A method for fitting an error ellipse of any tightness to a planar coordinate point cloud

Abstract

The present application relates to geodetic and engineering surveying technical field, specifically to a kind of plane coordinate point cloud fitting arbitrary close degree error ellipse in control network design stage.The method includes: the pre-processing of control point coordinate point cloud, removes abnormal value;Accurate spatial precision evaluation density field is constructed to calculate grid point density value;According to the specified confidence, the density threshold value is calculated, the contour is extracted and the error ellipse parameter is fitted.The present application solves the ellipse fitting problem of any specified confidence of non-normal distribution point cloud by adaptive bandwidth kernel density estimation, and significantly improves the reliability of precision evaluation under complex scene.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of geodetic surveying and engineering surveying, and in particular to a method for fitting an error ellipse of any tightness from a planar coordinate point cloud in the design stage of a control network. BACKGROUND

[0002] In geodetic surveying and engineering surveying, the point position error ellipse is an important way to describe the accuracy of the control points. In the design stage of a surveying control network, the accuracy of the designed control network is often evaluated by precision estimation methods such as simulation method to determine whether the design scheme can meet the predetermined accuracy requirements.

[0003] In the control network accuracy estimation method, the traditional simulation method only performs one or a small number of simulations, and the estimation results have certain randomness. Although the robust precision estimation method performs multiple batches of a large number of simulations, it can only calculate the mean error or point position error of the control points in the coordinate axis direction, and cannot obtain the point position error ellipse of any tightness.

[0004] However, in many cases, the accuracy of the control point position in some special directions needs to be studied, and the point position error ellipse can clearly show the accuracy of the point position in different directions and different confidence levels.

[0005] Although the Chinese patent CN120232410A can obtain the major axis and minor axis of the ellipse using the point cloud and the probability density function, it needs to assume that the data follows a normal distribution by using Kalman filtering technology, and can only output the position estimation of a fixed confidence interval, and needs to rely on dynamic sensor fusion. In the static control network accuracy evaluation, the point cloud data may not follow a normal distribution, and the error ellipse of any confidence level cannot be generated. SUMMARY

[0006] Based on a large number of planar control point position coordinates obtained by the robust precision estimation method, the present application proposes a method for fitting an error ellipse of any tightness from a planar coordinate point cloud, solves the problem that the existing method cannot generate a robust error ellipse of any confidence level in a non-normal distribution point cloud, and is particularly suitable for static control network accuracy evaluation of major projects such as bridges, tunnels, and dams.

[0007] The technical solution of the present application: a method for fitting an error ellipse of any tightness from a planar coordinate point cloud, specifically comprising the following steps:

[0008] Step 1: Preprocessing of the control point coordinate point cloud to remove outliers;

[0009] A large number of coordinate point clouds may contain a small number of outliers. The median and median absolute deviation of the coordinates are calculated, and then the standardized residual is calculated. The weight of the coordinates is calculated using a weight function, and the outliers in the point cloud are removed by iterative reweighting.

[0010] Step 2, constructing a precise spatial precision evaluation density field to calculate the grid point density value;

[0011] Calculating the local density of each coordinate point, calculating the adaptive bandwidth matrix according to the median absolute deviation and the local density, and constructing the kernel density function;

[0012] Step 3, calculating the density threshold according to the specified confidence, extracting the contour and fitting the error ellipse parameters;

[0013] Different confidence corresponds to different size of error ellipse, specified confidence, extract the contour, based on the point set on the longest closed contour, calculate the long semi-axis, short semi-axis and long semi-axis azimuth of the corresponding error ellipse by solving the eigenvalue and eigenvector of the covariance matrix, get the error ellipse of any tightness.

[0014] The number of coordinates of the plane coordinate point cloud in step 1 is greater than 10000.

[0015] The outlier rejection in step 1 is based on the weight function iteration, and the condition for judging whether it is an outlier is the standardized residual of the coordinate , inequality holds, that is, the coordinate is an outlier, inequality does not hold, that is, the coordinate is not an outlier.

[0016] In step 2, the adaptive bandwidth matrix is introduced, and the local density is dynamically adjusted.

[0017] The kernel density function in step 2 is The form is:

[0018] Wherein, , det() is the calculation of determinant, n is the number of coordinate point cloud, i is the coordinate serial number, is the inverse matrix of the adaptive bandwidth matrix corresponding to coordinate i.

[0019] The calculation of error ellipse parameters in step 3 is based on the point set on the longest closed contour corresponding to the confidence of α , wherein t is the number of coordinates contained in the closed contour, and i is the coordinate serial number.

[0020] The step 3 can generate multiple confidence error ellipses, and the ellipse parameters are calculated by eigenvalue decomposition of covariance matrix and chi-square distribution , the error ellipse parameters obtained by calculation are as follows:

[0021] Long semi-axis Short semi-axis Long semi-axis azimuth Wherein, And are two eigenvalues of the covariance matrix , and , the corresponding eigenvectors are , the covariance matrix is calculated as follows:

[0022] wherein, is the mean value of the point set on the longest closed contour, and is calculated as follows:

[0023] wherein, t is the number of coordinates contained by the closed contour, and respectively represent the x and y coordinates of the i-th point on the longest closed contour.

[0024] The application solves the ellipse fitting problem of any specified confidence of non-normal distribution point cloud by adaptive bandwidth kernel density estimation, and significantly improves the reliability of precision evaluation in complex scenes. BRIEF DESCRIPTION OF DRAWINGS

[0025] Figure 1 is a flowchart of the application;

[0026] Figure 2 is a normal distribution point cloud and a 95%, 68% confidence error ellipse obtained by fitting of an embodiment of the application.

[0027] Figure 3 is a non-normal distribution point cloud, a 95% density contour, and a 95% confidence error ellipse obtained by fitting of different methods of an embodiment of the application. DETAILED DESCRIPTION

[0028] The technical solutions of the application are described below in conjunction with the drawings and specific embodiments. Obviously, the described embodiments are part of the embodiments of the application, rather than all the embodiments.

[0029] EMBODIMENT

[0030] Figure 1 An embodiment of the application provides a method for fitting an arbitrary density error ellipse of a planar coordinate point cloud, which specifically comprises the following steps:

[0031] Step 1: Preprocess the coordinate point cloud of the control point P to eliminate outliers. First, calculate the standardized residual of each coordinate point, calculate the coordinate weight by using a weighted function, and gradually eliminate outliers by using the iterative reweighting method to ensure the accuracy of the point cloud data.

[0032] (1) The coordinates of the control point P are represented as (x, y), and the coordinate point cloud of P contains m results, m is greater than 10000;

[0033] (2) The median of the coordinates of the control point P is calculated, denoted as M x and M y , and the calculation method is as follows:

[0034] M x =median(x)

[0035] M y =median(y);

[0036] (3) The absolute deviation of the median of the coordinates of the control point P is calculated, and the calculation method is as follows:

[0037] ;

[0038] (4) The standardized residual of each coordinate is calculated, and the calculation method is as follows:

[0039] ;

[0040] (5) The weight of each coordinate is calculated, and the calculation method is as follows:

[0041] ;

[0042] (6) According to the weight calculated, the weighted mean is calculated according to the following formula, and the weighted mean is also denoted as M x and M y :

[0043] ;

[0044] (7) Using the above obtained M x and M y , the absolute deviation of the median of the coordinates is calculated again, and the calculation of (3)~(6) above is iterated for 3 times;

[0045] (8) Remove the outliers of the coordinate point cloud, the outliers are corresponding coordinate data, after removing the abnormal data, the coordinate point cloud of P contains n results.

[0046] Step 2, calculate the local density value according to the point cloud data, use the median absolute deviation method to calculate the adaptive bandwidth matrix, and construct the kernel density function. This method can automatically adjust the bandwidth according to the change of local data, so as to improve the accuracy of density estimation.

[0047] (1) According to the k-neighbor distance d k (i) Calculate the number of points in unit area , the recommended k value is 50, The calculation method is as follows:

[0048] ;

[0049] (2) Calculate the adaptive bandwidth matrix H i :

[0050] ;

[0051] Wherein: is the mean value of , I is the unit matrix;

[0052] The existing method (Chinese patent CN120232410A) directly locates the target by using Gaussian kernel function, while the present application introduces adaptive bandwidth matrix, adjusts the bandwidth dynamically through local density , and solves the problems of oversmoothing or underfitting under complex point cloud distribution.

[0053] (3) Construct the kernel density function f(x,y):

[0054] ;

[0055] Wherein: , det() is the calculation of determinant, n is the number of coordinate points, i is the coordinate number, is the inverse matrix of the adaptive bandwidth matrix corresponding to coordinate i.

[0056] Step 3, according to the specified confidence value, extract the corresponding contour line, calculate the long axis, short axis and azimuth angle of error ellipse by using the eigenvalue and eigenvector of covariance matrix, so as to obtain the accurate error ellipse parameters. This method can provide error analysis under different confidence degrees, and meet the complex control measurement demand.

[0057] (1) Set the grid resolution N, N is a positive integer, calculate the grid density value f(x g ,y g ) according to the density function, and calculate the grid area A:

[0058] ;

[0059] (2) Calculate the density threshold value corresponding to the specified confidence degree a , Should meet: Wherein: ;

[0060] (3) Use the rectangular grid contour generation algorithm to extract the contour of , a total of s contour lines Select the longest closed contour ;

[0061] (4) The point set on the longest closed contour C is also denoted as C, and the point set contains t coordinate data. ;

[0062] (4) Calculate the error ellipse parameters, fit the ellipse, and calculate the mean. Covariance matrix Eigenvalues ​​are obtained by performing eigendecomposition on the covariance matrix Q. and ,and , The corresponding feature vector is Then the ellipse parameters are:

[0063] Long half shaft ;

[0064] short half shaft ;

[0065] Major semi-axis azimuth ;

[0066] in: It is the square of the chi-square distribution value when the degrees of freedom are 2 and the confidence level is α.

[0067] Existing methods (Chinese Patent CN120232410A) employ Kalman filtering, which requires the assumption that the data follows a normal distribution. In contrast, this invention extracts contour lines C and combines them with a chi-square distribution. It can directly calculate the parameters of an ellipse with arbitrary confidence levels (major axis, minor axis, and azimuth angle of the major axis), breaking through the limitations of the normal distribution.

[0068] This invention can obtain point cloud ellipses with arbitrary confidence levels. Figure 2 The diagram illustrates the 95% and 68% confidence error ellipses obtained by the method of this invention when the point cloud follows a normal distribution.

[0069] This invention does not require the assumption that the data follows a normal distribution. Figure 3 This illustrates the 95% confidence error ellipse obtained by the method of this invention and the conventional method when the data points do not follow a normal distribution. Figure 3 It can be seen that, compared with the 95% density contour lines, the 95% confidence error ellipse obtained by the traditional method is significantly smaller and does not conform to reality, and there are still a large number of point clouds outside the ellipse.

[0070] This invention introduces adaptive kernel density estimation into the design of measurement control networks for the first time, solving the technical problem of generating ellipses with arbitrary confidence error, and providing a quantifiable design basis for the simulation coordinate point cloud in the design stage of control networks for major projects.

Claims

1. A method for fitting an ellipse with arbitrary fit error to a planar coordinate point cloud, characterized in that: Specifically, the following steps are included: Step 1: Preprocess the control point coordinate point cloud to remove outliers: Calculate the median and absolute deviation of the coordinates, then calculate the standardized residuals, use the weighting function to calculate the coordinate weights, iterate and reweight, and remove outliers from the point cloud. Step 2: Construct an accurate spatial density field to calculate the density values ​​at the grid points. Calculate the local density at each coordinate point, and then calculate the adaptive bandwidth matrix based on the median absolute deviation and the local density to construct the kernel density function. Step 3: Calculate the density threshold based on the specified confidence level, extract the contour lines, and fit the error ellipse parameters: By specifying the confidence level, extract the contour lines, and based on the point set on the longest closed contour line, calculate the major semi-axis, minor semi-axis, and azimuth angle of the corresponding error ellipse by solving the eigenvalues ​​and eigenvectors of the covariance matrix, and obtain the error ellipse with arbitrary fit. Step 3 can generate a multi-confidence error ellipse, the ellipse parameters of which are obtained through covariance matrix eigenvalue decomposition and chi-square distribution. The calculated error ellipse parameters are as follows: Long half shaft short half shaft Major semi-axis azimuth in, and It is the covariance matrix Two eigenvalues, and , The corresponding feature vector is covariance matrix The calculation method is as follows: Where t is the number of coordinates contained in the closed contour. It is the mean of the set of points on the longest closed contour, calculated as follows: Where t is the number of coordinates contained in the closed contour. and Let x and y represent the x and y coordinates of the i-th point on the longest closed contour, respectively.

2. The method for fitting an arbitrary fit error ellipse to a planar coordinate point cloud according to claim 1, characterized in that: In step 1, the number of coordinates in the planar coordinate point cloud is greater than 10,000.

3. The method for fitting an arbitrary fit error ellipse to a planar coordinate point cloud according to claim 1, characterized in that: In step 1, outlier removal is based on weight function iteration, and the condition for determining whether a value is an outlier is the standardized residual of the coordinates. If the inequality holds true, the coordinate is an outlier; if the inequality does not hold true, the coordinate is not an outlier.

4. The method for fitting an arbitrary fit error ellipse to a planar coordinate point cloud according to claim 1, characterized in that: In step 2, an adaptive bandwidth matrix is ​​introduced, which is achieved through local density. Dynamically adjust bandwidth.

5. The method for fitting an arbitrary fit error ellipse to a planar coordinate point cloud according to claim 1, characterized in that: In step 2, the kernel density function The form is: in, `det()` calculates the determinant, where `n` is the number of points in the coordinate cloud and `i` is the coordinate index. It is the inverse of the adaptive bandwidth matrix corresponding to coordinate i.

6. The method for fitting an arbitrary fit error ellipse to a planar coordinate point cloud according to claim 1, characterized in that: In step 3, the error ellipse parameters are calculated based on the longest closed contour corresponding to a confidence level of α. The process is performed, where t is the number of coordinates contained in the closed contour, and i is the coordinate index.

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