CN111127544B - Parameterized geometric ellipse fitting method based on short axis regression - Google Patents

Parameterized geometric ellipse fitting method based on short axis regression Download PDF

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CN111127544B
CN111127544B CN201911408099.8A CN201911408099A CN111127544B CN 111127544 B CN111127544 B CN 111127544B CN 201911408099 A CN201911408099 A CN 201911408099A CN 111127544 B CN111127544 B CN 111127544B
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张越
徐洪贺
彭锐涛
王涛
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Xiangtan University
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Abstract

The invention discloses a parameterized geometric ellipse fitting method based on short axis regression, which comprises four main steps of establishing short axis regression parameters of a target, establishing an initial value mathematical model, solving geometric distance errors and self-adaptive iterative optimization of a short axis; in theory, any radial section of the cylindrical target is an ellipse, and the ellipse outline is shielded by the ellipse outline, so that only a small amount of ellipse arc segment information can be obtained. Therefore, it is difficult to accurately fit the actual elliptical model with only the information of the small arc segment, and the existing fitting efficiency is low. The invention takes circle fitting as an initial value of geometric iteration of the ellipse, and adopts the known target radius of the cylinder as a short axis regression parameter, thereby greatly improving the efficiency of geometric iteration and realizing high-precision fitting even under the actual noise-containing condition.

Description

Parameterized geometric ellipse fitting method based on short axis regression
Technical Field
The invention discloses a short axis regression-based parameterized geometric ellipse fitting method, and belongs to the technical field of precision testing and contour analysis.
Background
An ellipse is the most basic geometry. Theoretically, a circle only belongs to a special form of an ellipse, and all circular or cylindrical physical materials in real life can be represented by an ellipse model. In short, the high-precision ellipse fitting can be applied to product quality monitoring, instrument calibration, accurate positioning analysis and the like. Taking the stamping connection technology in the field of material connection as an example, the two plates can form a cylindrical groove under the action of a punch and a die, and the materials at the connected part are deformed to form an inner lock and an inlay, so that a connection point capable of providing certain strength is formed. In this process, any cross section of the cylindrical groove can be represented as an ellipse due to residual stress and inclined placement of the sheet material. Therefore, to accurately describe the center of the imprint, an ellipse fitting is required to the extracted cross-sectional profile.
The current analysis methods for ellipse fitting mainly include: a least square method, which minimizes the sum of squares of errors of the prediction model and the measurement points; the Hough transform method is used for searching a local peak value in an accumulator space through feature voting to determine parameters of an ellipse; and the matrix method is used for decomposing the generalized characteristic system of a discrete matrix and obtaining the ellipse parameters. The least squares method is generally classified into an algebraic method and a geometric method according to the definition of the error distance and the constraint condition. The geometric method is the method with the highest fitting accuracy in the methods, but the research on geometric fitting of the ellipse is slow in the last two decades.
The ellipse geometric fitting method mainly has the following aspects to be promoted: 1) and (4) fitting efficiency. With the development of batch, scale and intellectualization in recent years, in actual detection, application and even basic research, the requirement on the positioning accuracy of the ellipse is high, the fitting efficiency is emphasized more and more, and the fitting times or the fitting time is generally used as a quantization index; 2) and noise resistance. The discrete ellipse data in the actual measurement contains noise, and only a small part of ellipse arc data is obtained. Noise and small arc section factors can greatly influence the fitting precision of the ellipse and even cause the inaccuracy or complete failure of ellipse fitting; 3) and (4) self-adapting. In the existing gauss-newton fitting method, due to poor adaptability, the singular value processing is very rough, and the situations of non-convergence and meaningless fitting exist, so that the application and popularization of ellipse fitting in various fields are not facilitated.
Disclosure of Invention
Aiming at the difficulties in the existing ellipse fitting method, the invention provides a parameterized geometric ellipse fitting method based on short axis regression. The method takes the known target radius of a cylinder as a regression parameter, takes geometric circle fitting as an initial value of an ellipse, and adopts self-adaptive fitting calculation. The new basic research method can be applied to the fields of precision testing technology and contour analysis technology, and can also be applied to a plurality of fields of image processing technology and the like.
The technical scheme adopted by the invention is a parameterized geometric ellipse fitting method based on short axis regression, which comprises the following steps:
t1: short axis regression parameters
In the basic study of ellipses, using parameter vectors
Figure BDA0002349221990000025
The actual elliptical model is accurately described. The physical meanings corresponding to the 5 parameters in the parameter vector are as follows: xCAnd YCRespectively representing coordinate values of the center of the ellipse on the X axis and the Y axis in the XY coordinate system, a and b respectively representing the sizes of the major axis and the minor axis of the ellipse,
Figure BDA0002349221990000026
the size of the included angle between the major axis of the ellipse model and the X axis in the OXY coordinate system is also called the rotation angle of the ellipse, and represents the direction of the ellipse.
In the fitting solution, the 5-term parameters of the elliptical model form numerical adjustment in the corresponding dimension. So in practice the ellipse fitting is to solve for the optimal values in 5 dimensions from the known discrete points Xj. Then, according to the principle of ellipse formation, the minor axis b of the 5 dimensions is the radius of the forming cylinder, and the nominal value of the radius of the forming cylinder is known, as: the stamped punch radius or the calibrated cylinder radius and other objects have accurate radius indexes. Regression parameter equation:
f(k)＝b-constant＝0 (1)
therefore, the short axis b can be accurately obtained through actual measurement and set as a regression parameter, and the calculation amount in the dimension can be reduced to improve the fitting efficiency.
T2: initial elliptic mathematical model
In the ellipse geometric fitting, an initial ellipse mathematical model needs to be established first, and fitting optimization is carried out on the basis. Since the circle fitting has only three dimensions
Figure BDA0002349221990000021
The calculated amount of the method has stronger robustness. Therefore, the discrete point X is equivalently calculated by adopting Gaussian circle fittingjThe initial elliptic mathematical model of (2) is performed according to the following procedure.
1) Calculating an initial circle
Center of gravity of discrete points arbitrarily in the OXY coordinate system
Figure BDA0002349221990000022
And roundInitial radius r0Expressed as:
Figure BDA0002349221990000023
Figure BDA0002349221990000024
wherein the center of gravity is adjusted
Figure BDA0002349221990000031
Set as the center of the initial circle. m represents the number of discrete points, X1Representing the first discrete point, XmFor the last discrete point, XjRefers to the jth discrete point.
2) Orthogonal distance vector epsilon of circlec
At any discrete point XjTo the initial circular center of gravity XAThe intersection point of the connecting lines on the circle is called the orthogonal point Di. The physical meaning of the orthogonal points is understood to mean the circle and the discrete points XjThe nearest corresponding point.
Figure BDA0002349221990000032
Thus, an arbitrary discrete point XjTo an orthogonal point DiIs measured in discrete points XjTo the initial circular center of gravity XAFor vector direction, define the orthogonal distance vector ε of the circlec。
Figure BDA0002349221990000033
3) Gaussian circle fitting
Orthogonal pointDiFor the points on the initial circle, a linear equation system related to the orthogonal distance vector can be established according to the Jacobian matrix of the orthogonal points.
Figure BDA0002349221990000034
The left matrix being each quadrature pointDiFor three dimensional parameters of initial circle
Figure BDA0002349221990000035
The partial derivatives of (1). The gaussian fit direction of the linear equation is:
Figure BDA0002349221990000036
the step length of fitting is 1 in the formula, and the orthogonal distance vector mode | | | epsilon is continuously reduced every fitting adjustmentcL. Because only three dimensions are provided, the Gaussian circle fitting can be optimized for a plurality of times, and finally the circle center of the optimal circle model is obtained
Figure BDA0002349221990000037
The radius r.
4) Calculating an initial rotation angle
Figure BDA0002349221990000038
According to the optimized circle model center
Figure BDA0002349221990000041
Center of gravity with discrete points
Figure BDA0002349221990000042
A connecting line, the included angle between the connecting line and the X axis in the XY coordinate system is defined as the initial rotation angle
Figure BDA0002349221990000043
The solution is performed by the arctan function:
Figure BDA0002349221990000044
wherein the content of the first and second substances,
Figure BDA0002349221990000045
and
Figure BDA0002349221990000046
representing the center of gravity XAThe values on the X-axis and Y-axis in the OXY coordinate system,
Figure BDA0002349221990000047
and
Figure BDA0002349221990000048
respectively represent optimized circles XBValues on the X-axis and Y-axis in the OXY coordinate system.
5) Initial elliptic mathematical model
The optimal circle center solved in the previous three steps
Figure BDA0002349221990000049
Set as the initial ellipse center
Figure BDA00023492219900000410
The radius r is set to the major and minor axes (a) of the ellipse0＝b0R), plus the initial circular rotation angle
Figure BDA00023492219900000411
That is, five initial parameters based on discrete points are obtained
Figure BDA00023492219900000412
Establishing a parameterized initial elliptical mathematical model:
Figure BDA00023492219900000413
wherein beta isjRepresenting the jth discrete point in the initial ellipse X(k0)And determining the corresponding centrifugal angle on the model, wherein the angle needs the elliptic point with the minimum orthogonal distance.
T3: geometric distance error vector solving
For the same reason of fitting orthogonal distance to circleAn off-vector quantity. On the ellipse model, any discrete point XjAll have a unique orthogonal point X corresponding theretoP. For distinguishing the two, the discrete point X is used for easy understandingjTo the orthogonal point X of the elliptic mathematical modelPIs defined as the geometric distance epsilonp. By elliptical centrifugal angle betajFrom a geometric distance epsilonpThe mapping relation between discrete points is represented as follows:
εp＝Xj-XP|β (10)
whether the elliptical model is optimized in fitting can be determined by the geometric property delta0(β) quantitative characterization:
δ0(β)＝||Xj-XP|β||＝||(fx,fy)|| (11)
wherein the geometric distance error (f)x,fy) Representing from discrete points XjTo an orthogonal point XPThe vector of (2). Due to the discrete point XjFive parameters of the initial ellipse are known and have been estimated. Then, the only unknown angle of eccentricity βjAn objective function F can be established:
Figure BDA0002349221990000051
wherein the angle of rotation
Figure BDA0002349221990000052
The centrifugal angle beta corresponding to the minimum time of the target function F can be quickly solved by the widely used least square methodj. Then substituting into the formulas (9), (10) and (11), the orthogonal point X on the ellipse model can be calculatedPOrthogonal distance epsilonpGeometric performance function delta corresponding to elliptical model0The value of (. beta.).
T4: regression constrained LM geometry fitting
1) Elliptic Jacobian matrix and its equation system
In the ellipse fitting, arbitrarily modifying a basis vector must be pairedThe vector of which is adjusted exactly, which would otherwise lead to divergence of the parameter space. The regression parameter equation f (k) is weighted to 5 parameters of the ellipse
Figure BDA0002349221990000053
Figure BDA0002349221990000054
And the partial derivatives and the elliptic Jacobian matrix jointly form a left matrix. For each discrete point XjThe following system of linear equations is established:
Figure BDA0002349221990000055
wherein w is 1.0 × 1012And mu represents a damping coefficient used for adaptively adjusting the change of the elliptical model in five dimensions. I is a 5 × 5 identity matrix.
Figure BDA0002349221990000056
For a parameterized elliptic Jacobian matrix JijInverse matrix of, JijCan be obtained by partial derivation of equation (9):
Figure BDA0002349221990000057
2) initial value and direction of LM fitting
In equation (13), the left matrix added to the identity matrix is positive, so it has a unique solution, thus ensuring that it is invertible. The initial damping coefficient range is mu epsilon [1, N*]And carrying out global search of five parameters of the ellipse by taking the value mu as 50. According to the elliptical geometric property function delta0(beta), LM self-adaptive regulation strategy is carried out on the damping coefficient mu,
Figure BDA0002349221990000058
wherein, k is ∈ [0.25,0.75 ]]，μnextRepresenting the next fitDamping coefficient, | εp+1And | | l represents the magnitude of the geometric property corresponding to the next elliptical model in the fitting. The direction of LM fitting is:
kv+1＝kv+Δk (16)
in the formula, the parameter vector kvThe middle subscript v is belonged to [0, N ∈]Is a natural number, representing a vector of five parameters, k, of the elliptical modelv+1The physical meaning of the representation is the next elliptical model parameter vector to be solved by fitting. And deltak is a vector which needs to be accurately adjusted in five dimensions of the elliptical model.
3) Termination condition of LM fitting
The termination criterion reflects the corresponding geometric performance function delta of the elliptical model0(β) is a global minimum. Therefore, taking the LM fitting step size, the steepest descent rate (modulo of the gradient) and the fitting ceiling as termination criteria,
Figure BDA0002349221990000061
Figure BDA0002349221990000062
v≥vmax (19)
wherein e is1Subdividing the upper limit, e, for the step size2Is the sum of the upper limit of the gradient mode and vmaxThe upper limit of the number of fits. That is, it is determined that the optimization of the elliptical model is achieved as long as any one of the above three upper limit conditions is achieved in the adaptive fitting process.
The parameterized geometric ellipse accurate fitting method has the following advantages:
1. solving of the initial ellipse value is carried out on the basis of Gaussian circle fitting, the calculation amount of preprocessing is reduced, the calculation efficiency is high, and the robustness is strong;
2. the short axis is used as a regression parameter, and weighting is carried out, so that the problem of dimension space divergence caused by modifying ellipse fitting parameters can be effectively avoided, and the calculated amount can be saved;
3. the regression-constrained LM geometric fitting has strong self-adaptive capacity, and the condition of non-convergence in fitting can be effectively avoided;
4. the method has strong noise immunity, can be suitable for extracting the elliptic features with noise in practical application, and has potential application prospects in the fields of intelligent manufacturing, quality monitoring, three-dimensional reconstruction, robot and instrument calibration and the like.
Drawings
FIG. 1 orthogonal distance vectors fitted to a Gaussian circle;
FIG. 2 calculation of initial rotation angle
Figure BDA0002349221990000071
FIG. 3 ellipse angle definition and calculation of orthogonal distances;
FIG. 4 LM geometry fitting parameters for short axis regression;
FIG. 5 LM geometry fit three-dimensional plot of short axis regression;
FIG. 6 LM geometric ellipse fitting of short axis regression;
figure 7 is a flow of parameterized geometric ellipse fitting for short axis regression.
Detailed Description
As an example, a short axis regression parameterized geometric ellipse fitting method is further illustrated with the use of discrete point data with a short axis of 3.03184 (table 1) in conjunction with the figures.
Table 18 discrete points data for the examples
Figure BDA0002349221990000072
In the basic study of ellipses, using parameter vectors
Figure BDA0002349221990000074
The actual elliptical model is accurately described. The physical meanings corresponding to the 5 parameters in the parameter vector are as follows: xCAnd YCCoordinate values of the center of the ellipse on the X-axis and the Y-axis in the XY coordinate system, respectively, and a and b represent the major axis and the minor axis of the ellipse, respectivelyThe size of the capsule is determined by the size of the capsule,
Figure BDA0002349221990000075
the size of the included angle between the major axis of the ellipse model and the X axis in the OXY coordinate system is also called the rotation angle of the ellipse, and represents the direction of the ellipse.
In the fitting solution, the 5-term parameters of the elliptical model form numerical adjustment in the corresponding dimension. So in practice the ellipse fit is based on the known discrete points XjThe optimal values in 5 dimensions are solved. Then, according to the principle of ellipse formation, the minor axis b of the 5 dimensions is the radius of the forming cylinder, and the nominal value of the radius of the forming cylinder is known, as: the stamped punch radius or the calibrated cylinder radius and other objects have accurate radius indexes. Regression parameter equation:
f(k)＝b-3.03184＝0 (20)
therefore, by setting the short axis 3.03184, which is known in actual measurement, as a regression parameter, the calculation amount in the dimension can be reduced to increase the fitting efficiency.
Initial elliptic mathematical model
In the ellipse geometric fitting, an initial ellipse mathematical model needs to be established first, and fitting optimization is carried out on the basis. Since the circle fitting has only three dimensions
Figure BDA0002349221990000073
The calculated amount of the method has stronger robustness. Therefore, the discrete point X is equivalently calculated by adopting Gaussian circle fittingjThe initial elliptic mathematical model of (2) is performed according to the following procedure.
Calculating an initial circle
Center of gravity of discrete points arbitrarily in the OXY coordinate system
Figure BDA0002349221990000081
And the initial radius r of the circle0Expressed as:
Figure BDA0002349221990000082
Figure BDA0002349221990000083
wherein, the center of gravity XA(5.1250, 5.7500) is set as the center of the initial circle. 8 denotes the number of discrete points, X1Representing the first discrete point, X8As the last discrete point, XjRefers to the jth discrete point.
Orthogonal distance vector epsilon of circlec
As shown in fig. 1, at arbitrary discrete points XjTo the initial circular center of gravity XAThe intersection point of the connecting lines on the circle is called the orthogonal point Di. The physical meaning of the orthogonal points is understood to mean the circle and the discrete points XjThe nearest corresponding point.
Figure BDA0002349221990000084
Thus, an arbitrary discrete point XjTo an orthogonal point DiIs measured in discrete points XjTo the initial circular center of gravity XAFor vector direction, define the orthogonal distance vector ε of the circlec. The calculated orthogonal distance vectors of the initial circle are shown in table 2.
Figure BDA0002349221990000085
TABLE 2 orthogonal distance vector ε of initial circlec
Figure BDA0002349221990000086
Gaussian circle fitting
Orthogonal point DiFor the points on the initial circle, a linear equation system related to the orthogonal distance vector can be established according to the Jacobian matrix of the orthogonal points.
Figure BDA0002349221990000091
The left matrix is three dimensional parameters of each orthogonal point Di to the initial circle
Figure BDA0002349221990000092
The partial derivatives of (1). The gaussian fit direction of the linear equation is:
Figure BDA0002349221990000093
the step length of fitting is 1 in the formula, and the orthogonal distance vector mode | | | epsilon is continuously reduced every fitting adjustmentcL. The circle center X of the optimal circle model is obtained through 12 sub-optimization of Gaussian circle fittingB(3.3911,4.8399) and the radius r is 4.7967.
Calculating an initial rotation angle
Figure BDA0002349221990000094
As shown in fig. 2, according to the optimized circle model center
Figure BDA00023492219900000917
Center of gravity with discrete points
Figure BDA0002349221990000095
A connecting line, the included angle between the connecting line and the X axis in the XY coordinate system is defined as the initial rotation angle
Figure BDA0002349221990000096
The solution is performed by the arctan function:
Figure BDA0002349221990000097
wherein the content of the first and second substances,
Figure BDA0002349221990000098
and
Figure BDA0002349221990000099
representing the center of gravity XAThe values on the X-axis and Y-axis in the OXY coordinate system,
Figure BDA00023492219900000910
and
Figure BDA00023492219900000911
respectively represent optimized circles XBValues on the X-axis and Y-axis in the OXY coordinate system.
Initial elliptic mathematical model
The optimal circle center solved in the previous three steps
Figure BDA00023492219900000912
Set as the initial ellipse center
Figure BDA00023492219900000913
The radius r is set to the major and minor axes (a) of the ellipse0＝b04.7967) plus the initial circular rotation angle
Figure BDA00023492219900000914
That is, five initial parameters based on discrete points are obtained
Figure BDA00023492219900000915
Establishing a parameterized initial elliptic mathematical model:
Figure BDA00023492219900000916
wherein, betajRepresenting the jth discrete point in the initial ellipse X(k0)And determining the corresponding centrifugal angle on the model, wherein the angle needs the elliptic point with the minimum orthogonal distance.
Geometric distance error vector solving
As shown in fig. 3, the same is true for circle fitting orthogonal distance vectors. On the ellipse model, any discrete point XjAll have a unique orthogonal point X corresponding theretoP. It is convenient here to distinguish between the twoUnderstanding that the discrete point X will bejTo the orthogonal point X of the elliptic mathematical modelPIs defined as the geometric distance epsilonp. By elliptical centrifugal angle betajFrom a geometrical distance epsilonpThe mapping relation between discrete points is represented as follows:
εp＝Xj-XP|β (29)
whether the elliptical model is optimized in fitting can be determined by the geometric property delta0(β) quantitative characterization:
δ0(β)＝||Xj-XP|β||＝||(fx,fy)|| (30)
wherein the geometric distance error (f)x,fy) Representing from discrete points XjTo an orthogonal point XPThe vector of (2). Due to the discrete point XjFive parameters of the initial ellipse are known and have been estimated. Then, the only unknown angle of eccentricity βjAn objective function F can be established:
Figure BDA0002349221990000101
wherein the angle of rotation
Figure BDA0002349221990000102
The centrifugal angle beta corresponding to the minimum time of the target function F can be quickly solved by the widely used least square methodj. Then substituting into the formulas (28), (29) and (30), the orthogonal point X on the ellipse model can be calculatedPOrthogonal distance epsilonpGeometric performance function delta corresponding to elliptical model0The value of (. beta.).
Regression constrained LM geometry fitting
Elliptic Jacobian matrix and its equation system
In ellipse fitting, any modification of a basis vector must make an exact adjustment to its vector, otherwise it will result in divergence of the parameter space. The regression parameter equation f (k) is weighted to 5 parameters of the ellipse
Figure BDA0002349221990000103
Figure BDA0002349221990000104
And the partial derivatives and the elliptic Jacobian matrix jointly form a left matrix. For each discrete point XjThe following system of linear equations was established:
Figure BDA0002349221990000105
wherein w is 1.0 × 1012And mu represents a damping coefficient used for adaptively adjusting the change of the elliptical model in five dimensions. I is a 5 × 5 identity matrix.
Figure BDA0002349221990000111
For a parameterized elliptic Jacobian matrix JijInverse matrix of, JijCan be obtained by partial derivation of equation (28):
Figure BDA0002349221990000112
initial value and direction of LM fitting
In equation (32), the left matrix added to the identity matrix is positive, so it has a unique solution, thus ensuring that it is invertible. The initial damping coefficient range is mu epsilon [1, N*]And carrying out global search of five parameters of the ellipse by taking the value mu as 50. According to the elliptical geometric property function delta0(beta), LM self-adaptive regulation strategy is carried out on the damping coefficient mu,
Figure BDA0002349221990000113
wherein, k is ∈ [0.25,0.75 ]]，μnextDamping coefficient, i | epsilon, representing the next fitp+1And | | l represents the magnitude of the geometric property corresponding to the next elliptical model in the fitting. The direction of LM fitting is:
kv+1＝kv+Δk (35)
in the formula, the parameter vector kvThe middle subscript v is belonged to [0, N ∈]Is a natural number, representing a vector of five parameters, k, of the elliptical modelv+1The physical meaning of the representation is the next elliptical model parameter vector to be solved by fitting. And deltak is a vector which needs to be accurately adjusted in five dimensions of the elliptical model, and the adjustment process is shown in figure 4.
Termination condition of LM fitting
The termination criterion reflects the corresponding geometric performance function delta of the elliptical model0(β) is a global minimum. Therefore, taking the LM fitting step size, the steepest descent rate (modulo the gradient) and the fitting ceiling as termination criteria,
Figure BDA0002349221990000114
Figure BDA0002349221990000115
v≥vmax (38)
wherein e is1Subdividing the upper limit, e, for the step size2Is the sum of the upper limit of the gradient mode and vmaxThe upper limit of the number of fits. That is, it is determined that the optimization of the elliptical model is reached as long as any one of the above three upper limit conditions is reached in the adaptive fitting process, as shown in fig. 5.
As shown in fig. 6, after fitting for the 21 st time, the best fit solution can be obtained, and the fitting efficiency is improved by 53.8% compared with that of non-short axis regression. The final corrected vector norm Δ k | | | 8.5665 × 10-7Obtaining a stable orthogonal point set { (X)P,YP) }, geometric Performance function δ01.17188, locating the center of the ellipse (2.69995, 3.81612), and calculating the rotation angle
Figure BDA0002349221990000122
The results of each exact fit are shown in table 3.
TABLE 3 results of LM Adaptation calculations
Figure BDA0002349221990000121
And printing a fitting result report according to the self-adaptive fitting result.
The detailed measurement flow is shown in fig. 7.

Claims (1)

1. A parameterized geometric ellipse fitting method based on short axis regression is characterized in that: the method comprises the following steps of,
t1: short axis regression parameters
In the basic study of ellipses, using parameter vectors
Figure FDA0003533831490000011
Accurately describing an actual elliptical model; the physical meanings corresponding to the 5 parameters in the parameter vector are as follows: xCAnd YCRespectively representing coordinate values of the center of the ellipse on the X axis and the Y axis in the XY coordinate system, a and b respectively representing the sizes of the major axis and the minor axis of the ellipse,
Figure FDA0003533831490000012
the size of an included angle between the long axis of the ellipse model and the X axis in the XY coordinate system is also called as the rotation angle of the ellipse, and the direction of the ellipse is represented;
in the fitting solution, 5 parameters of the ellipse model form numerical adjustment on corresponding dimensionality; so in practice the ellipse fit is based on the known discrete points XjSolving the optimal values in 5 dimensions; then, according to the principle of ellipse formation, the minor axis b of the 5 dimensions is the radius of the forming cylinder, the nominal value of which is known, the radius of the forming cylinder being the radius of the stamped punch, which has an accurate radius index; regression parameter equation:
f(k)＝k-constant＝0 (1)
therefore, the short axis b can be accurately obtained through actual measurement and set as a regression parameter, so that the calculation amount in the dimension can be reduced, and the fitting efficiency can be improved;
t2: initial elliptic mathematical model
In the ellipse geometric fitting, an initial ellipse mathematical model is required to be established first, and fitting optimization is carried out on the basis; since the circle fitting has only three dimensions
Figure FDA0003533831490000013
The calculated amount of the method has stronger robustness; therefore, the discrete point X is equivalently calculated by adopting Gaussian circle fittingjThe initial ellipse mathematical model is carried out according to the following process;
1) calculating an initial circle
Center of gravity of discrete points arbitrarily in the OXY coordinate system
Figure FDA0003533831490000014
And the initial radius r of the circle0Expressed as:
Figure FDA0003533831490000015
Figure FDA0003533831490000016
wherein the center of gravity is adjusted
Figure FDA0003533831490000017
Setting as the center of an initial circle; m represents the number of discrete points, X1Representing the first discrete point, XmFor the last discrete point, XjRefers to the jth discrete point;
2) orthogonal distance vector epsilon of circlec
At any discrete point XjTo the initial circular center of gravity XAThe intersection point of the connecting lines on the circle is called the orthogonal point Di(ii) a The physical meaning of the orthogonal points is understood to mean the circle and the discrete points XjThe nearest corresponding point;
Figure FDA0003533831490000021
thus, an arbitrary discrete point XjTo an orthogonal point DiIs measured in discrete points XjTo the initial circular center of gravity XAFor vector direction, define the orthogonal distance vector ε of the circlec；
Figure FDA0003533831490000022
3) Gaussian circle fitting
Orthogonal point DiEstablishing a linear equation set related to the orthogonal distance vector for the point on the initial circle according to the Jacobian matrix of the orthogonal point;
Figure FDA0003533831490000023
the left matrix being each quadrature point DiFor three dimensional parameters of initial circle
Figure FDA0003533831490000024
Partial derivatives of (a); the gaussian fit direction of the linear equation is:
Figure FDA0003533831490000025
the step length of fitting is 1 in the formula, and the orthogonal distance vector mode | | | epsilon is continuously reduced every fitting adjustmentcL; because only three dimensions are provided, the Gaussian circle fitting can be optimized for a plurality of times, and finally the circle center of the optimal circle model is obtained
Figure FDA0003533831490000026
A radius r;
4) computingInitial rotation angle
Figure FDA0003533831490000027
According to the optimized circle model center
Figure FDA0003533831490000028
Center of gravity with discrete points
Figure FDA0003533831490000029
A connecting line, the included angle between the connecting line and the X axis in the XY coordinate system is defined as the initial rotation angle
Figure FDA00035338314900000210
The solution is performed by the arctan function:
Figure FDA0003533831490000031
wherein the content of the first and second substances,
Figure FDA0003533831490000032
and
Figure FDA0003533831490000033
representing the center of gravity XAThe values on the X-axis and Y-axis in the OXY coordinate system,
Figure FDA0003533831490000034
and
Figure FDA0003533831490000035
respectively represent optimized circles XBValues on the X-axis and Y-axis in the OXY coordinate system;
5) initial elliptic mathematical model
The optimal circle center solved in the previous three steps
Figure FDA0003533831490000036
Set as the initial ellipse center
Figure FDA0003533831490000037
The radius r is set to the major and minor axes (a) of the ellipse0＝b0R), plus the initial circular rotation angle
Figure FDA0003533831490000038
That is, five initial parameters based on discrete points are obtained
Figure FDA0003533831490000039
Establishing a parameterized initial elliptical mathematical model:
Figure FDA00035338314900000310
wherein, betajRepresenting the jth discrete point in the initial ellipse X(k0)The corresponding centrifugal angle on the model is determined by the elliptic point with the minimum orthogonal distance;
t3: geometric distance error vector solving
Fitting the orthogonal distance vector to a circle in the same way; on the ellipse model, any discrete point XjAll have a unique orthogonal point X corresponding theretoP(ii) a For distinguishing the two, the discrete point X is used for easy understandingjTo the orthogonal point X of the elliptic mathematical modelPIs defined as the geometric distance epsilonp(ii) a By elliptical centrifugal angle betajFrom a geometric distance epsilonpThe mapping relation between discrete points is represented as follows:
εp＝Xj-XP|β (10)
whether the elliptical model is optimized in fitting can be determined by the geometric property delta0(β) quantitative characterization:
δ0(β)＝||Xj-XP|β||＝||(fx,fy)|| (11)
whereinGeometric distance error (f)x,fy) Representing from discrete points XjTo an orthogonal point XPThe vector of (a); due to the discrete point XjFive parameters of the initial ellipse are known and have been estimated; then, the only unknown angle of eccentricity βjAn objective function F can be established:
Figure FDA00035338314900000311
wherein the angle of rotation
Figure FDA00035338314900000312
The centrifugal angle beta corresponding to the minimum time of the target function F can be quickly solved by the widely used least square methodj(ii) a Then substituting into the formulas (9), (10) and (11), the orthogonal point X on the ellipse model can be calculatedPOrthogonal distance epsilonpGeometric performance function delta corresponding to elliptical model0A value of (β);
t4: regression constrained LM geometry fitting
1) Elliptic Jacobian matrix and its equation system
In the ellipse fitting, any modification of a basic vector must perform precise adjustment on the vector, otherwise, the divergence of a parameter space is caused; the regression parameter equation f (k) is weighted to 5 parameters of the ellipse
Figure FDA0003533831490000041
Figure FDA0003533831490000042
Partial derivatives, which form a left matrix together with the elliptic Jacobian matrix; for each discrete point XjThe following system of linear equations is established:
Figure FDA0003533831490000043
wherein w is 1.0 × 1012Mu represents a damping coefficient and is used for adaptively adjusting the change of the elliptical model in five dimensions; i is a 5 × 5 identity matrix;
Figure FDA0003533831490000044
for a parameterized elliptic Jacobian matrix JijInverse matrix of, JijCan be obtained by partial derivation of equation (9):
Figure FDA0003533831490000045
2) initial value and direction of LM fitting
In equation (13), the left matrix added to the identity matrix is positive, and thus it has a unique solution, thus ensuring its reversibility; the initial damping coefficient range is mu epsilon [1, N*]Carrying out global search of five elliptic parameters by taking the value of mu as 50; according to the elliptical geometric property function delta0(beta), LM self-adaptive regulation strategy is carried out on the damping coefficient mu,
Figure FDA0003533831490000046
wherein, k is ∈ [0.25,0.75 ]]，μnextDamping coefficient, i | epsilon, representing the next fitp+1The | | represents the magnitude of the geometric performance corresponding to the ellipse model of the next time in the fitting; the direction of LM fitting is:
kv+1＝kv+Δk (16)
in the formula, the parameter vector kvThe middle subscript v is belonged to [0, N ∈]Is a natural number, representing a vector of five parameters, k, of the elliptical modelv+1The physical meaning of the expression is the parameter vector of the next ellipse model for fitting solution; delta k is a vector which needs to be accurately adjusted in five dimensions of the elliptical model;
3) termination condition of LM fitting
The termination criterion reflects the corresponding geometric performance function delta of the elliptical model0(beta) is a global maximumA small value; therefore, taking the LM fitting step size, the steepest descent rate, i.e., the modulus of the gradient, and the fitting ceiling as termination criteria,
Figure FDA0003533831490000051
Figure FDA0003533831490000052
v≥vmax (19)
wherein e is1Subdividing the upper limit, e, for the step size2Is the sum of the upper limit of the gradient mode and vmaxIs the upper limit of the fitting times; that is, it is determined that the optimization of the elliptical model is achieved as long as any one of the above three upper limit conditions is achieved in the adaptive fitting process.
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