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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. 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, 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.

The 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 [

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 [

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 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. Therefore, if the fractional order differential is introduced into the first-order 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 second-order Laplacian edge detection operator is extended to fractional order pattern, which is used to extract the edge feature of coal-rock 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.

The image reconstruction model is uniformly described as follows [

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 (

In the MATLAB 7.0 environment, the Tikhonov regularization method is used to reconstruct the coal-rock infrared image.

Figure

Coal-rock fracture image. (a) Infrared image. (b) Reconstructed image. (c) Gray-level transformation image.

Sobel edge detection method is a typical first-order gradient, it uses a pair of

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

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.

By introducing differential order from first order to fractional order, a fractional Sobel operator is proposed, whose differential form is along the

Fractional differential operators

Then, the fractional gradient components along the

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

Fractional order Sobel template. (a)

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 [

Fractional order differential operator

A fractional order Laplacian convolution template based on the above approximation is presented, as shown in Figure

Fractional order Laplacian template.

The image fractional order gradient amplitude is selected as the basis of judging the image edge point [

This measure gives the maximum change rate of

In order to investigate the fractional order differential template size (fractional differential expansion term

Edge extraction results of fractional order Sobel operator. (a)

Edge extraction results of fractional order Laplacian operator. (a)

In Figures

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. 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:

Comparison of edge extraction results under different orders Sobel on a coal-rock image. (a)

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. 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:

Comparison of edge extraction results under different orders Laplacian on a coal-rock image. (a)

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

Localization errors of fractional Sobel detection.

Order | Missed detection rate | False detection rate |
---|---|---|

0.1 | 0.3574 | 0.6481 |

0.2 | 0.3315 | 0.6328 |

0.3 | 0.3104 | 0.6266 |

0.4 | 0.3098 | 0.6099 |

0.5 | 0.3032 | 0.6057 |

0.6 | 0.3001 | 0.6009 |

0.7 | 0.3256 | 0.6233 |

0.8 | 0.3463 | 0.6446 |

0.9 | 0.3519 | 0.6499 |

Localization errors of fractional Laplacian detection.

Order | Missed detection rate | False detection rate |
---|---|---|

1.1 | 0.4216 | 0.4785 |

1.2 | 0.4183 | 0.4611 |

1.3 | 0.4112 | 0.4589 |

1.4 | 0.4085 | 0.4510 |

1.5 | 0.4002 | 0.4497 |

1.6 | 0.3991 | 0.4399 |

1.7 | 0.4741 | 0.4871 |

1.8 | 0.4824 | 0.5024 |

1.9 | 0.5019 | 0.5112 |

And it can be shown that the missed detection rate and false detection rate from Table

Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (

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:

The quality factor is calculated according to equation (

Quality factor. (a) Fractional order Sobel detection algorithm. (b) Fractional order Laplacian detection algorithm.

And it can be shown that the quality factor from Figure

Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (

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

Execution time and memory footprint of fractional Sobel detection.

Order | Execution time (s) | Memory footprint (MB) |
---|---|---|

0.1 | 0.1766 | 33.92 |

0.2 | 0.1665 | 33.01 |

0.3 | 0.1559 | 32.65 |

0.4 | 0.1407 | 31.27 |

0.5 | 0.1297 | 30.89 |

0.6 | 0.1288 | 29.76 |

0.7 | 0.1441 | 31.77 |

0.8 | 0.1626 | 32.18 |

0.9 | 0.1936 | 34.48 |

Execution time and memory footprint of fractional Laplacian detection.

Order | Execution time (s) | Memory footprint (MB) |
---|---|---|

1.1 | 0.2098 | 37.12 |

1.2 | 0.2056 | 36.95 |

1.3 | 0.2011 | 36.07 |

1.4 | 0.1992 | 35.19 |

1.5 | 0.1881 | 34.89 |

1.6 | 0.1772 | 33.91 |

1.7 | 0.1956 | 35.22 |

1.8 | 0.2077 | 36.18 |

1.9 | 0.2103 | 38.35 |

In Table

And it can be shown that the execution time and memory footprint from Table

Taking into account the above analysis, we can draw a conclusion that the optimal fractional orders (

The 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

The comparison of integer order and fractional order methods. (a)

Figure

We can find from Figures

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

Localization errors of integer order and fractional order methods.

Evaluation metrics | The first-order Sobel operator ( |
The fractional order Sobel operator ( |
The second-order Laplacian operator ( |
The fractional order Laplacian operator ( |
---|---|---|---|---|

Missed detection rate | 0.3677 | 0.3001 | 0.5136 | 0.3991 |

False detection rate | 0.6541 | 0.6009 | 0.5228 | 0.4399 |

Execution time and memory footprint of integer order and fractional order methods.

Evaluation metrics | The first-order Sobel operator ( |
The fractional order Sobel operator ( |
The second-order Laplacian operator ( |
The fractional order Laplacian operator ( |
---|---|---|---|---|

Execution time (s) | 0.2059 | 0.1288 | 0.2577 | 0.1772 |

Memory footprint (MB) | 35.29 | 29.76 | 40.02 | 33.91 |

The quality factor of integer order and fractional order methods.

Evaluation metrics | The first-order Sobel operator ( |
The fractional order Sobel operator ( |
The second-order Laplacian operator ( |
The fractional order Laplacian operator ( |
---|---|---|---|---|

The quality factor | 0.59 | 0.99 | 0.62 | 0.98 |

It is clearly seen from Tables

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:

The steps of Canny operator edge detection are as follows:

Step 1. The original image is smoothed or convoluted by using two-dimensional Gauss filtering function, which can reduce noise.

Step 2. Calculating the partial derivative

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 B-spline 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

Step 2. According to the detection edge result

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

The comparison of different edge detection methods. (a) Canny method. (b) B-spline wavelet transform. (c) Multidirection fuzzy morphological method. (d) Present method.

Figure

We can find from Figures

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

Localization errors under different edge detection methods.

Evaluation metrics | Canny | B-spline wavelet transform | Multidirection fuzzy morphological | Present |
---|---|---|---|---|

Missed detection rate | 0.4836 | 0.4012 | 0.3378 | 0.3001 |

False detection rate | 0.7977 | 0.7152 | 0.6815 | 0.6009 |

Execution time and memory footprint under different edge detection methods.

Evaluation metrics | Canny | B-spline wavelet transform | Multidirection fuzzy morphological | Present |
---|---|---|---|---|

Execution time (s) | 0.2133 | 0.1906 | 0.1577 | 0.1288 |

Memory footprint (MB) | 40.65 | 38.33 | 32.14 | 29.76 |

The quality factor under different edge detection methods.

Evaluation metrics | Canny | B-spline wavelet transform | Multidirection fuzzy morphological | Present |
---|---|---|---|---|

The quality factor | 0.67 | 0.79 | 0.90 | 0.99 |

It is clearly seen from Tables

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 (first-order Sobel and second-order 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, B-spline 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.

No data were used to support this study.

The authors declare that there are no conflicts of interest.

This work was supported by the Chinese National Natural Science Foundation under Contract nos. 51674106 and 51274091.