Journal of Electrical and Computer Engineering

Volume 2019, Article ID 9594301, 13 pages

https://doi.org/10.1155/2019/9594301

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

^{1}Heilongjiang University of Science and Technology, Harbin 150022, China^{2}College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Chunping Ren; moc.anis@nipnuhcner

Received 29 January 2019; Revised 13 March 2019; Accepted 11 April 2019; Published 2 May 2019

Academic Editor: Sos S. Agaian

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.

#### Abstract

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.

#### 1. Introduction

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 [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 first-order 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]. 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 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 first-order 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 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.

#### 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 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 [20–24]: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 coal-rock infrared image.

Figure 1(a) shows the coal-rock fracture infrared image [25–27], 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. 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 gray-level transformation images as input images.