows: (5)

Second-Order Laplacian
Operator.According to the calculus theory, the local maxima of the first-order derivative correspond to the zero crossing of the second-order derivative, so the image edge can be detected by the zero crossing point of the image second-order derivative.For a continuous image u(x, y), the second-order derivative of position (x, y) can be defined as the following form [29]: 2 Journal of Electrical and Computer Engineering where G 2 x , G 2 y represents the second-order derivative along the X-axis and the Y-axis, respectively.In an image, if Δx and Δy are computed according to the number of pixels between two pixels, let Δx � Δy � 1, and the differential form of the second-order derivative is represented as follows:

Fractional Order Edge Detection Operator.
Considering that the integer order differential can only deal with the image information in the eight neighborhoods, however, the fractional differential can deal with the image global information.In this paper, the traditional integer differential operator is extended to the fractional order pattern, which is beneficial to extract more image edge features.

Fractional Order Sobel
Operator.By introducing differential order from first order to fractional order, a fractional Sobel operator is proposed, whose differential form is along the X-axis and the Y-axis [30][31][32]: Fractional differential operators G α x and G α y are discretized by using fractional differential G-L.We assume the image size is M × N, and then, fractional gradient at (x, y) is represented as follows: where K ≥ 3 is the integer constant, and w (α) j can be derived from the following recurrence formula: en, the fractional gradient components along the Xaxis and the Y-axis can be approximated as follows:

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A fractional Sobel convolution template based on the above approximation is presented, as shown in Figure 2.

Fractional Order Laplacian Algorithm.
e differential order is extended from the second order to the fractional order, a fractional Laplacian operator is proposed, whose differential form can be defined as follows [33,34]: Fractional order differential operator G α is discretized using fractional order differential G-L.
en, fractional gradient is represented as follows: A fractional order Laplacian convolution template based on the above approximation is presented, as shown in Figure 3.

reshold Selection.
e image fractional order gradient amplitude is selected as the basis of judging the image edge point [35,36]: is measure gives the maximum change rate of u(x, y) in the unit distance along the ∇ α u direction, where the threshold of the edge point is set based on the average fractional gradient, let where T means the threshold, τ ≥ 1 is a preset parameter, M × N represents the image size, and pixels can be labeled as edge points when ∇ α u > T. Parameters are as follows: τ � 1.5, using fractional order Sobel operator α � 0.6, and using fractional order Laplacian operator α � 1.6.In Figures 4 and 5, we can see that the edges of the image can be identified, but with the increasing of the template size, the edge of the recognition is confused, and the useless edge information is doped, which leads to the fact that the useful real edges are hidden.In other words, the fractional order differential template size is too large, there will be a lot of noise in the edge detection pattern when dealing with the noisy image, and the edge recognition will also consume a lot of computing time.

Test Results and Analysis
erefore, a small template size can effectively identify the true edges of an image, and it is appropriate to select the 3 × 3 template.

Fractional Image Edge Detection Results.
Taking into account the frequency characteristics of the fractional differential and the global range of the domain, the classical first-order Sobel operator is extended to the fractional order pattern.en, the performance and superiority of fractional Sobel operator edge detection are tested and analyzed by numerical experiments.e experimental parameters are set as follows: K � 3 and differential order α ∈ (0, 1).Firstly, the coal-rock fracture image in Figure 4(a) is chosen as the test image; let τ � 3.In Figure 6, the results of fractional Sobel operators under several typical fractional orders are given.

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Experimental results show that, compared with the classical first-order Sobel operator, the fractional Sobel operator is more robust to noise and can effectively suppress the interference of irrelevant details.For fractional Sobel operators, with the increasing of the fractional order, the edge detail extraction capability of the image is enhanced, but the sensitivity to noise is also increased, and a large number of noise components are left in the image.is result agrees with the frequency characteristic of the fractional differential.
e performance and superiority of fractional Laplacian operator edge detection are tested and analyzed.
e experimental parameters are set as follows: K � 3 and differential order α ∈ (1.0, 2.0).e results of fractional Sobel operators under several typical fractional orders are shown in Figure 7.

Quantitative Analysis.
In order to quantitatively analyze the effect of fractional order edge detection, we give the following evaluation criteria including the edge localization criterion and location accuracy, the quality factor, execution time, and memory footprint.
Fractional differential algorithm performance is quantitatively measured by the edge localization criterion and location accuracy, and the noise effect is investigated.
e edge localization criterion can detect a given object edge according to the edge detection algorithm, and location errors include real edge error detection and nonedge error detection, that is, missed detection and false detection.
Tables 1 and 2 show the results of the missed detection rate and false detection rate under different detection operators.In Table 1, the missed detection rate and false detection rate of 0.1, 0.2, 0.3, 0.4, and 0.5 are larger than that of 0.6.However, the missed detection rate and false detection rate of 0.6 are smaller than that of 0.7, 0.8, and 0.9.Hence, we find from Table 1 that the fractional order α � 0.6 has the lowest missed detection rate and false detection rate.
And it can be shown that the missed detection rate and false detection rate from Table 2 under fractional order Laplacian detection algorithm and the missed detection rate and false detection rate of 1.1, 1.2, 1.3, 1.4, and 1.5 are larger than that of 1.6.However, the missed detection rate and false detection rate of 1.6 are smaller than that of 1.7, 1.8, and 1.9.Hence, we find from Table 2 that the fractional order α � has the lowest missed detection rate and false detection rate.
Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (α � 0.6 and α � 1.6) using the above methods make them more successful to identify the edge of coal-rock fractures well.
In order to compare the performance of the two operators from the point of view of localization accuracy, the quality factor for edge detection is described as follows: where n a and n d represent the actual edge number and the detected edge number, respectively, d represents the distance between the real edge and the detected edge, f is the constant factor, and f � 0.05.If Q is relatively larger, it means that localization accuracy of the edge points is higher; therefore, edges can be clearly detected.e quality factor is calculated according to equation and the result is shown in Figure 8.In Figure 8(a), the quality factors of 0.1, 0.2, 0.3, 0.4, and 0.5 are smaller than that of 0.6.However, the quality factor of 0.6 is larger than that of 0.7, 0.8, and 0.9.Hence, we find from Figure 8(a) that the fractional order α � 0.6 has the largest quality factor.
And it can be shown that the quality factor from Figure 8(b) under fractional order Laplacian detection algorithm and the quality factor of 1.1, 1.2, 1.3, 1.4, and 1.5 are smaller than that of 1.6.However, the quality factor of 1.6 is larger than that of 1.7, 1.8, and 1.9.Hence, we find from Figure 8(b) that the fractional order α � 1.6 has the largest quality factor.
Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (α � 0.6 and α � 1.6) using the above methods make them more successful to identify the edge of coal-rock fractures well from the point of view of location accuracy.
Execution time and memory footprint not only reflect the algorithm computational complexity and execution efficiency but also reflect the algorithm performance.All codes were written in MATLAB 7.0 and run on a HP with 2.0 GB RAM and Windows 7 operating system.e test results of execution time and memory footprint are shown in Tables 3 and 4.
In Table 3, execution time and memory footprint of 0.1, 0.2, 0.3, 0.4, and 0.5 are larger than that of 0.6.However, the quality factor of 0.6 is smaller than that of 0.7, 0.8, and 0.9.Hence, we find from Table 3 that the fractional order α � 0.6 has the smallest execution time and memory footprint.
And it can be shown that the execution time and memory footprint from Table 4 under fractional order Laplacian detection algorithm and the execution time and memory footprint of 1.1, 1.2, 1.3, 1.4, and 1.5 are smaller than that of 1.6.However, the execution time and memory footprint of 1.6 are larger than that of 1.7, 1.8, and 1.9.Hence, we find from Table 4 that the fractional order α � 1.6 has the smallest execution time and memory footprint.
Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (α � 0.6 and α � 1.6) using the above methods make them more successful to identify the edge of coal-rock fractures well.
e fractional order edge detection method in this paper is compared with the classical first-order Sobel operator and second-order Laplacian edge detection operator, and the detection results of those methods are shown in Figures 9(a)-9(d), respectively.
Figure 9(a) explains that the detection result is gained using the classical first-order Sobel operator method at α � 1.0; Figure 9(b) explains that the detection result is given by the fractional order Sobel operator method at α * � 0.6; Figure 9(c) shows that the detection result is gained using the second-order Laplacian edge detection operator at α � 2.0; and the detection result at α * � 1.6 is shown in Figure 9(d) using the fractional order Laplacian edge detection operator in the paper.
We can find from Figures 9(a)-9(d) that the four methods can all detect the image edge; however, fractional order Sobel operator method and the fractional order Laplacian edge detection operator in the paper can successfully and effectively detect the detailed information of the image edge in comparison with the classical first-order Sobel operator and second-order Laplacian edge detection operator method.
In order to quantitatively analyze the effect of fractional order and integer order edge detection, we still give the following evaluation criteria including the localization errors, execution time and memory footprint, and the quality factor, which are shown in Tables 5-7.   5 and 6 that the errors (missed detection rate and false detection rate) and execution time and memory footprint of fractional order are less than that of the integer order detection algorithm.In Table 7, we can see that the quality factor of fractional order is large than that of the integer order detection algorithm.erefore, we conclude that the fractional order detection algorithm is better than the integer order detection algorithm which is more conducive to detect image edge.

Comparison of Different Edge Detection Methods
In order to compare the proposed method with other edge detection methods, this paper presents three recent edge detection methods, including Canny algorithm, B-spline wavelet transform, and multidirection fuzzy morphological edge detection method; the following are the steps of three edge detection algorithms:   Step 3. We can get the direction of the edge by step 2. e direction of edge gradient is divided into four directions: horizontal direction, vertical direction, 45 degree direction, and 135 degree direction.
Step 4. Nonmaximum suppression of image: if the gray value difference between a certain pixel and two adjacent pixels is not very large, then it is determined that the pixel is not an edge.
Step 5. Two thresholds are obtained by using cumulative histogram, including high threshold T1 and low threshold T2.If the value is greater than the high threshold T1, the edge is determined; if the value is smaller than the low threshold T2, the edge is not determined; and if the value is between the high threshold T1 and the low threshold T2, the two adjacent pixels are considered.If the two adjacent pixels are larger than the high threshold T1, the edge is determined; otherwise, the point is not considered as the edge point.e steps of B-spline wavelet transform edge detection are as follows: Step 1. e original image is transformed by wavelet transform, and the module image and phase image cluster are obtained.
Step 2. Find out the local modulus maximum point in the modulus image cluster and get the edge image to be selected.
Step 3. e mean value of modulus maxima is obtained by dividing blocks and weighted summation, and the threshold of each block is obtained.
Step 4. Remove false edges.Judging whether the selected edge point is the real edge point according to the threshold value obtained by each block, retaining the point whose modulus is greater than the threshold value, removing the false edge point which is less than the threshold value, and obtaining the real edge image.e steps of multidirectional fuzzy morphological edge detection are as follows: Step 1. Edge E i � μ (F•B i )⊗B i − μ (F•B i )⊗B i in all directions was detected.
Step 2. According to the detection edge result E i , the final result E �  4 i�1 ω i E i is calculated by using composite weighting method.
Step 3. e detection results are binarized by using the Otsu method.e proposed method is compared with other edge detection methods, the detection results of those methods are shown in Figures 10(a)-10(d), respectively.
Figure 10(a) explains that the detection result is gained using the classical Canny edge detection method; Figure 10(b) explains that the detection result is given by the B-spline wavelet transform edge detection method; Figure 10(c) shows that the detection result is gained using the multidirection fuzzy morphological edge detection method; and the detection result is shown in Figure 10(d) using the proposed edge detection operator in the paper.
We can find from Figures 10(a)-10(d) that the four methods can all detect the image edge; however, the proposed edge detection method in the paper can successfully and effectively detect the detailed information of the image edge in comparison with the classical Canny edge detection method, B-spline wavelet transform, and multidirection fuzzy morphological method.
In order to quantitatively analyze the effect of different edge detection methods, we still give the following evaluation criteria including the localization errors, execution time and memory footprint, and the quality factor, which are shown in Tables 8-10.
It is clearly seen from Tables 8 and 9 that the location errors (missed detection rate and false detection rate) and execution time and memory footprint of the proposed edge detection method in the paper are less than that of other methods.In Table 10, we can see that the quality factor of the

Table 1 :
Localization errors of fractional Sobel detection.

Table 2 :
Localization errors of fractional Laplacian detection.

Table 3 :
Execution time and memory footprint of fractional Sobel detection.

Table 4 :
Execution time and memory footprint of fractional Laplacian detection.

Table 5 :
Localization errors of integer order and fractional order methods.

Table 6 :
Execution time and memory footprint of integer order and fractional order methods.

Table 7 :
e quality factor of integer order and fractional order methods.