0 Introduction

Edge detection is very important for the processing and interpreting of potential field data. The edge of anomaly is determined by certain method based on density difference and magnetic difference between geological bodies. In the recent years, plenty of methods have been studied to identify the edge of geological bodies. Horizontal derivative and vertical derivative are widely used in edge enhancement. Nabighian (1972) put forward the analytical signal amplitude (ASM) method, in which maximum value reflects the edge position of objects. Liu et al. (2013) used analytical signal and the horizontal derivative to process magnetic data derived from Liangcheng iron ore in Inner Mongolia, and delineated the edge position of ultralow- grade magnetite accurately. Miller and Singh (1994) proposed the tilt angle (Tilt) method, which is the first equalization filter. It can balance anomalies with different amplitude. Ma et al. (2012) presented enhanced balancing filter, which can effectively improve the accuracy of edge detection. Wijins et al. (2005) put forward the theta map (Theta) method, which also uses maximum value to determine the edge position. Yu et al. (2015) utilized normalized differential and theta map to ascertain the locations of the Daxinganling fault zone and the Heihe-Qiqihar-Baicheng fault, which verified the existence of the Heihe-Qigihar-Baicheng fault. The conventional edge detection methods can classify the edge of geological bodies quickly and effectively, but they are still susceptible to noise interference. Furthermore, error information would be produced when positive and negative anomalies exist at the same time. With the continuous development of image processing technology, image edge detection methods have been popularized and applied into many fields. Zhao et al. (2008) applied Canny operator into geological mapping of remote sensing. This method can effectively improve the accuracy of edge extraction and divide the boundaries among faults and strata in the study area. Optimized Canny operator (Yang, 2015) was used for the edge detection of gravity and magnetic anomalies, by which the edge points and the noise points were divided effectively, and that the results were ideal. Liu (2016) applied differential operator, LoG operator and Canny operator into the extraction of geological boundary and made a comparison of these methods. The result showed that the color image detection method based on Canny operator can clearly identify the geological boundary and achieve satisfactory effect.

In this paper, three image processing methods are studied and applied to the edge detection of model data and real data. At the same time, three conventional edge detection methods are selected to make a comparison and analysis on the detection effect with image processing methods. These three operators could effectively overcome the defects of conventional methods and get better edge detection results.

1 Methodology

1.1 Canny operator

In image processing, the edge is defined as the position of obvious variations in the grayscale image, and Canny operator is a multilevel edge detection algorithm (Canny, 1986). As the image edge points to different directions, four masks are used to detect image edges in horizontal, vertical and diagonal directions. The same as image processing, Canny algorithm includes four steps for edge detection of potential field data.

(1) Denoising

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Noise in original data would seriously interfere edge detection, therefore, the first step is to convolute the original data with 2D Gaussian filter and get a sliding average. Assuming the potential field function is f(x,y), the corresponding Gaussian filter function is

3)在所设所选磁钢性能、漆包线规格、气隙长度等其他相关参数相同的情况下，以传统作圆求交点法所制梯形槽电枢齿部磁密高于理想梯形槽带电枢齿部磁密；在额定点，以传统法所制梯形槽电机效率比理想梯形槽电机效率低0.24%。

稀土元素是一组能很好揭示成矿物质来源及成矿条件的示踪元素，对热液型铅锌硫化物矿床的成矿物质来源和矿床成因有很大的指示意义[7-8]。矿物稀土元素配分模式是对成矿流体稀土元素组成特征的直接反应[9-10]。不同标高采集的4件新鲜矿石样品其稀土元素含量见表1，用W.V.Boynton球粒陨石标准化后作出其稀土元素分布型式图(图9)。

(1)

Where σ is the standard deviation of the Gaussian filter, which controls the smoothness of the filter. The convolution of Gaussian filter and f(x,y) is calculated and the smooth filtering result is obtained

g(x,y) =G(x,y;σ)*f(x,y)

(2)

A gravity forward model is built. It contains five prisms with uniform residual density, and there is no overlap among the prisms. The starting coordinates, dimensions and residual density parameters are shown in Table 1, where the unit of length is km, and the unit of density is kg/m3. Fig.3 shows the spatial distribution of the prisms. Gravity anomaly of the model is calculated and random noise with power equal to 5% of the peak-to-peak vaule is added. Then the model gravity anomaly is calculated again. The model gravity anomaly and the anomaly with random noise are shown in Fig.4.

size=2·ceil(2σ)+1

(3)

Where the ceil operator takes the smallest integer greater than or equal to the constant in the parenthesis. The size of Gaussian kernel matrix has an important impact on the smoothness of the original data. As it is far less than the size of the data, convolution is carried out in the sliding window. Equation (3) calculates the size of the sliding window. The noise interference is lower with a bigger window, but less details are reflected. On the contrary, the data obtained from the smaller window is susceptible to noise.

(1) Implementing LoG convolution with the original data.

First order finite difference is used to approximately calculate the partial derivatives of the smoothed data in x and y directions. The gradient intensity and direction angle in the data matrix can be obtained respectively by the partial derivatives as

(4)

Where M reflects edge intensity of the data, θ reflects the edge direction, and the θ that makes M the local maximum is the edge direction.

(3) Non-maximum suppression

Theta (Wijins et al., 2005) obtains the angular cosine between the analytic signal vector and the horizontal plane, and the formula is

This step is to make the blurred edge of the first step clear. The procedure includes: firstly, the gradient direction of a data point is approximated to one of the angle (0, 45, 90, 135, 180, 225, 270, 315). Then comparing gradient intensity of the point with that of other points in the gradient direction, if the gradient intensity is the maximum, it will be retained, otherwise it is set to zero. Fig.1 shows a detailed description of the process.

The numbers represent gradient intensity of the data points. The arrow points to the gradient direction. The gradient intensity of four points with gray background in the second row and third row is larger than that of the other points along the gradient direction. Therefore, these numbers will be retained, and the other points will be set to zero.

(4) Detection and connection of edge points with dual threshold algorithm

In edge detection of the potential field data, the influence of points in the neighborhood of each data point has different weights. The further the distance is, the smaller the influence is. The weighted sum can be represented by weight function in x and y directions as shown in Fig.2 (Sobel & Feldman, 1968).

There is still much noise in the data after the non-maximum suppression, and the dual threshold technique in Canny algorithm is used to suppress the noise. After setting upper bound and lower bound of threshold, the data point is regarded as the edge (strong edge) if it is greater than the upper bound. If the point is less than the lower bound, it is not the edge, and it will be considered as an option (weak edge) between the two bounds and needs to be further verified.

In image edge detection, two thresholds, thr_upper and thr_lower are used to process the non-maximum suppression image, and thr_upper=2.5×thr_lower. The gray value of the pixels with gradient less than thr_lower pixel is set to zero, and the result is map A. The gray value of the pixels with gradient less than thr_upper is set to zero, and the result is map B. The threshold of map B is higher, in which most of the noise is suppressed and useful edge information would also be lost. The threshold of map A is lower, and more information is retained. Therefore, map A is made as a supplement to connect the image edges based on map B.

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当关键工序的类型为块首工序或块尾工序时，如图10a所示，将块首工序后移至工序块其他工序以及同机器其他工序之后，将块尾工序前移至工序块其他工序以及同机器其他工序之前，不一定满足式(3)。

本文主要讨论北部湾经济区四座主要城市的区内贸易额(区内出口额和区内进口额之和)在该市GDP中所占的比重，通过计算这个系数来判断北部湾经济区主要城市的国民经济对区内贸易的依赖程度。通过对南宁、北海、钦州、防城港四市2008年-2011年数据的选取和处理，得到如下表格：

The size of the Gaussian filter: in the first step, the smooth filter can directly affect the results of Canny algorithm. Smaller filters could produce less blur effect, which can detect smaller and more distinct edges. Larger filters could produce more blur effect, which can detect larger and smooth edges.

Canny algorithm contains many adjustable parameters, which will affect the computing time and the effect of the algorithm.

Selection of dual threshold: using two thresholds is more flexible than only one, but details would be lost if the threshold is too large, and noise would surpass information of details when the threshold is too small.

Eshaghzadeh and Salehyan (2016) obtained the empirical formula of optimal upper bound threshold after many experiments.

(5)

Where SM, AM and MaxM represent the standard deviation, average and maximum of the data gradient intensity, respectively. In Canny algorithm, the ratio of the upper bound threshold over the lower bound is generally within [2, 3] (Xiao et al., 2011).

1.2 Laplacian-of-Gaussian operator

Laplacian-of-Gaussian (LoG) operator is an effective edge detection algorithm. First of all, the image is denoised with Gaussian filter and the second derivative is calculated, the edge position of the image is then obtained by detecting zero crossing points of the calculated results (Marr & Hildreth, 1980).

In image edge detection with LoG operator, the Gaussian filter is the same as that of Canny operator. When calculating the partial derivative of the filtered data, the following relationship exists

*f(t)]=f(t)*

(6)

Where t=(x,y). Therefore, the partial derivative of the Gaussian function can be calculated first, and then the convolution would be carried out. According to equation (1), the Gaussian kernel function in convolution can be obtained by simple derivation as

(7)

The 2D LoG operator can be approximated by the following 5×5 convolution kernel

It is noteworthy that the sum of all elements of the square convolution kernel should be set to zero to ensure that the convolution of the arbitrary homogeneous region is always zero. In the application of edge detection in potential field data, the main steps of the algorithm include:

(2) Calculating the amplitude and direction of the data gradient

(2) Identification of zero crossing points (i.e. transition points from negative to positive or reverse).

(3) Zero crossing points thresholding to suppress weak zero crossing points that may be caused by noise.

1.3 Sobel operator

In the edge detection of image, Sobel operator is also a common method. Sobel operator has two detection directions, one detects the edge of the horizontal direction, and the other detects the edge of the vertical direction. When processing images, Sobel operator weighs the position effect of the pixel points, which can effectively reduce the edge blur to get better effect.

如果测得质点的速度和P波的速度，测得煤岩的密度和泊松比，可以计算出动载荷的大小。以潞新矿区为例，根据大量监测数据，产生的煤炮所能监测到的能量为1×104～1×106 J，质点震动的峰值速度为0.5～3.5 m/s，煤的泊松比取0.2，密度取1 400 kg/m3，P波的波速取3 500 m/s，S波的波速取1 800 m/s。由此可计算得σdp= 2.21～15.44 MPa，σds=1.26～8.82 MPa。这个量级的动载荷相对于其原岩最大主应力16.5 MPa和最小主应力9.1 MPa而言，是非常大的。

废气再循环(Exhaust Gas Recirculation，EGR)是将排气中的一部分废气引进到进气管，同新鲜空气混合后进入气缸重新参与燃烧的过程。[4]EGR技术能有效降低柴油机最高燃烧温度和缸内氧浓度，从而降低NOx污染物的生成，目前,已成为控制船舶柴油机NOx排放物的有效手段之一。[1-8]

Where the x direction matrix is used as a horizontal operator and signed as Wx. The vertical operator in the y direction is signed as Wy. By convoluting the two operators with the original data respectively, the intensity difference approximation in horizontal and vertical directions can be obtained. Assuming that the original data is D, the data obtained by convolution in horizontal and vertical directions is

(8)

The transverse and longitudinal gradient approximation and gradient direction of each data point are respectively

(9)

The advantage of Sobel operator is the simple and fast calculation. However, the edge detection effect is not working well for data with complex edge details as only templates along horizontal and vertical directions are used.

1.4 Numerical calculation methods of edge detection

In order to compare the result of edge detection in the potential field data with the three image processing methods mentioned above, three conventional methods of edge detection are introduced, ASM, Tilt and Theta.

ASM calculates the total derivative modulus of the potential field data along three directions (Nabighian, 1972), which is the combination of horizontal and vertical derivatives. Assuming that the data is V and the expression is

(10)

The maximum of ASM delineates the edge position of the source, and its main advantage is that the second order derivative of the anomaly is calculated with relatively low noise interference. In addition, the magnetization direction has a low influence on the method for magnetic data. But its shortcoming is also obvious, i.e. the resolution is insufficient.

Tilt is the normalization result of horizontal derivative over vertical derivative (Miller & Singh, 1994), and its expression is

(11)

This method can effectively overcome the problem that the detection effect decreases with the increasing of depth of source. Tilt is usually positive above the source, zero on the edge and negative outside the source, so the edge position can be identified.

2.5.3 网关。Zig bee模块接收传感器发送来的数据，通过串口透传将数据包再发送 给Wi-Fi模块，从而连接外网。

(12)

The range of cosine value is between [0, 1], and its maximum value corresponds to the source edge. This method can enhance weak anomalies and play a role on balancing anomalies.

The three commonly used edge detection methods have achieved good results in practical applications, but there are still some problems. Therefore, they can be combined with other methods, such as total horizontal derivatives, to achieve better effect. Imaging processing methods are compared with conventional edge detection methods in this paper to verify their effect.

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2 Model test

The size of Gaussian filter kernel matrix is determined by the standard deviation (Eshaghzadeh & Salehyan, 2016).

The forward model includes positive and negative anomalies, which conforms to the real geological conditions, and the contact area among the prism edges increases the difficulty of edge detection. In addition, adding random noise can reasonably verify the ability of each method on suppressing noise interference. The image processing methods and the numerical edge detection methods are applied to the noisy data, and the Among them, LoG operator is better in identifying relatively isolated prisms edges, such as the prism 1, 2 and 3. But for the prisms with close contact, fake edge information is introduced. As shown in Fig.5b, the contact area between prism 4 and 5 has divergent fake edge points, which correspond to the edges of positive and negative anomalies. In addition, there are obvious fake edges at the positive and negative edges of the prisms 2, 4 and 3, 5. It shows that LoG operator is not very effective in identifying the edges between positive and negative anomalies with close distance. However, the overall effect is satisfactory and it is not sensitive to noise. In Fig.5c, although there is no fake edge in the result of Sobel operator, the edge points of prism 2 and 3 are not focused and it is susceptible to noise. As a more complex algorithm, Canny operator's edge detection results are evaluated to be better. As shown in Fig.5a, Canny operator dose not introduce fake edges and is less affected by noise, compared with LoG operator and Sobel operator.

results are shown in Fig.5. It can be seen that the three operators can effectively detect the edge of the prisms.

The edge detection results of ASM, Tilt and Theta are shown in Fig.5d-f. It can be seen that all of the methods have basically detected the true edge of the prisms, but they are affected by noise and thus the resolution is insufficient. Especially for ASM, the effect is not ideal in the edge detection of the small prism 4 and 5. Through comparison, it shows that the three operators in this paper achieve better results than conventional methods. Especially for Canny operator, it does not introduce error edge information and is less affected by noise.

3 Application of real data

In order to verify the edge detection effect of image processing methods in real data application, gravity anomaly data in Sichuan basin and magnetic data in Zhurihe region are used.

First of all, the above methods are used to process gravity data in Sichuan Basin and the results are shown in Fig.6. Fig.6a is the Bouguer anomaly, in which the horizontal positions of known faults are marked with white curve (Ma et al., 2012). In Fig.6c-d, the edge points identified by LoG operator and Sobel operator are more dispersive and do not form continuous edge, which does not match well with known faults. And the results of LoG operator are more scattered in the edge position, which may result in fake edge information due to the noise interference. In Fig.6b, Canny operator has achieved better results. The detected edge points are continuous and correspond well with the known faults. The trend of faults and more small faults can be identified clearly, and more details are obtained. In Fig.6e-g, edge detection methods do not get clear edge information. Especially for Tilt and Theta, the resolution is insufficient, and the results are not well matched with known faults.

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一些鼓励和支持“特色景观旅游名镇”“体育旅游”等融合性发展项目的建设也被写到了政策中，伴随着多项政策的实施，据国家旅游局统计，2016年我国旅游业实现总收入4.65万亿，旅游业对GDP的综合贡献为10.8%；到2020年旅游业总收入将达到7.0万亿，旅游投资规模达到2万亿，旅游业对GDP的综合贡献达到12%，而体育旅游所创造的价值大约占旅游总收入的25%[3]。

Magnetic data in Zhurihe region need to be reduced to pole firstly and then the proposed method is used to process the data. As shown in Fig.7. Fig.7a is a magnetic anomaly map after being reduced to pole, and it can be seen that the range of the metallogenic belt is clear. According to the application effect of the methods, the resolution of ASM in Fig.7e is insufficient, which cannot determine accurate edge positions of metallogenic belt. In Fig.7g, Theta magnifies the noise and introduces fake edge information. In Fig.7f, Tilt gets better edge detection results, but there are also fake edge points at the upper left corner. From Fig.7c-d, LoG operator and Sobel operator get better edge position. However, fake information is also introduced and some edge points are scattered. As shown in Fig.7b, Canny operator clearly delineates the edge of metallogenic belt and corresponds to the known information well.

对马克思共产主义思想的研究同样属于马克思哲学超越维度研究的传统内容，但是长期以来由于缺乏对于马克思的“现实的人”和人的全面发展的真实关注，存在着对共产主义的诸多认识误区及其研究理路模式化、样板化的不足。这一局面已经随着对“人”的真切关注得到了一定程度的纠正和改善。总体而言，学界主要在两个向度上展开思考和研究。一方面，对原有认识的反思和批判，主要表现为对于以往共产主义单纯强调经济发展和制度建设的片面性认识的澄清和纠正，这在一定程度上已然是学界的共识。另一方面，从人的生成性视域，确立起共产主义对人全面自由发展的本真意义和终极旨归，表现为从多重维度构建起对马克思共产主义思想的理解和诠释。

4 Conclusions

(1) The resolution of conventional edge detection methods is usually insufficient. When derivative is further calculated, noise would be magnified, which results in blurry edge position and introduces fake edge information.

(2) The image processing methods discussed in this paper detect edges through calculating horizontal gradient intensity and direction angle of image and determining the extreme points or zero points with threshold. Therefore, the edge detection effect for potential field anomalies with horizontal distribution is better.

(3) Canny operator and LoG operator convolute the original data with 2D Gaussian filter before calculating gradient intensity, which reduces the influence of noise effectively. Sobel operator weighs the position effect of the data points, which can get clearer edge detection results.

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