Anomaly detection method for UAV power line inspection based on point cloud data

Abstract

The present invention discloses a method for detecting abnormalities in power line inspection by unmanned aerial vehicles based on point cloud data, which belongs to the field of point cloud information processing and includes the following steps: A. Data acquisition to obtain the three-dimensional point cloud spatial position information and color information of the transmission tower; B. Filtering, downsampling and registration processing of the acquired point cloud data to remove noise, improve accuracy and ensure data consistency; C. Target segmentation of the point cloud data according to the color characteristics of the insulator chain to extract the insulator chain area; D. Fitting the insulator chain point cloud data using the least squares method to determine the axial plane and generate a two-dimensional plane projection; E. Performing time series abnormality detection to identify and mark the position of the self-explosive insulator. The present invention realizes the efficient detection of power line insulator abnormalities by unmanned aerial vehicles through the combination of point cloud data processing and intelligent algorithms, significantly improves the automation and accuracy of power inspection, and is suitable for large-scale power equipment inspection tasks.

Description

Technical Field

[0001] The present invention relates to the field of point cloud information processing, and in particular to a method for detecting abnormalities in power line inspections by unmanned aerial vehicles based on point cloud data. Background Art

[0002] The equipment and facilities of the power industry are often distributed over a wide geographical area. How to quickly and accurately detect equipment failures and deal with them in a timely manner has always been the focus of power industry workers.

[0003] Insulators are important components to ensure the normal operation of transmission lines. The traditional method of detecting insulator defects through manual visual judgment consumes materials and labor. In recent years, drone power inspections, which have been vigorously promoted, are the mainstream inspection method. They take images from multiple angles, and then manually analyze the images, mark the problems, and finally provide feedback. However, due to the large number of images taken by drones, a single overhead tower takes more than 300 images. It takes 5 to 10 minutes to manually mark a picture, which is a huge workload. At the same time, the relevant personnel who perform the marking work are prone to eye fatigue, which leads to missed labels and wrong labels.

[0004] In recent years, the use of image processing and deep learning methods to annotate pictures, pre-process and enhance the captured images, and train deep learning neural networks has improved the recognition efficiency to a certain extent. Yun Weiguo et al. proposed a power grid defect detection and recognition method based on YOLOv5-EM, and trained the YOLOv5-EM target detection model to detect test set pictures and randomly shot videos of network drones. However, the recognition process will be affected by the shooting situation. Insulators may be blocked, and the three-dimensional information of the transmission tower cannot be fully analyzed from multiple pictures in time. In addition, the computing center built by the deep learning network has high computing cost and poor unified information after multiple pictures are recognized. Therefore, it is difficult to establish a complete inspection system for insulator detection based on image recognition and accurately locate the position of the self-explosive insulator in the entire transmission tower.

[0005] With the continuous development of 3D scanning technology, the rapid and accurate identification and positioning of various objects can be achieved by using depth cameras and Intel T265 space inertial navigation cameras for airborne scanning. On the other hand, Intel T265 space inertial navigation cameras can achieve precise positioning and tracking of objects. By combining Intel T265 space inertial navigation cameras with depth cameras, point cloud scanning of objects can be achieved.

[0006] The power line anomaly inspection method based on point cloud data can integrate the point cloud data information of multiple sampling scans. The point cloud data contains the point cloud spatial distribution of the scanned object and the surrounding environment. The point cloud is segmented by the insulator chain features to obtain the target insulator area. Xu Lianggang et al. proposed a method for detecting insulator strings in three-dimensional laser point cloud data, and detected the insulator string point cloud using an insulator string point cloud detection method based on a hybrid voxel network. However, this method is insufficient in noise processing and occlusion separation in complex environments. Therefore, the present invention effectively improves the detection accuracy and stability by improving data preprocessing and anomaly detection algorithms. Summary of the invention

[0007] The purpose of the present invention is to develop a UAV power line inspection anomaly detection method based on point cloud data. By scanning the transmission tower area with a UAV and processing the point cloud data, combined with an intelligent algorithm, anomaly detection of transmission tower insulators can be achieved, thereby improving inspection efficiency and accuracy.

[0008] To achieve the above object, the present invention provides the following technical solution: a method for detecting abnormalities in UAV power line inspection based on point cloud data, which includes the following steps:

[0009] A. Data collection;

[0010] B. Data preprocessing;

[0011] C. Insulator chain target segmentation;

[0012] D. Plane fitting and projection of insulator chain;

[0013] E. Timing anomaly detection;

[0014] The specific steps are as follows:

[0015] A. Data collection: Obtain the three-dimensional point cloud spatial position information and color information of the transmission tower.

[0016] B1. Filter the point cloud data;

[0017] B2. Downsampling the point cloud data;

[0018] B3. Perform registration processing on the point cloud data collected multiple times to eliminate the errors between the scanning data at different positions.

[0019] C1. Identify and segment the insulator chain according to its color features in the point cloud;

[0020] C2. Further segmentation is performed based on the geometric features of the insulator chain so that the point cloud data only contains the point cloud of the insulator chain.

[0021] D1. Use the least squares method to fit the point cloud data of the insulator chain and generate the axial plane equation of the insulator chain z=ax+by+c;

[0022] D2, projecting the fitted 3D point cloud data onto a 2D image plane to generate a 2D image of the point cloud projected on the plane;

[0023] D3. After generating the two-dimensional projection image, the amount of point cloud data in each interval is counted, and a distribution histogram data sequence is created using the generated two-dimensional projection data.

[0024] E1: Generate a time series, segment the point cloud data along the axial direction of the insulator chain, generate equally spaced intervals, and for each interval i, count the maximum value of the point cloud data in the interval and minimum value , and calculate the difference , through this difference, generate the histogram data sequence of each interval, and convert the The difference data are arranged in order to form a complete time series;

[0025] E2: Construct a time series anomaly detection model combined with the geometric features of the insulator chain, establish OneClass SVM for sequence detection, and build a time series anomaly detection model based on the generated time series as training data. Use OneClass SVM as the core algorithm for anomaly detection and use radial basis function RBF as the kernel function. Its form is: Among them, γ=0.001, γ is the kernel parameter; after training, the new time series is input into the OneClass SVM model for detection. By calculating the distance of each point relative to the hyperplane, if the distance exceeds a certain threshold, the point is judged as an outlier;

[0026] E3: Determine the continuous abnormal interval according to the statistical formula: > , in It is a set minimum threshold, which indicates the minimum number of abnormal points contained in each continuous interval. Intervals with the number of abnormal points greater than the threshold are selected as potential abnormal defect locations.

[0027] E4: Three-dimensional restoration and marking of abnormal points. The abnormal points in the two-dimensional plane are mapped back to the three-dimensional spatial coordinates of the insulator chain through the projection equation in step D: z=ax+by+c to determine the specific location of the self-explosive insulators in the insulator chain; these abnormal points are marked in the three-dimensional point cloud model of the transmission tower and an abnormality report is generated.

[0028] Furthermore, in step A, a drone equipped with an Intel D435i depth camera and a T265 inertial navigation camera is used to perform an all-round scan of the transmission tower.

[0029] Furthermore, the filtering process in step B1 uses Gaussian filtering technology to remove noise and error points in the point cloud data while maintaining important edge information, and the downsampling process in step B2 uses a voxel grid downsampling method to reduce the computational load by reducing the density of data points while maintaining the structural characteristics of the point cloud data.

[0030] Further, the least squares fitting step in step D1: First, by extracting the coordinate point set of the insulator chain from the point cloud data, a linear equation system is constructed: Ax=b, where A is the coefficient matrix, x is the coefficient vector to be solved, and b is the constant vector. Based on the following linear relationship, a specific equation system can be constructed: Next, the above equations are solved to obtain the coefficients a, b and c of the plane equation: Finally, the axial plane equation of the insulator chain z = ax + by + c is generated.

[0031] Furthermore, the least squares fitting in step D performs straight line detection to ensure the axial projection fitting accuracy of the insulator chain.

[0032] The beneficial effects of the present invention are:

[0033] The present invention can quickly construct a three-dimensional model of the transmission tower through efficient point cloud data collection and processing technology, and realize high-precision power line inspection;

[0034] The present invention adopts an insulator chain segmentation method based on color and geometric features, which can accurately extract point cloud data of the insulator chain and improve the accuracy of segmentation;

[0035] The present invention combines the least square method and plane projection technology to ensure the accurate fitting of the insulator chain axial direction and provide reliable data support for subsequent abnormality detection;

[0036] The present invention adopts the OneClass SVM time series anomaly detection algorithm to effectively identify and locate self-exploding insulators, thereby improving the automation level and detection accuracy of power line inspection;

[0037] The present invention provides direct support for subsequent maintenance work by automatically marking abnormal points and generating inspection reports, significantly improving the efficiency and intelligence of power line inspections;

[0038] The present invention integrates drone inspection, point cloud data processing, anomaly detection and remote maintenance system to build a complete power inspection automation solution, which is suitable for large-scale power line inspection tasks, helping to improve the inspection efficiency of the power industry and reduce maintenance costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1This is a flow chart of a method for detecting abnormalities in UAV power line inspection based on point cloud data in one embodiment of the present invention;

[0040] Figure 2 3D point cloud expansion diagram of insulator chain;

[0041] Figure 3 Overlay of 3D point cloud and 2D projection histogram of insulator chain;

[0042] Figure 4 Global point cloud and height distribution map of insulator chain;

[0043] Figure 5 Insulator chain point cloud data and abnormal interval marking map. DETAILED DESCRIPTION

[0044] In order to make the purpose, technical solutions and advantages of the present invention clearer, the following Figure 1-5 The specific steps are further described in detail for the UAV power line inspection anomaly detection method based on point cloud data of the present invention. Figure 1 As shown, the present invention provides a method for scanning a transmission tower using a drone equipped with a depth camera and an inertial navigation camera. The method mainly includes the following steps:

[0045] Step A1: Figure 2 and Figure 3 As shown in the figure, a drone equipped with an Intel D435i depth camera and a T265 inertial navigation camera is used to perform a full-scale scan of the transmission tower from multiple angles. Through these devices, the three-dimensional point cloud spatial position information and color information of the transmission tower are obtained. This step ensures the comprehensiveness and accuracy of data collection, especially under different angles and lighting conditions, it can capture the complete geometric structure of the insulator on the transmission tower, providing a reliable data basis for subsequent processing.

[0046] Steps B1-B3: Preprocess the collected point cloud data, including filtering, downsampling and registration. Figure 3 As shown in the figure, the point cloud data is filtered using Gaussian filtering technology to remove noise and error points in the point cloud data while maintaining important edge information and improving data quality. Then downsampling is performed. The downsampling process uses a voxel grid downsampling method to reduce the computational load by reducing the density of data points while maintaining the structural characteristics of the point cloud data, thereby improving the computational efficiency of subsequent processing. Finally, the registration process eliminates the differences between multiple scans to ensure the spatial consistency between all scan data, laying the foundation for the subsequent insulator chain segmentation.

[0047] When a drone scans the same transmission tower area multiple times, there is a certain spatial deviation between the point cloud data obtained from each scan due to the differences in the position, attitude angle, lighting conditions and other factors of the aircraft. Therefore, in order to ensure that the point cloud data collected multiple times can be accurately aligned in the same coordinate system, registration processing must be performed. Through registration, the differences between different scan data can be eliminated, and the spatial consistency of each scan data can be ensured, thereby generating a complete and accurate three-dimensional model, providing a reliable data basis for subsequent insulator chain segmentation and anomaly detection.

[0048] B31. Initial alignment (rough registration): First, use the drone’s own posture sensor to perform preliminary alignment on the point cloud data collected at different times. Using the position information provided by the sensor, the point cloud of each scan is roughly placed in the same coordinate system to ensure the initial consistency of the data.

[0049] B32. Feature matching: Based on the preliminary alignment, feature point matching is performed by analyzing the geometric features in the point cloud (such as corner points, boundary points, planes, etc.). The SIFT (Scale Invariant Feature Transform) algorithm is used to extract feature points, and the point cloud data collected at different times are matched to find the corresponding points.

[0050] B33. Iterative Closest Point (ICP) algorithm for fine registration: In order to further improve the alignment accuracy, the ICP (Iterative Closest Point) algorithm is used for fine registration. The algorithm is implemented through the following steps:

[0051] Nearest neighbor matching: In the source point cloud and the target point cloud, find the nearest neighbor of each point for pairing.

[0052] Calculate the transformation matrix: Based on the paired points, calculate the rotation matrix and translation vector to minimize the error between the two sets of point clouds.

[0053] Iterative optimization: Repeat the nearest neighbor matching and transformation matrix calculation steps until the registration error meets the predetermined convergence condition, ensuring accurate alignment.

[0054] Step C1-C2: Figure 4 As shown, the point cloud data is initially segmented according to the color characteristics of the insulator chain (such as red or green for 220KV suspension insulators and 500KV standard suspension insulators) to extract the area of ​​the insulator chain.

[0055] The geometric features of the insulator chain are combined for further segmentation to ensure the segmentation accuracy of the insulator chain. The segmented results will be used for subsequent plane fitting and anomaly detection.

[0056] D. Insulator chain plane fitting and projection: including the following sub-steps:

[0057] In this step, the least squares method is used to fit the three-dimensional point cloud data of the insulator chain to determine the axial plane of the insulator chain. The specific steps are as follows: First, the coordinate point set of the insulator chain is extracted from the point cloud data to construct a linear equation system: Among them, A is the coefficient matrix, x is the coefficient vector to be solved, and b is the constant vector. Based on the following linear relationship, a specific set of equations can be constructed: , solve the above equations to obtain the coefficients a, b and c of the plane equation:

[0058] .

[0059] D2. Plane projection of 3D point cloud data

[0060] After completing the fitting, the plane equation of the insulator chain is obtained: z=ax+by+c. This equation represents the axial plane of the insulator chain in three-dimensional space. Next, the three-dimensional point cloud data of the insulator chain is projected onto the plane to simplify the data representation and facilitate subsequent anomaly detection. Assuming that P(x,y,z) is a point in the point cloud, its projection P′(x′,y′,z′) on the plane can be expressed as: z′=ax′+by′+c. Through this projection process, the three-dimensional geometric structure of the insulator chain can be simplified to a representation on a two-dimensional plane.

[0061] After the two-dimensional projection image is generated, the distribution histogram is created to describe the density distribution of the insulator chain in all directions. By counting the amount of point cloud data in each segment of the projection image, a histogram data sequence can be generated to represent the spatial distribution of the point cloud data. As shown in the formula, the interval distribution in the histogram sequence is calculated: , by calculating the number of data points in each projection area of ​​the insulator chain and generating a histogram data sequence, further preparation is made for time series anomaly detection. This process can significantly improve the accuracy of subsequent anomaly detection. This data sequence provides basic data for the subsequent time series anomaly detection algorithm. By analyzing the histogram data of the insulator chain, anomalies in the insulator chain can be effectively detected, such as the location and identification of self-exploding insulators.

[0062] E. Timing anomaly detection: includes the following sub-steps:

[0063] 1. Data axial segmentation:

[0064] In order to detect anomalies in the insulator chain, the point cloud data is first segmented along the axial direction of the insulator chain (such as the Y axis). A fixed step size dn is set to divide the insulator chain into multiple intervals i at equal intervals. These intervals are used to analyze the distribution characteristics of the point cloud data in different segments. This segmentation operation helps to convert the structural information of the insulator chain into quantifiable sequence data, which is convenient for subsequent anomaly detection.

[0065] 2. Count the maximum and minimum values ​​of each interval: For each interval i, count the maximum value of the point cloud data in the Y-axis direction within the interval and minimum value These values ​​reflect the point cloud height distribution characteristics of the insulator chain within this interval.

[0066] 3. Calculate the difference and generate a histogram data sequence: For each interval, calculate the difference between the maximum and minimum values ​​of the point cloud data: , this difference It reflects the vertical distribution of point cloud data in each interval. The values ​​can generate an interval distribution histogram data sequence. This sequence will become the input data for time series anomaly detection, helping the system analyze the abnormal conditions of the insulator chain.

[0067] Constructing a time series:

[0068] For each interval The difference data are arranged in order to form a complete time series. This time series reflects the geometric changes of the insulator chain at different positions. By analyzing the abnormal fluctuations of this series, possible faults or abnormalities in the insulator chain can be identified.

[0069] 1. Construction of time series anomaly detection model:

[0070] Based on the generated time series data, a time series anomaly detection model is constructed using the geometric characteristics of the insulator chain (such as the axial length and the height change of each interval). Since a normal insulator chain has relatively uniform geometric characteristics and regular distribution, any area that deviates significantly from these characteristics may be an abnormal area.

[0071] 2. Introduction to OneClass SVM model:

[0072] OneClass SVM is an algorithm specifically designed for unsupervised anomaly detection. It identifies anomalies that deviate from the normal distribution by learning the distribution boundary of normal points. The OneClass SVM model can effectively detect time series data points in abnormal areas of insulator chains. The model uses radial basis function (RBF) as the kernel function, and the formula is as follows: Wherein, γ is a kernel parameter, which controls the amplitude width of the kernel function. In the present invention, the width parameter of the kernel function is usually set to γ=0.001, which makes the model more sensitive to outliers.

[0073] 3. Model training and parameter selection:

[0074] The training data consists of normal insulator chain time series. By training the OneClass SVM model, the distribution boundary of normal data points can be learned. The goal of the model is to enclose most normal points within the boundary, and any point outside the boundary is considered an outlier. The hyperparameter ν is a parameter that controls the proportion of outliers, which is usually set to a small value (such as ν=0.01), indicating that only a very small number of points are allowed to be outliers. The model obtains a support vector hyperplane through training to distinguish normal points from outliers.

[0075] 4. Model application:

[0076] After training, the new time series is input into the OneClass SVM model for detection. By calculating the distance of each point relative to the hyperplane, if the distance exceeds a certain threshold, the point is judged as an outlier. This detection process can identify self-exploding insulators or other damaged locations in the insulator chain.

[0077] Detection of continuous abnormal intervals:

[0078] After OneClass SVM detects the abnormal points, the next step is to analyze the distribution characteristics of these abnormal points. If the abnormal points are concentrated in certain intervals, it may reflect the actual damage of the insulator chain. Therefore, the system performs cluster analysis on all detected abnormal points and combines adjacent abnormal points into continuous abnormal intervals.

[0079] For each outlier point, the system will determine whether it belongs to the same continuous outlier interval based on its coordinates and the positions of adjacent points. By analyzing the distance and distribution density between outliers, the area where outliers are concentrated can be identified (such as Figure 5 as shown).

[0080] 3. Interval statistics and screening:

[0081] All detected abnormal intervals are counted to filter out intervals containing multiple abnormal points. The statistical formula is: > , in It is a set minimum threshold, which indicates the minimum number of abnormal points contained in each continuous interval. Intervals with a number of abnormal points greater than the threshold are selected as potential abnormal defect locations.

[0082] 1. Three-dimensional spatial restoration of outliers:

[0083] After identifying the abnormal points, the next step is to restore these abnormal points from the two-dimensional projection to the three-dimensional point cloud data. Through the projection equation in step D: z=ax+by+c, the abnormal points in the two-dimensional plane can be mapped back to the three-dimensional spatial coordinates of the insulator chain, and the specific location of the self-explosive insulator in the insulator chain can be accurately determined.

[0084] 2. 3D point cloud model marking:

[0085] After determining the 3D coordinates of each abnormal point, the system marks these abnormal points in the 3D point cloud model of the transmission tower. Through visualization tools, operation and maintenance personnel can clearly see the specific location of each abnormal point on the transmission tower insulator chain.

[0086] 3. Generate exception report:

[0087] The system automatically generates a detailed inspection report, which includes: the specific location and three-dimensional coordinates of the abnormal interval; the number and type of abnormal points (self-explosion, wear, other abnormalities); the total length of each abnormal interval and the distribution of faults. This report helps maintenance personnel to understand the health status of the insulator chain in a timely manner and plan subsequent maintenance work.

[0088] Through these steps, the system can efficiently detect anomalies in the insulator chain and generate a detailed inspection report, providing a basis for subsequent maintenance and repair.

[0089] In addition, the drone inspection system of the present invention has autonomous flight capability and can automatically adjust the flight trajectory according to the pre-set inspection path to ensure full coverage and real-time detection of the insulator chain. The drone is equipped with a real-time data transmission module, which can transmit point cloud data and detection results to a remote data center in real time for inspection personnel to analyze and process. This part is common knowledge and will not be described in detail.

[0090] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, it should be emphasized that the technical solutions of the present invention are not limited to the above-described embodiments. Professionals and technicians in this field should understand that they can modify or replace the technical solutions of the present invention while still complying with the purpose and scope of the technical solutions. These modifications or equivalent replacements should be covered within the scope of the claims of the present invention.

Claims

1. A method for detecting abnormalities in UAV power line inspection based on point cloud data, characterized in that: The method comprises the following steps: A. Data collection to obtain the three-dimensional point cloud spatial position information and color information of the transmission tower B. Data Preprocessing C. Insulator chain target segmentation D. Plane fitting and projection of insulator chain D1. Use the least squares method to fit the point cloud data of the insulator chain and generate the axial plane equation of the insulator chain z=ax+by+c; D2, projecting the fitted 3D point cloud data onto a 2D image plane to generate a 2D image of the point cloud projected on the plane; D3. After generating the two-dimensional projection image, the amount of point cloud data in each interval is counted, and a distribution histogram data sequence is created using the generated two-dimensional projection data; E. Timing anomaly detection E1: Generate time series The point cloud data is segmented along the axial direction of the insulator chain to generate equally spaced intervals. For each interval i, the maximum value of the point cloud data in the interval is counted. and minimum value , and calculate the difference , through this difference, generate the histogram data sequence of each interval, and convert the The difference data are arranged in order to form a complete time series; E2: Construct a time series anomaly detection model combining the geometric features of the insulator chain and establish a OneClass SVM for sequence detection Based on the generated time series as training data, a time series anomaly detection model is constructed, using OneClass SVM as the core algorithm for anomaly detection and radial basis function RBF as the kernel function, which is in the form of: Among them, γ=0.001, γ is the kernel parameter, which controls the amplitude width of the kernel function; after the training is completed, the new time series is input into the OneClass SVM model for detection. By calculating the distance of each point relative to the hyperplane, if the distance exceeds a certain threshold, the point is judged as an abnormal point; E3: Determine continuous abnormal intervals If multiple outliers appear in adjacent intervals, these outliers are combined into continuous outlier intervals, and the intervals containing multiple outliers are screened out. The statistical formula is: > ,in It is a set minimum threshold, which indicates the minimum number of abnormal points contained in each continuous interval. Intervals with the number of abnormal points greater than the threshold are selected as potential abnormal defect locations. E4: 3D restoration and marking of outliers Through the projection equation in step D: z=ax+by+c, the abnormal points in the two-dimensional plane are mapped back to the three-dimensional spatial coordinates of the insulator chain to determine the specific position of the self-explosive insulator in the insulator chain; after determining the three-dimensional coordinates of each abnormal point, these abnormal points are marked in the three-dimensional point cloud model of the transmission tower, and an abnormal report is generated.

2. The method for detecting abnormalities in UAV power line inspection based on point cloud data according to claim 1, characterized in that: In step A, a drone equipped with an Intel D435i depth camera and a T265 inertial navigation camera is used to perform a full-scale scan of the transmission tower.

3. The method for detecting abnormalities in UAV power line inspection based on point cloud data according to claim 1 is characterized in that: Step B includes: B1. Filter the point cloud data using Gaussian filtering technology to remove noise and error points in the point cloud data while retaining important edge information; B2. Downsampling the point cloud data, using voxel grid downsampling method to reduce the computational load by reducing the density of data points while maintaining the structural characteristics of the point cloud data; B3. Perform registration processing on the point cloud data collected multiple times to eliminate the errors between the scanning data at different positions.

4. The method for detecting abnormalities in UAV power line inspection based on point cloud data according to claim 1 is characterized in that: Step C includes: C1. Identify and segment the insulator chain according to its color features in the point cloud; C2. Further segmentation is performed based on the geometric features of the insulator chain so that the point cloud data only contains the point cloud of the insulator chain.

5. The method for detecting abnormalities in UAV power line inspection based on point cloud data according to claim 1 is characterized in that: The least squares fitting step in step D1: First, by extracting the coordinate point set of the insulator chain from the point cloud data, a linear equation system is constructed: Ax=b, where A is the coefficient matrix, x is the coefficient vector to be solved, and b is the constant vector. Based on the following linear relationship, a specific equation system is constructed: Next, the above equations are solved to obtain the coefficients a, b and c of the plane equation: Finally, the axial plane equation of the insulator chain z = ax + by + c is generated.

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