Research Article  Open Access
Research on CoalRock Fracture Image Edge Detection Based on Tikhonov Regularization and Fractional Order Differential Operator
Abstract
Aiming at the conventional image edge detection algorithm, the firstorder differential edge detection method is easy to lose the image details and the secondorder differential edge detection method is more sensitive to noise. To deal with the problem, the Tikhonov regularization method is adopted to reconstruct the input coalrock 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 firstorder Sobel and secondorder 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 coalrock fracture edge features identification. The 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, Bspline wavelet transform, and multidirection fuzzy morphological edge detection method, can detect more coalrock fracture infrared image edge details, and is more robust to noise.
1. Introduction
The coalrock fracture detection is an effective means to help coal seam gas development, once coalrock image fractures are accurately detected, which will play an important role in further exploiting coal seam gas [1, 2]. In the final analysis, the coalrock 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 firstorder derivative, and the boundary is usually positioned in the direction of the maximum gradient [3, 4]. Another kind is the secondorder derivative method; when the image firstorder derivative is taken as the maximum, the secondorder derivative is zero, so we can find the boundary by looking for the zero crossing point of the image secondorder derivative [5]. These 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. The firstorder derivative method is easy to produce thicker edge, which results in the loss of part of the image information details. And the secondorder 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.
The 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 firstorder Sobel and the secondorder 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. Therefore, if the fractional order differential is introduced into the firstorder and the secondorder differential operators, it will inherit the advantages of the firstorder and secondorder 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 firstorder 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 coalrock fracture infrared image. The results show that, compared with integer order differential, fractional order differential can detect more image edge detail features and is more robust to noise.
2. Image Reconstruction
The image reconstruction model is uniformly described as follows [15–19]:where represents a matrix of dimension , represents an dimensional vector denoting the gray level values, and represents an dimensional vector denoting the normalized capacitance values.
Image reconstruction is an illposed problem, and it is generally known that Tikhonov regularization is an efficient way to solve illposed problems. Its basic idea is to transform equation (1) into an optimization problem [20–24]:where is the regularization parameter and denotes 1 norm. Then, minimizing equation (2) yields the solution:
In the MATLAB 7.0 environment, the Tikhonov regularization method is used to reconstruct the coalrock infrared image.
Figure 1(a) shows the coalrock fracture infrared image [25–27], Figure 1(b) represents the reconstructed image, and Figure 1(c) represents a graylevel 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. Therefore, our conclusion is that the Tikhonov regularization method can reduce the noise interference, which lays the foundation for further image edge detection. The edge detection mentioned below uses graylevel transformation images as input images.
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3. Image Edge Detection
3.1. Integer Order Edge Detection Operator
3.1.1. FirstOrder Sobel Operator
Sobel edge detection method is a typical firstorder gradient, it uses a pair of 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 , the gradient of position can be defined as the following form [28]:where represents the gradient along the Xaxis and the Yaxis, respectively. In an image, if and are computed according to the number of pixels between two pixels, let , and the differential form of the gradient component is represented as follows:
3.1.2. SecondOrder Laplacian Operator
According to the calculus theory, the local maxima of the firstorder derivative correspond to the zero crossing of the secondorder derivative, so the image edge can be detected by the zero crossing point of the image secondorder derivative. For a continuous image , the secondorder derivative of position can be defined as the following form [29]:where represents the secondorder derivative along the Xaxis and the Yaxis, respectively. In an image, if and are computed according to the number of pixels between two pixels, let , and the differential form of the secondorder derivative is represented as follows:
3.2. 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.
3.2.1. 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 Xaxis and the Yaxis [30–32]:
Fractional differential operators and are discretized by using fractional differential GL. We assume the image size is , and then, fractional gradient at is represented as follows:where
is the integer constant, and can be derived from the following recurrence formula:
Then, the fractional gradient components along the Xaxis and the Yaxis can be approximated as follows:
A fractional Sobel convolution template based on the above approximation is presented, as shown in Figure 2.
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3.2.2. Fractional Order Laplacian Algorithm
The 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 is discretized using fractional order differential GL. Then, fractional gradient is represented as follows:
A fractional order Laplacian convolution template based on the above approximation is presented, as shown in Figure 3.
3.2.3. Threshold Selection
The image fractional order gradient amplitude is selected as the basis of judging the image edge point [35, 36]:
This measure gives the maximum change rate of in the unit distance along the direction, where the threshold of the edge point is set based on the average fractional gradient, letwhere means the threshold, is a preset parameter, represents the image size, and pixels can be labeled as edge points when .
4. Test Results and Analysis
4.1. Selecting Fractional Order Differential Template Size
In order to investigate the fractional order differential template size (fractional differential expansion term ) influence on the results of image edge detection, we take the graylevel transformation coalrock fracture image (size is ) as the test image. Figures 4 and 5 show the edge detection results of fractional order Sobel operator and fractional order Laplacian operator under the action of , , , , , and templates, respectively. Parameters are as follows: , using fractional order Sobel operator , and using fractional order Laplacian operator .
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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. Therefore, a small template size can effectively identify the true edges of an image, and it is appropriate to select the template.
4.2. Fractional Image Edge Detection Results
Taking into account the frequency characteristics of the fractional differential and the global range of the domain, the classical firstorder Sobel operator is extended to the fractional order pattern. Then, the performance and superiority of fractional Sobel operator edge detection are tested and analyzed by numerical experiments. The experimental parameters are set as follows: and differential order . Firstly, the coalrock fracture image in Figure 4(a) is chosen as the test image; let . 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 firstorder 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. This result agrees with the frequency characteristic of the fractional differential.
The performance and superiority of fractional Laplacian operator edge detection are tested and analyzed. The experimental parameters are set as follows: and differential order . The results of fractional Sobel operators under several typical fractional orders are shown in Figure 7.
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4.3. 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. The 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 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 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 ( and ) using the above methods make them more successful to identify the edge of coalrock 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 and represent the actual edge number and the detected edge number, respectively, represents the distance between the real edge and the detected edge, is the constant factor, and . If is relatively larger, it means that localization accuracy of the edge points is higher; therefore, edges can be clearly detected.
The quality factor is calculated according to equation (17), 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 has the largest quality factor.
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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 has the largest quality factor.
Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders ( and ) using the above methods make them more successful to identify the edge of coalrock 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. The 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 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 has the smallest execution time and memory footprint.
Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders ( and ) using the above methods make them more successful to identify the edge of coalrock fractures well.
The fractional order edge detection method in this paper is compared with the classical firstorder Sobel operator and secondorder Laplacian edge detection operator, and the detection results of those methods are shown in Figures 9(a)–9(d), respectively.
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Figure 9(a) explains that the detection result is gained using the classical firstorder Sobel operator method at ; Figure 9(b) explains that the detection result is given by the fractional order Sobel operator method at ; Figure 9(c) shows that the detection result is gained using the secondorder Laplacian edge detection operator at ; and the detection result at 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 firstorder Sobel operator and secondorder 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.



It is clearly seen from Tables 5 and 6 that the location 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. Therefore, we conclude that the fractional order detection algorithm is better than the integer order detection algorithm which is more conducive to detect image edge.
5. 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, Bspline wavelet transform, and multidirection fuzzy morphological edge detection method; the following are the steps of three edge detection algorithms:
The steps of Canny operator edge detection are as follows: Step 1. The original image is smoothed or convoluted by using twodimensional Gauss filtering function, which can reduce noise. Step 2. Calculating the partial derivative of gray image, the gradient , and the direction of the gradient. Step 3. We can get the direction of the edge by step 2. The 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.
The steps of Bspline wavelet transform edge detection are as follows: Step 1. The 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. The 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.
The steps of multidirectional fuzzy morphological edge detection are as follows: Step 1. Edge in all directions was detected. Step 2. According to the detection edge result , the final result is calculated by using composite weighting method. Step 3. The detection results are binarized by using the Otsu method.
The proposed method is compared with other edge detection methods, the detection results of those methods are shown in Figures 10(a)–10(d), respectively.
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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 Bspline 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, Bspline 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 proposed edge detection method in the paper is large than that of other methods. Therefore, we conclude that the proposed edge detection method in the paper is better than Canny method, Bspline wavelet transform, and multidirection fuzzy morphological edge detection method which is more conducive to detect image edge.
6. Conclusions
This paper presents a new pattern of fractional order differential image edge detection method to detect the edge of coal rock fracture. Considering the frequency characteristics and long memory properties of fractional differential, integer order differential edge detection methods (firstorder Sobel and secondorder Laplacian) are extended to the fractional order pattern. The test 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, Bspline wavelet transform, and multidirection fuzzy morphological edge detection method; the proposed method in this paper based on Tikhonov regularization and fractional order differential operator is an effective method for image edge detection.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the Chinese National Natural Science Foundation under Contract nos. 51674106 and 51274091.
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Copyright
Copyright © 2019 Chunsheng Liu and Chunping Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.