Power line fitting method for LiDAR robot point cloud power inspection

Abstract

The method for fitting transmission lines with LiDAR robot point clouds in this application mounts the LiDAR system on the robot, processes its scanning accuracy and three-dimensional point clouds, and explores a new way of inspecting transmission lines with high economic efficiency and adaptability to future development. First, the data measured and collected by the LiDAR robot is analyzed, the geometric positioning principle and the conversion relationship of the coordinate systems of each component are established, then the work process of the airborne lidar measurement system is designed, and the accuracy analysis and calibration method of the LiDAR robot point cloud data are proposed. Component models for ranging error, instantaneous scan angle error, system installation error, and attitude angle error are quantitatively analyzed in the context of the work of the line inspection robot. By performing in-flight calibration on the airborne lidar measurement system, the relative accuracy of the point cloud model is improved. The power line equation is fitted using the LiDAR robot point cloud data, realizing safe, fast, accurate, and economical inspection of the power corridor.

Description

Technical Field

[0001] This application relates to a method for fitting power transmission lines with LiDAR robot point clouds, and particularly to a method for fitting power transmission lines with LiDAR robot point clouds for power line inspection, belonging to the technical field of LiDAR point cloud power transmission line fitting. Background Art

[0002] Power transmission lines are the most core components of the power grid system. However, since power transmission lines are usually erected in the wild with complex terrain and harsh conditions in a bare form, they are affected by factors such as strong winds, rain, snow, lightning strikes, sand and dust, sunlight, birds, and the surrounding environment for a long time, resulting in the aging and rusting of power lines and components. Therefore, for a large-scale power grid, regularly inspecting power transmission lines to ensure the safety and stability of power lines is an important task in current power grid maintenance.

[0003] Manual inspection is a traditional inspection method that requires maintenance personnel to climb mountains and work in the wild for a long time. The working environment in the wild is hard, and coupled with the long inspection time and running around in the wild all year round, the work efficiency is relatively low. In addition, once special situations occur, such as entering uninhabited mountains or when power transmission lines cross large rivers, the inspection difficulty is greatly increased, bringing pressure to inspection workers. Therefore, considering that manual inspection is restricted by the external environment and humans and is prone to failing to detect potential safety hazards in power transmission lines in a timely manner, ultimately causing disasters, a new method should be adopted to inspect power transmission lines to reduce potential safety hazards in power transmission lines. The traditional manual inspection method can no longer meet the development and safe operation needs of modern power grids.

[0004] For aerial inspection, in order to maintain a safe distance from the line towers, it is impossible to conduct a close inspection of the line towers. At the same time, it is greatly affected by weather factors and cannot be inspected under harsh conditions such as fog and strong winds. In addition, helicopter inspection involves many issues such as personnel, navigation, ground maintenance, fuel, communication, weather, and repair. Drone inspection is small and flexible, can detect the line at close range with high precision, but its payload is limited and it cannot carry too many detection devices. At the same time, it is restricted by battery factors and has a short endurance time.

[0005] Robot inspection uses intelligent robots as carriers and the phase wires or ground wires of power transmission lines as operation paths. By carrying detection equipment, it inspects the corridors of power transmission lines. Since it approaches the power transmission lines at close range, the detection accuracy is high. In addition to being able to conduct routine inspections on lines and equipment, it can also perform various functions such as fault repair of power lines, replacement of spacer dampers, and pruning of vegetation near power lines through different additional structures. It has the advantages of powerful inspection functions, low maintenance costs, and long endurance time.

[0006] The laser detection and ranging system LiDAR can accurately measure the distance from the detector to an object or the ground surface. By processing the data generated during the ranging process, a digital elevation model (DEM) can be obtained. Through point cloud data, the three-dimensional space model of the object to be measured can be constructed. This is beneficial for collecting the coordinates along the power grid, obtaining a high-precision point cloud model of the transmission corridor, including the actual geometric data of transmission wires, towers, vegetation, buildings, etc. The point cloud data can be recorded in a short time, so that the distance between the wire and the ground object, or the distance between the wire and the adjacent wire, can be further automatically measured. By analyzing the data, the spacing distance between wires, the sag of transmission wires, or the span between power towers can be obtained, generating a series of data to determine whether the tension of the power optical cable meets the requirements, so as to take measures in advance to prevent damage to the power optical cable. Using the data obtained by LiDAR measurement, the monitoring and management data of the power grid safety status can be obtained, reducing the costs of inspection and safety status management.

[0007] However, there are many problems and key technical difficulties that need to be solved in the power line inspection and transmission line fitting method of the existing technology, including:

[0008] (1) For the manual inspection of transmission lines in the existing technology, maintenance personnel need to climb mountains and cross ridges and work outdoors for a long time. It is hard work and the inspection time is long, so the work efficiency is relatively low. Once entering the wild mountains and crossing large rivers, the inspection difficulty is greatly increased, bringing huge pressure to the inspection workers and making it easy to fail to detect the potential safety hazards of the transmission line in time, ultimately causing disasters. For aerial inspection, in order to maintain a safe distance from the line tower, it is impossible to closely inspect the line tower. At the same time, it is impossible to conduct inspections in bad weather such as heavy fog and strong winds. In addition, helicopter inspection involves many problems such as personnel, navigation, ground maintenance, fuel, communication, weather, and repair. UAV inspection is small and flexible, but its payload is limited and it cannot carry too many detection devices. At the same time, restricted by battery factors, the endurance time is short. There is a lack of applications in the existing technology that combine the inspection of power line robots with the collection of point cloud data of the LiDAR system for power line inspection, which cannot improve the data collection and analysis ability, and the power grid detection speed is slow, the accuracy is low, and the cost is high.

[0009] (2) Introducing the LiDAR system to be configured on the line inspection robot faces a series of difficulties. The positioning rules of the LiDAR system need to be defined and tested on the power line inspection robot. The point cloud measurement accuracy of the transmission line inspection needs to be further improved. The inspection error of the LiDAR robot for power transmission lines needs to be analyzed and reduced. It is impossible to extract the power line equation using the point cloud data. Since the traditional airborne lidar measurement system is a LiDAR system mounted on an aircraft, and the lidar measurement system is mounted on the inspection robot, the geometric positioning and the conversion relationship of the coordinate systems of each component need to be recalibrated. The working process of the airborne lidar measurement system will also change. There are many sources of inspection errors for the LiDAR robot in power line inspection. It is necessary to model the ranging error, instantaneous scanning angle error, system installation error (including installation angle error and eccentricity measurement error), system installation error, and attitude angle error, and quantitative analysis needs to be carried out in the context of the line inspection robot's work. It is necessary to perform in-flight calibration on the airborne lidar measurement system to improve the relative accuracy of the point cloud model and meet the requirements of the robot's point cloud power transmission line inspection and fitting.

[0010] (3) Introducing the LiDAR system to be configured on the line inspection robot poses new challenges to the positioning, point cloud measurement accuracy, and error analysis and correction of the LiDAR system. At the same time, due to the limitations of the catenary equation of the overhead line in practical applications, the existing technology lacks a method for extracting and fitting the power transmission line from the point cloud of the LiDAR robot for power line inspection. It is impossible to fit the catenary equation of the overhead line based on the LiDAR robot system. It is impossible to combine the point cloud model of the transmission line generated by the airborne lidar measurement equipment. It is impossible to use the elevation projection method for filtering, Canny operator edge detection, and improved Hough line detection methods to extract the power line. It is impossible to fit the power line equation using the point cloud data of the power line, resulting in low accuracy and reduced efficiency in generating the power line equation using the point cloud model, and ultimately leading to the inability to inspect the power corridor safely, quickly, accurately, and economically. Summary of the Invention

[0011] This application mounts a LiDAR system on a robot for the inspection of power corridors. By analyzing its scanning accuracy and processing data using 3D point clouds, a new, more economically viable, and future - adaptable way of inspecting transmission lines is explored. First, the data measured and collected by the LiDAR robot is analyzed, and the geometric positioning principle and the conversion relationship of the coordinate systems of each component are established. Then, the workflow of the airborne lidar measurement system is designed, and methods for analyzing the accuracy and calibration of the LiDAR robot's point cloud data are proposed. Error models for ranging error, instantaneous scanning angle error, system installation error (including installation angle error and eccentricity measurement error), and attitude angle error are constructed and quantitatively analyzed in the context of the line - inspection robot's operation. By performing in - flight calibration on the airborne lidar measurement system, the relative accuracy of the point cloud model is improved. The power line equation is fitted using the LiDAR robot's point cloud data, enabling safe, fast, accurate, and economical inspection of power corridors.

[0012] To achieve the above technical effects, the technical solutions adopted in this application are as follows:

[0013] A method for fitting power lines with the LiDAR robot's point cloud in power inspection. The LiDAR system is introduced and configured on the line - inspection robot. Based on the positioning of the LiDAR system by the intelligent robot, the point cloud measurement accuracy, and error analysis and correction, the power line equation is extracted using the point cloud data.

[0014] S1: Construct a LiDAR robot power inspection platform mounted on an intelligent robot, and form a complete workflow for data collection using the airborne lidar measurement system and generating a point cloud model of the transmission line.

[0015] S2: Establish multiple error models for configuring the LiDAR system on the line - inspection robot, quantitatively analyze the influence of multiple errors on the accuracy of the final laser foot point coordinates, and based on the comprehensive error, significantly reduce the accuracy of the laser foot point coordinates. Integrate multiple error sources and perform integrated calibration on the system.

[0016] S3: Based on the characteristics of the intelligent robot inspection of the transmission line, perform in - flight calibration on the airborne lidar measurement equipment. By correcting the roll angle and heading angle, the generated point cloud model image is clear, and the relative accuracy is improved.

[0017] S4: Establish a method for fitting the catenary equation of overhead lines based on a LiDAR robot system. Combine the point cloud model of the transmission line generated by the airborne lidar measurement device to construct a power line equation fitting framework. Extract the elevation distribution characteristics and plane projection characteristics of the LiDAR power line scan data. Use the elevation projection method for filtering, elevation projection and resampling, Canny operator edge detection and improved Hough line detection methods to extract the power line. Through clustering of adjacent pixel points, perceptual organization of lines, and line segment detection based on free probability transformation, fit the power line equation using the point cloud data of the power line, and conclude that it is more accurate and fast to generate the power line equation using the point cloud model.

[0018] Preferably, the power line equation fitting framework: Use the catenary equation of overhead power lines to calculate the shape of the power line. The integral form of the catenary equation of overhead power lines is:

[0019]

[0020] where σ0 is the horizontal stress of the overhead line, γ is the specific load, and C1 and C2 are constants, and the values thereof are related to the origin position of the coordinate system;

[0021] Use the catenary equation of overhead lines with unequal suspension points as the mathematical model of the overhead ground wire. The height difference between the two suspension points A and B is the height difference h, and the angle with the horizontal plane is the height difference angle Φ. Take point A as the coordinate origin:

[0022]

[0023] where, represents the catenary length within the span of the overhead line with equal suspension points;

[0024] The catenary shape of the overhead line is related to As long as the ratio is the same, the final suspension curve of the overhead line will be the same. And as long as is certain, the final suspension curve of the overhead line will be certain. In the formula, y is the only influencing factor;

[0025] Use the lidar measurement system to scan the power line to generate a three-dimensional point cloud model of the power line. Use the point cloud model to accurately determine the spatial three-dimensional coordinates of any point on the power line, the specific shape of the power line, generate a power line equation consistent with the actual situation, and feedback it to the catenary equation of the overhead line, so as to provide accurate data parameters for the research of the power line.

[0026] Preferably, extraction of elevation distribution characteristics: Based on the power line structure, within a certain range, ensure that there is a difference between the elevation of the power line and the ground elevation, and use the elevation difference to preliminarily distinguish the ground points and the power line;

[0027] Among the measured data points, the vast majority of the data points are concentrated in the area with relatively small elevation values, and this part is the ground points. Only a small number of data points are concentrated in the area with relatively large elevation values, and this part is the power line points. And the points scattered between the ground points and the power line points are the tower points and other feature points; within a certain range, the power lines have basically the same elevation distribution or are approximately on the same horizontal plane.

[0028] Preferably, for planar projection feature extraction: The point cloud data obtained after filtering includes the following types: power line point cloud data, tower point cloud data, and part of the feature point cloud data that has not been filtered out. Among these point cloud data, the features that have not been filtered out may be relatively close to the power lines, less than the safety distance. The projection of the point cloud data on the horizontal plane after filtering shows that the power lines are presented as parallel lines spliced by discrete discontinuous points, which are extracted by linear fitting to obtain the power line extraction data; both the towers and the potentially dangerous features are irregular closed figures, which are separated by mathematical morphology methods. In addition, the potentially dangerous features are vegetation or buildings, and the projected area is smaller than that of the towers. A set area threshold is used to separate the two to distinguish the towers and the potentially dangerous features.

[0029] Preferably, for the power line fitting algorithm based on elevation projection: At first, the filtering analysis is completed by using the elevation threshold segmentation method. First, some ground points are removed, and the remaining data is the power line point cloud data, tower point cloud data, and part of the feature point cloud data that has not been filtered out; then, the above data is transformed into a two-dimensional elevation value projection through the elevation projection method and resampling method, and the power lines are extracted and fitted by using the linear fitting method to obtain the power line-related data; finally, the obtained image is processed to facilitate the separation of the towers and the potentially dangerous features.

[0030] Preferably, for the filtering of the power line fitting algorithm based on elevation projection: According to the distribution characteristics of the data, the power line point cloud data is obtained, and then the ground points are removed by using the elevation threshold segmentation method to reduce the data and improve the fitting degree and fitting efficiency. The optimal iterative threshold method is used to achieve filtering, and the specific calculation steps are as follows:

[0031] Step 1: Select the average elevation T0 as the initial threshold for this calculation, that is, T k = T0;

[0032] Step 2: According to the threshold T k , then the data is divided into two subsets, and the average elevations of the two subsets are calculated in turn, which are T A and T B respectively. Assume that T A < T k , T B > T k ;

[0033] Step 3: Obtain the critical value T according to Equation 3 k+1 :

[0034]

[0035] Step 4: Replace T k+1 with T k ;

[0036] Step 5: Repeatedly repeat the above process until T k converges. Numerically, T k+1 = T k , then T k is the selected segmentation critical value T;

[0037] Step 6: Divide the data into two parts according to the finally obtained segmentation critical value T.

[0038] Preferably, elevation projection and resampling: Use the point cloud projection and resampling method to obtain the elevation value image, calculate the coverage range and elevation range of the dataset on the XOY plane; Map the non-ground points to the XOY plane, and calculate the projection sampling distance d through Equation 4 based on the width W of the elevation value image:

[0039]

[0040] Convert the planar coordinates (x, y) of the point into its corresponding grid row and column numbers (r, c) through Equation 5:

[0041]

[0042] Apply resampling point by point. If there is no laser point in the grid point, it is represented as a gray value of zero. If there is a laser point in the grid point, normalize the maximum elevation value to the gray range of [a, b] using the formula and use it as the gray value gray:

[0043]

[0044] Finally, obtain the elevation value image, thereby realizing the conversion of the point cloud data from three-dimensional space to two-dimensional plane.

[0045] Preferably, clustering of adjacent pixel points: After performing filtering, segmentation, thinning, edge and other processing on the actually collected image, cluster the adjacent pixel points in the image to reduce the interference of small short lines formed by noise on detection. The specific steps are as follows:

[0046] First step: First select a point P0 in the binary image, and then determine whether the point is 1, that is, whether P0 is a feature pixel point. If the obtained P0 ≠ 1, then select a point again in the binary image. If P0 = 1, then enter the next stage;

[0047] Step 2: Record the position of the obtained characteristic pixel points as P, put it at one end of the set of characteristic points S+P→S, and set the pixel value of this point to P = 0;

[0048] Step 3: Judge all the neighborhoods of point P to see if it is a characteristic point. If the final judgment result is a characteristic point, repeat Step 2. However, if the final judgment result is not a characteristic point, proceed to the next step;

[0049] Step 4: Search the data in the neighborhood of point P0 again to judge whether these data have connected characteristic pixel points. If the judgment result is yes, execute Step 5. If the judgment result is no, end the execution;

[0050] Step 5: Put the characteristic point at the other end of the set P+S→S, and the remaining operations are the same as in Step 2;

[0051] Step 6: Continue to judge whether these data are characteristic points. If the judgment result is yes, execute Step 5. If the judgment result is no, end the execution;

[0052] After the above processing, the pixel points are aggregated. In addition, those independent pixel points are also removed, and the pixel points are regarded as similar to straight lines, so as to further determine the detection object and separate parallel lines with very close intervals.

[0053] Preferably, perception grouping of straight lines: Further subdivide the broken line, and the obtained broken line will be closer to a straight line. (x1, y1) and (x2, y2) represent the coordinates of the two endpoints of the short line, and B is the straight line passing through (x1, y1) and (x2, y2). (x3, y3) represents the coordinates of the point where the line segment A deviates from the straight line B by a deviation distance of d, and d is calculated by Equation 7:

[0054] d = [(y3 - y1)(x2 - x1) - (x3 - x1)(y2 - y1)] / L Equation 7

[0055] In the formula: Let:

[0056] R = L / maxd Equation 8

[0057] This ratio R is only related to the straight line itself, has nothing to do with the obtained image size, and does not change with the size of the image. Use the point with the largest offset distance for binary segmentation to obtain the ratio R i (i = 1, 2), compare it with the ratio of the original line segment. If the ratio R i has a larger value, replace the original line segment with two sub-line segments; otherwise, keep the original line segment. In this way, the obtained original line segment is closer to a straight line and no further subsequent processing is required;

[0058] Through calculation and analysis, each line segment will be closer to a straight line, determining the detected straight line and improving the detection accuracy.

[0059] Preferably, for the detection of line segments based on free probability transformation: successively detect the approximate line segments, and arbitrarily select two points (x p , y p ) and (x q , y q ) on the approximate line segment, generate a straight line using the two points, and the straight line variable generated is obtained by the following formula:

[0060]

[0061] ρ = x p cosθ + y p sinθ or ρ = x q cosθ + y q sinθ Formula 10

[0062] By using the processing of clustering and perceptual organization, the effectiveness can be better guaranteed. Therefore, when performing the Hough transform on the above image, only a very small critical value needs to be reached to be valid. The above analysis operation results are as follows:

[0063] Step 1: Regard all the obtained pixel points d i (x i , y i ) as the set D;

[0064] Step 2: In the above set D, select d p , d q and calculate whether they meet the requirements of the point pair. If the two selected d p , d q points do not meet the requirements of the point pair, then re-select d p , d q ;

[0065] Step 3: Obtain the corresponding points (θ, ρ) of d p , d q and accumulate them to 1 at the corresponding positions;

[0066] Step 4: If the value at a certain point in the parameter space reaches the critical value, stop the above calculation, and the straight line corresponding to this point is the required straight line;

[0067] Step 5: Use the opencv software to process the elevation projection. Through software calculation, the projections of most power lines are detected;

[0068] Sort the lengths \(p\) of the detected straight lines in the polar coordinate system, and manually select the power line matching straight line that best meets the requirements by comparing the straight lines corresponding to different \(p\) values;

[0069] Finally, separate the point cloud data of the matched power line to obtain the three-dimensional coordinates of the point cloud, and input the point cloud coordinates into MATLAB software to display its spatial position and shape.

[0070] Compared with the prior art, the innovation points and advantages of this application are as follows:

[0071] (1) This application mounts the LiDAR system on a robot to conduct inspections of the power corridor, processes the scanning accuracy and three-dimensional point cloud data, and explores a new, more economically efficient, and future-developed way of inspecting transmission lines. First, analyze the data collected by the LiDAR robot measurement, establish its geometric positioning principle and the conversion relationship of the coordinate systems of each component, then design the workflow of the airborne lidar measurement system, and propose the accuracy analysis and calibration method for the point cloud data of the LiDAR robot. Model the ranging error, instantaneous scanning angle error, system installation error (including installation angle error and eccentricity measurement error), system installation error, and attitude angle error, and quantitatively analyze them in the context of the line inspection robot's work. By conducting in-flight calibration on the airborne lidar measurement system, improve the relative accuracy of the point cloud model. Use the point cloud data of the LiDAR robot to fit the power line equation, and achieve safe, fast, accurate, and economical inspection of the power corridor.

[0072] (2) This application introduces the LiDAR system and configures it on the line inspection robot. Based on the positioning of the LiDAR system by the intelligent robot, the point cloud measurement accuracy and error analysis and correction, extract the power line equation using the point cloud data; construct a lidar measurement system mounted on the intelligent robot, and form a complete set of workflows for data collection using the airborne lidar measurement system and generating the point cloud model of the transmission line. Detail the construction of various error models, and combined with the characteristics of the high-voltage transmission line inspection by the line inspection robot, quantitatively analyze the influence of various errors on the accuracy of the final laser foot point coordinates. It is found that the comprehensive error will significantly reduce the accuracy of the laser foot point coordinates, but its influence is not a simple linear superposition of various errors, and its influence mechanism is very complex. It is necessary to comprehensively consider multiple error sources to perform integrated calibration on the system. For the characteristics of the high-voltage transmission line inspection by the line inspection robot, conduct in-flight calibration on the airborne lidar measurement equipment. By correcting the roll angle and heading angle, finally make the point cloud model image generated by the measurement clear and improve the relative accuracy. Due to the limitations of the catenary equation of the overhead line in practical applications, a method for fitting the catenary equation of the overhead line based on the LiDAR robot system is proposed, and it is more accurate and fast to generate the power line equation using the point cloud model.

[0073] (3) This application proposes a method for fitting the catenary equation of overhead lines based on a LiDAR robot system, which solves the limitations of the catenary equation of overhead lines in practical applications. By combining the point cloud model of the transmission line generated by the airborne lidar measurement device, the method uses the elevation projection method for filtering, Canny operator edge detection, and improved Hough line detection to extract the power line, and fits the power line equation using the point cloud data of the power line, obtaining the conclusion that it is more accurate and fast to generate the power line equation using the point cloud model. This method has outstanding advantages such as safe driving along the line (orbit), reaching wherever the line goes, strong environmental adaptability, easy on-line power supply, high inspection efficiency, good close-range inspection effect, unmanned or few-person operation, and low cost. In addition, as a three-dimensional measurement system, the LiDAR system can quickly obtain the three-dimensional point cloud data of the line environment, realize the safety status monitoring and management of the power grid, and greatly reduce the inspection and safety status management costs. Mounting the LiDAR system on an intelligent robot for the inspection of transmission lines reduces the inspection and safety status management costs, and is a new type of transmission line inspection method with high economic benefits and suitable for future development. Description of the Drawings

[0074] Figure 1 It is a schematic diagram of the catenary of overhead lines with unequal suspension points.

[0075] Figure 2 It is a schematic diagram of the elevation of the power line scan data.

[0076] Figure 3 It is a histogram of the elevation distribution of the power line scan data.

[0077] Figure 4 It is a schematic cross-sectional view of the ground point filtering result.

[0078] Figure 5 It is a flow chart of the clustering algorithm for adjacent pixel points.

[0079] Figure 6 It is a schematic diagram of calculating the maximum offset distance of a straight line.

[0080] Figure 7 It is a schematic diagram of software solution to extract straight line features.

[0081] Figure 8 It is a schematic diagram of single power line matching.

[0082] Figure 9 It is a schematic diagram of separating the point cloud three-dimensional coordinates obtained from the matching power line point cloud data. Detailed Implementation Manner

[0083] The following further describes the technical solution of the transmission line fitting method for LiDAR robot point cloud power line inspection provided by this application in conjunction with the accompanying drawings, so that those skilled in the art can better understand this application and be able to implement it.

[0084] This application introduces the LiDAR system to be configured on the line inspection robot. For the positioning, point cloud measurement accuracy and error analysis and correction of the LiDAR system, the power line equation is extracted using point cloud data.

[0085] 1. In combination with the high-voltage transmission line inspection robot platform, the working principle of the airborne lidar measurement system is explained, and a complete set of work processes for data collection using the airborne lidar measurement system and generating a transmission line point cloud model are formed.

[0086] 2. Study the error sources of airborne lidar measurement, construct various error models in detail, and in combination with the characteristics of high-voltage transmission line inspection by the inspection robot, quantitatively analyze the influence of various errors on the accuracy of the final laser foot point coordinates. It is found that the comprehensive error will significantly reduce the accuracy of the laser foot point coordinates, but its influence is not a simple linear superposition of various errors, and its influence mechanism is very complex. It is necessary to comprehensively consider multiple error sources to integrate and calibrate the system.

[0087] 3. For the inspection characteristics of the high-voltage transmission line inspection robot, on-air calibration of the airborne lidar measurement equipment is carried out. By correcting the roll angle and heading angle, the point cloud model image generated by the measurement is finally clear, and the relative accuracy is improved.

[0088] 4. Due to the limitations of the catenary equation of the overhead line in practical applications, a method for fitting the catenary equation of the overhead line based on the LiDAR robot system is proposed. In combination with the transmission line point cloud model generated by the airborne lidar measurement equipment, the elevation projection method is used for filtering, Canny operator edge detection and improved Hough line detection methods to extract the power line. Finally, the power line equation is fitted using the point cloud data of the power line, and the conclusion is drawn that it is more accurate and fast to generate the power line equation using the point cloud model.

[0089] I. Power line equation fitting framework

[0090] The sag and line length of the overhead power line are key parameters, and traditional methods cannot accurately measure these parameters. The shape of the power line is calculated using the catenary equation of the overhead power line. The integral form of the catenary equation of the overhead power line is:

[0091]

[0092] Among them, σ0 is the horizontal stress of the overhead line, γ is the specific load, and C1 and C2 are constants, and the values of which are related to the origin position of the coordinate system.

[0093] The undulation of the terrain or the different heights of the erected poles and towers will result in unequal suspension point heights. Using the overhead line equation with unequal suspension points as the mathematical model for the overhead ground wire, the height difference between the two suspension points A and B is the elevation difference h, and the angle with the horizontal plane is the elevation angle Φ. Taking point A as the coordinate origin, as Figure 1 :

[0094]

[0095] Among them, represents the catenary length within the span of the overhead line with equal suspension points;

[0096] The catenary shape of the overhead line is related to As long as the ratio is the same, the final suspension curve of the overhead line will be the same, and as long as is certain, the final suspension curve of the overhead line will be certain. In the formula, y is the only influencing factor.

[0097] The horizontal stress is determined based on the ultimate strength of the conductor, considering the increase in the suspension point stress, acceptable extreme meteorological conditions, and safety factor. The horizontal stress is affected by the final elastic coefficient of the overhead line, the coefficient of thermal expansion, the cross-sectional area of the wire, and the temperature. At the same time, the specific weight γ changes in different climate environments, ultimately resulting in different sag of the overhead line, that is, the position change of each point on the overhead line. The power line equation obtained using the catenary equation of the overhead line is an equation under specific conditions, with an error from the actual situation.

[0098] Using a lidar measurement system to scan the power line, generating a three-dimensional point cloud model of the power line. Using the point cloud model, accurately determine the spatial three-dimensional coordinates of any point on the power line, the specific shape of the power line, generate a power line equation consistent with the actual situation, and feedback it to the catenary equation of the overhead line, thereby providing accurate data parameters for the research of the power line.

[0099] II. Feature Extraction of LiDAR Power Line Scanning Data

[0100] (1) Extraction of Elevation Distribution Features

[0101] Based on the power line structure ( Figure 2 ), there is an obvious height between the power line and the ground objects, ensuring the safety of people and livestock walking under the high-voltage overhead transmission line and preventing it from causing harm to them. That is, within a certain range, it is necessary to ensure a difference between the elevation of the power line and the ground. Therefore, the elevation difference can be used to preliminarily distinguish the ground points and the power line (including the transmission line tower).

[0102] From Figure 3It can be seen that among the measured data points, the vast majority of the data points are concentrated in the area with a relatively small elevation value, and this part is the ground points. Only a small number of data points are concentrated in the area with a relatively large elevation value, and this part is the power line points. And the points scattered between the ground points and the power line points are the pole tower points and other ground object points. From Figure 2 and Figure 3 It can also be studied and concluded that within a certain range, the power lines have basically the same elevation distribution or are similar to being on the same horizontal plane, which is exactly the algorithm basis of this application.

[0103] (2) Plane projection feature extraction

[0104] The point cloud data obtained after filtering includes the following types: power line point cloud data, pole tower point cloud data, and part of the ground object point cloud data that has not been filtered out. Among these point cloud data, the ground objects that have not been filtered out may be relatively close to the power lines, less than the safe distance, which is likely to cause danger. The projection of the point cloud data on the horizontal plane after filtering shows that the power lines are presented as parallel lines spliced by discrete discontinuous points, and are extracted by linear fitting to obtain the power line extraction data; the pole towers and possible dangerous ground objects are both irregular closed figures, and are separated by mathematical morphology methods. In addition, the possible dangerous ground objects are vegetation or buildings, and the projected area is smaller than that of the pole towers. Therefore, a set area threshold is used to separate the two to distinguish the pole towers and possible dangerous ground objects.

[0105] III. Power line fitting algorithm based on elevation projection

[0106] At first, the filtering analysis is completed by using the method of elevation threshold segmentation. First, some ground points are removed, and the remaining data are power line point cloud data, pole tower point cloud data, and part of the ground object point cloud data that has not been filtered out; then, the above data are transformed into two-dimensional elevation value projections through the elevation projection method and the resampling method, and the power lines are extracted and fitted by using the linear fitting method to obtain the power line-related data; finally, the obtained image is processed to facilitate the separation of the pole towers and possible dangerous ground objects.

[0107] (1) Filtering

[0108] According to the distribution characteristics of the data, the power line point cloud data is obtained, and then the ground points are removed by using the method of elevation threshold segmentation to reduce the data and improve the fitting degree and fitting efficiency. The optimal iterative threshold method is used to achieve filtering, and the specific calculation steps are as follows:

[0109] Step 1: Select the average elevation T0 as the initial threshold for this calculation, that is, T k = T0;

[0110] Step 2: According to the threshold T k, then divide the data into two subsets, and calculate the average elevation of the two subsets in turn, which are T A and T B , assuming T A <T k , T B >T k ;

[0111] Step 3: Obtain the critical value T k+1 :

[0112]

[0113] Step 4: Replace T k+1 with T k ;

[0114] Step 5: Repeat the above process repeatedly until T k converges. Numerically, T k+1 =T k , then T k is the selected segmentation critical value T;

[0115] Step 6: Divide the data into two parts according to the finally obtained segmentation critical value T;

[0116] Iteratively implement the filtering of ground points, Figure 4 is the ground points filtered after analysis. Among these points, the gray points represent non-ground points, and the black points represent ground points. Some tower points in the figure are often filtered as ground points, but it will not affect the subsequent analysis results.

[0117] (2) Elevation projection and resampling

[0118] Use the point cloud projection and resampling method to obtain the elevation value image, and calculate the coverage range and elevation range of the data set on the XOY plane; map the non-ground points to the XOY plane, and calculate the projection sampling distance d through Equation 4 according to the width W of the elevation value image:

[0119]

[0120] Convert the plane coordinates (x, y) of the point into its corresponding grid row and column numbers (r, c) through Equation 5:

[0121]

[0122] Perform resampling point by point. If there is no laser point in the grid point, it is represented as a gray value of zero. If there is a laser point in the grid point, normalize the maximum elevation value to the gray range of [a, b] using the formula and use it as the gray value gray:

[0123]

[0124] Finally, the elevation value image is obtained, and the point cloud data is transformed from three-dimensional space to two-dimensional plane.

[0125] (3) Power line extraction based on straight line detection

[0126] A point in the original image coordinate system corresponds to a straight line in the polar coordinate system. All points that appear as straight lines in the original coordinate system have the same slope and intercept, so these straight lines correspond to the same point in the polar coordinate system. After projecting each point in the original coordinate system to the polar coordinate system, we check whether there are any clustered points in the polar coordinate system. Such clustered points correspond to straight lines in the original coordinate system. In the elevation value image of the LiDAR power line point cloud data, the power lines are similar to line segments composed of a large number of discrete points, rather than complete straight lines. Hough transform is used to detect LiDAR data to prevent the power line data from being too discrete.

[0127] However, the traditional Hough transform first decomposes the variable plane into numerous squares at a certain step size. Then, using a many-to-one mapping, it derives the variable values of multiple pixels on the same line in the plane. If the result of the operation falls within a square in the variable plane, the squares are summed to 1, obtaining the variable of the line in the image space. This exhaustive search method is not only computationally intensive and memory-intensive, but also causes the square with the most votes in the variable domain to be surrounded by squares with fewer votes, which can easily interfere with the detection process and cause false or missed detection of the line. Therefore, when using line detection methods for power line detection, a more optimized detection algorithm is required.

[0128] (4) Power line extraction based on improved Hough line detection algorithm

[0129] First, the original image is preprocessed to remove all noise; then the object and the background are separated, and the object is refined based on this; then, a cluster analysis is performed on the adjacent pixels in the refined image, and then perceptual grouping is used to further subdivide the adjacent pixels so that they can be closer to a straight line; finally, the Hough transform is used to perform straight line detection on each segment, and the final fitting is used to obtain the detection effect.

[0130] ① Clustering of adjacent pixels

[0131] Under normal circumstances, the obtained straight line is not a smooth one. This is because external noise will interfere with the smoothness of the straight line, and at the same time, problems such as the performance of the image acquisition device will also interfere with the smoothness of the straight line, resulting in the obtained image not being as smooth as expected. After processing the actually acquired image through filtering, segmentation, thinning, edge processing, etc., the resulting image is still an unsmooth straight line, a broken line composed of unsmooth straight lines, and small short lines formed by noise. Therefore, if traditional Hough transform is used to process the data, it will be affected by these line segments, small short lines, etc., resulting in a decrease in detection accuracy. The clustering of adjacent pixel points in the image is to reduce the interference of these small short lines formed by noise on the detection. The specific steps are as follows:

[0132] Step 1: First, select a point P0 in the binary image, and then determine whether this point is 1, that is, whether P0 is a feature pixel point. If the obtained P0≠1, then re-select a point in the binary image. If P0 = 1, then enter the next stage;

[0133] Step 2: Record the position of the obtained feature pixel point as P, put it at one end of the feature point set S+P→S, and set the pixel value of this point to P = 0;

[0134] Step 3: Judge all the neighborhoods of point P to see if they are feature points. If the final judgment result is a feature point, then repeat Step 2. However, if the final judgment result is not a feature point, then enter the next step;

[0135] Step 4: Search the data in the neighborhood of point P0 again to determine whether these data have connected feature pixel points. If the judgment result is yes, then execute Step 5. If the judgment result is no, then end the execution;

[0136] Step 5: Put the feature points at the other end of the set P+S→S, and the remaining operations are the same as in Step 2;

[0137] Step 6: Continue to judge whether these data are feature points. If the judgment result is yes, then execute Step 5. If the judgment result is no, then end the execution.

[0138] Figure 5 It is the task diagram of the clustering process of adjacent pixel points. After the above processing, the pixel points are aggregated. In addition, those independent pixel points are also removed, and the pixel points are regarded as similar straight lines. This will further determine the detection object, separate parallel lines with very close intervals, prevent false detection and missed detection, and also reduce the scale of operation data.

[0139] ② Perceptual grouping of straight lines

[0140] For the convenience of subsequent processing, the polyline is further subdivided, and the resulting polyline will be closer to a straight line. As Figure 6 shown, (x1, y1) and (x2, y2) represent the coordinates of the two endpoints of the short line segment, and B is the straight line passing through (x1, y1) and (x2, y2). (x3, y3) represents the coordinates of the point where the deviation distance of line segment A from line B is d, and d is calculated by Equation 7:

[0141] d = [(y3 - y1)(x2 - x1) - (x3 - x1)(y2 - y1)] / L Equation 7

[0142] In the formula: Let:

[0143] R = L / maxd Equation 8

[0144] This ratio R is only related to the straight line itself and has nothing to do with the size of the obtained image, and does not change with the size of the image. The point with the maximum offset distance is used for binary segmentation to obtain the ratio R i (i = 1, 2), and compare it with the ratio of the original line segment. If the value of the ratio R i is larger, then replace the original line segment with two sub-line segments; otherwise, keep the original line segment. In this way, the obtained original line segment is closer to a straight line and no further subsequent processing is required.

[0145] Through calculation and analysis, each line segment will be closer to a straight line, the straight line to be detected is determined, and the detection accuracy is improved.

[0146] ③ Straight line segment detection based on free probability transformation

[0147] Detect the approximate straight line segments in sequence. Arbitrarily select two points (x p , y p ), (x q , y q ) on the approximate straight line segment, generate a straight line using the two points, and the generated straight line variable is obtained by the following formula:

[0148]

[0149] ρ = x p cosθ + y p sinθ or ρ = x q cosθ + y q sinθ Equation 10

[0150] By using the processing of clustering and perceptual grouping, the effectiveness can be better guaranteed. Therefore, when performing the Hough transform on the above image, only a very small critical value needs to be reached to hold. The above analysis operation results are as follows:

[0151] Step 1: Consider all the obtained pixel points d i (x i ,y i ) as the set D;

[0152] Step 2: In the above set D, select d p ,d q and calculate whether they meet the requirements of a point pair. If the two selected d p ,d q points do not meet the requirements of a point pair, then re-select d p ,d q ;

[0153] Step 3: Obtain the corresponding points (θ, ρ) of d p ,d q , and accumulate them to 1 at the corresponding positions;

[0154] Step 4: If the accumulation at a certain point in the parameter space reaches the critical value, stop the above calculation, and the line corresponding to this point is the required line.

[0155] Step 5: Use the opencv software to process the elevation projection, and its core code is:

[0156] if len(sys.argv) < 3:

[0157] print("Usage: python hough-lines.py [img_path] [output_path]")

[0158] sys.exit()

[0159] imgIn, imgOut = sys.argv[1], sys.argv[2]

[0160] img = cv2.imread(imgIn)

[0161] #img = cv2.imread('railroad.jpg')

[0162] gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)

[0163] edges = cv2.Canny(gray, 100, 200)

[0164] lines = cv2.houghLines(edges, 1, np.pi / 180, 200)

[0165] Through software calculation, the projections of the vast majority of power lines are detected. (Such as Figure 7 )

[0166] Sort the lengths p of the detected straight lines in the polar coordinate system, and manually select the power line matching straight line that best meets the requirements by comparing the straight lines corresponding to different p values. (Such as Figure 8 )

[0167] Finally, separate the point cloud data of the matched power line to obtain the three-dimensional coordinates of the point cloud (such as Figure 9 ), input the point cloud coordinates into MATLAB software to display its spatial position and shape;

[0168] Finally, transform the data on the X-axis, Y-axis, and Z-axis, and let Y * = Z, Fit the data, and finally obtain the equation of the point cloud data in the plane perpendicular to the power line.

Claims

1. A method for fitting a transmission line in LiDAR robot point cloud power inspection, characterized in that Introduce the LiDAR system to the line inspection robot. Based on the positioning of the LiDAR system by the intelligent robot, the point cloud measurement accuracy and error analysis and correction, extract the power line equation using the point cloud data; S1: Construct a LiDAR robot power line inspection platform carried on the intelligent robot, and form a complete workflow for data collection using the airborne lidar measurement system and generating a point cloud model of the transmission line; S2: Establish multiple error models for configuring the LiDAR system on the line inspection robot, quantitatively analyze the influence of multiple errors on the accuracy of the final laser foot point coordinates, and based on the comprehensive error, significantly reduce the accuracy of the laser foot point coordinates. Integrate and calibrate the system by integrating multiple error sources; S3: Based on the inspection characteristics of the transmission line intelligent robot, perform in-flight calibration on the airborne lidar measurement equipment. By correcting the roll angle and heading angle, make the point cloud model image generated by the measurement clear and improve the relative accuracy; S4: Establish a catenary equation fitting method for overhead lines based on the LiDAR robot system. Combine the point cloud model of the transmission line generated by the airborne lidar measurement equipment, construct a power line equation fitting framework, extract the elevation distribution characteristics and plane projection characteristics of the LiDAR power line scan data, and use the elevation projection method for filtering, elevation projection and resampling, Canny operator edge detection and improved Hough line detection methods to extract the power line. After clustering of adjacent pixel points, perceptual organization of lines, and line segment detection based on free probability transformation, use the point cloud data of the power line to fit the power line equation, and obtain that it is more accurate and fast to generate the power line equation using the point cloud model; Power line equation fitting: The catenary equation of the overhead power line is used to calculate the shape of the power line. σ0 is the horizontal stress of the overhead line, and γ is the specific load. The catenary shape of the overhead line is related to As long as the ratio is the same, the final suspension curve of the overhead line will be the same. And as long as is certain, the final suspension curve of the overhead line will be certain. The lidar measurement system is used to scan the power line to generate a three-dimensional point cloud model of the power line. Using the point cloud model, the spatial three-dimensional coordinates of any point on the power line can be accurately determined, and the specific shape of the power line can be obtained. A power line equation consistent with the actual situation is generated and fed back to the catenary equation of the overhead line, providing accurate data parameters for the research of the power line. The clustering steps of adjacent pixel points for power line extraction are as follows: The first step: First, select a point P0 in the binary image, and then judge whether the point is 1 and whether P0 is a feature pixel point. If the obtained P0≠1, then re-select a point in the binary image. If P0 = 1, then enter the next stage; The second step: Record the position of the obtained feature pixel point as P, put it into one end of the feature point set S+P→S, and set the pixel value of this point to P = 0; The third step: Judge all the neighborhoods of point P to see if it is a feature point. If the final judgment result is a feature point, repeat the second step, but if the final judgment result is not a feature point, then enter the next step; The fourth step: Search the data in the neighborhood of point P0 again to judge whether these data have connected feature pixel points. If the judgment result is yes, then execute the fifth step. If the judgment result is no, then end the execution; The fifth step: Put the feature points into the other end of the set P+S→S, and the rest of the operations are the same as the second step; The sixth step: Continue to judge whether these data are feature points. If the judgment result is yes, then execute step 5. If the judgment result is no, then end the execution; After the above processing, aggregate the pixel points; The analysis and calculation using clustering and perceptual organization are as follows: Step 1: Consider all the obtained pixel points d i (x i ,y i ) as the set D; Step 2: In the above set D, select d p , d q and calculate whether they meet the requirements of the point pair. If the two selected d p , d q points do not meet the requirements of the point pair, then re-select d p , d q ; Step 3: Obtain d p , d q corresponding points (θ, ρ), and accumulate them to 1 at the corresponding positions; Step four: If a certain point in the parameter space is increased to the critical value, then stop the above calculation, and the line corresponding to this point is the required line; Step 5: Use the opencv software to process the elevation projection. Through software calculation, the projections of the vast majority of power lines are detected; Sort the lengths p of the detected lines in the corresponding polar coordinate system. By comparing the lines corresponding to different p values, manually select the power line matching line that best meets the requirements; finally, separate the point cloud data of the matched power lines to obtain the three-dimensional coordinates of the point cloud, and input the point cloud coordinates into the MATLAB software to display its spatial position and shape.

2. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein Power line equation fitting framework: Use the catenary equation of overhead power lines to calculate the shape of power lines. The integral form of the catenary equation of overhead power lines is: where σ0 is the horizontal stress of the overhead line, γ is the specific load, and C1 and C2 are constants, and the values thereof are related to the origin position of the coordinate system; Use the overhead line equation with unequal suspension points as the mathematical model of the overhead ground wire. The height difference between the two suspension points A and B is the height difference h, and the angle with the horizontal plane is the height difference angle Φ. Take point A as the coordinate origin: Among them, represents the catenary length within the span of the overhead line with equal-height suspension points.

3. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein, Elevation distribution feature extraction: Based on the power line structure, within a certain range, ensure that there is a difference between the power line and the ground elevation, and use the elevation difference to preliminarily distinguish the ground points and the power lines; Among the measured data points, most of the data points are concentrated in the area with a smaller elevation value, and this part is the ground points. Only a small number of data points are concentrated in the area with a larger elevation value, and this part is the power line points. The points scattered between the ground points and the power line points are the tower points and other ground object points; within a certain interval, the power lines have basically the same elevation distribution or are approximately on the same horizontal plane.

4. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein Plane projection feature extraction: The point cloud data obtained after filtering includes the following types: power line point cloud data, tower point cloud data, and some ground object point cloud data that have not been filtered out. Among these point cloud data, the ground objects that have not been filtered out may be close to the power lines, less than the safety distance. The projection of the point cloud data on the horizontal plane after filtering shows that the power lines are presented as parallel lines spliced by discrete discontinuous points, which are extracted by linear fitting to obtain the power line extraction data; the towers and possible dangerous ground objects are all irregular closed figures, which are separated by mathematical morphology methods. In addition, the possible dangerous ground objects are vegetation or buildings, and the projected area is smaller than that of the towers. A set area threshold is used to separate the two to distinguish the towers and possible dangerous ground objects.

5. The method for fitting a transmission line in LiDAR robot point cloud power inspection according to claim 1, characterized in that, Power line fitting algorithm based on elevation projection: At first, use the method of dividing by the elevation threshold to complete the filtering analysis. First, remove some ground points, and the remaining data is the power line point cloud data, tower point cloud data, and some ground object point cloud data that have not been filtered out; then, convert the above data into two-dimensional elevation value projections through the elevation projection method and resampling method, and use the linear fitting method to extract and fit the power lines to obtain the power line related data; finally, process the obtained image to facilitate the separation of the towers and possible dangerous ground objects.

6. The method for fitting a power transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein Filtering of the power line fitting algorithm for elevation projection: According to the distribution characteristics of the data, the power line point cloud data is obtained, and then the ground points are removed by the method of elevation critical value segmentation to reduce the data and improve the fitting degree and fitting efficiency. The optimal iterative critical value method is used to implement the filtering. The specific calculation steps are as follows: Step 1: Select the average elevation T0 as the initial critical value for this calculation, i.e., T k = T0; Step 2: According to the critical value T k , then divide the data into two subsets, and calculate the average elevation of the two subsets from each other in turn, which are T A and T B respectively. Assume that T A < T k , T B > T k ; Step 3: Obtain the critical value T according to Equation 3 k+1 : Step 4: Replace T k+1 with T k ; Step 5: Repeat the above process repeatedly until T k converges. Numerically, T k+1 = T k , then T k is the selected segmentation critical value T; Step 6: According to the obtained final segmentation critical value T, the data is divided into two parts.

7. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein Elevation projection and resampling: The elevation value image is obtained by using the point cloud projection and resampling methods. Calculate the coverage range and elevation range of the data set on the XOY plane; Map the non-ground points to the XOY plane, and calculate the projection sampling distance d through Equation 4 according to the width W of the elevation value image: The plane coordinates (x, y) of the point are converted into the corresponding grid row and column numbers (r, c) through Equation 5: Resampling is applied point by point. If there is no laser point in the grid point, it is represented by a gray value of zero. If there is a laser point in the grid point, the maximum elevation value is normalized to the gray range of [a, b] using the formula and used as the gray value gray: Finally, the elevation value image is obtained, and thus the transformation of the point cloud data from the three-dimensional space to the two-dimensional plane is realized.

8. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein, Clustering of adjacent pixel points: After filtering, segmentation, thinning, and edge processing of the actually collected image, clustering of adjacent pixel points in the image is performed to reduce the interference of small short lines formed by noise on the detection; The pixel points are aggregated. In addition, those independent pixel points are also removed. The pixel points are regarded as similar straight lines, which will further determine the detection object and separate the parallel lines with very close intervals.

9. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein, Perceptual grouping of straight lines: Further subdivide the broken line, and the obtained broken line will be closer to a straight line. (x1, y1) and (x2, y2) represent the coordinates of the two endpoints of the short line, and B is the straight line passing through (x1, y1) and (x2, y2). (x3, y3) represents the coordinates of the point where the deviation distance of line segment A from the straight line B is d, and d is calculated through Equation 7: d = [(y3 - y1)(x2 - x1) - (x3 - x1)(y2 - y1)] / L Equation 7 In the formula: Let: R = L / maxd Equation 8 The ratio R is only related to the straight line itself and has nothing to do with the size of the obtained image, and does not change with the size of the image. The point with the maximum offset distance is used for binary segmentation to obtain the ratio R i (i = 1, 2), compare its ratio with that of the original line segment. If the ratio R i has a larger value, then replace the original line segment with two sub-line segments; Otherwise, retain the original line segment. In this way, the obtained original line segment is closer to a straight line and no further subsequent processing is required; Through calculation and analysis, each line segment will be closer to a straight line, determining the detected straight line and improving the detection accuracy.

10. The method for fitting a transmission line in the LiDAR robot point cloud power inspection according to claim 1, wherein, Line segment detection based on free probability transformation: successively detect approximate line segments, and arbitrarily select two points (x p , y p ) and (x q , y q ) on the approximate line segment, generate a straight line using the two points, and the straight line variable generated is obtained by the following formula: ρ = x p cosθ + y p sinθ or ρ = x q cosθ + y q sinθ Equation 10 By using the processing of clustering and perceptual grouping, the effectiveness can be better guaranteed. Therefore, when performing the Hough transform on the above image, only a very small critical value needs to be reached to hold.

Images