CN110163905A - A kind of high-precision ellipse fitting method - Google Patents
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Abstract
The invention proposes a kind of high-precision ellipse fitting methods, by absolute value error and minimize introducing elliptic parameter estimation frame, obtain an algebraic distance based on L1 and minimize cost function, overcome the problem of traditional L2 cost function is to noise-sensitive；The present invention can obtain high-precision elliptic parameter fitting precision for noise free data, Gauusian noise jammer data, non-Gaussian noise interference data.
Description
Technical field
The present invention relates to field of optical measurements more particularly to a kind of high-precision ellipse fitting methods.
Background technique
With the fast development of computer technology, three-dimensional measurement technology has become a kind of powerful detection instrument, extensively
Also have applied to numerous industrial circles such as aerospace, automobile manufacture, while in other fields such as cultural creative, education, medical treatment
Be widely applied.The use of circular index point is very universal during three-dimensional measurement, not only in camera calibration parametric solution rank
Section, needs accurately to estimate the center of circle of Circle in Digital Images shape index point.In particular for the split of wide view-field three-D point cloud data,
Need to rely on the center of circle reconstructed results of index point, is the weight for reducing big visual field point data splitting accumulated error using index point
Want means.Therefore the correctness and robustness of index point center of circle estimation are just particularly important.
Circular index point shows elliptical shape after projection in the picture, at present for the side of oval center of circle fitting
Method is very more, proposes a large amount of algorithm for inhibiting the interference of noise for the current domestic and foreign scholars of ellipse fitting, but
Existing result of study primarily directed to least square method improvement, by using means such as data normalization, stochastical samplings
The influence of noise is reduced to a certain extent.But for practical application scene, the usually complicated multiplicity of noise type is based on Gauss
There is inevitable defect, easily shadow of the amplification noise to estimated result in itself in the least square class method of noise modeling
It rings, to be difficult to obtain high-precision estimated result.For existing ellipse fitting method to non-Gaussian noise interference data fitting
The problem of effect low precision, the present invention provide a kind of high-precision ellipse fitting method, and it is dry can effectively to solve various types noise
Ellipse fitting problem under disturbing, arithmetic accuracy is high, adaptability is good.
Summary of the invention
In view of this, can effectively solve various types noise the invention proposes a kind of high-precision ellipse fitting method
Ellipse fitting problem under interference, arithmetic accuracy is high, adaptability is good.
The technical scheme of the present invention is realized as follows: the present invention provides a kind of high-precision ellipse fitting methods, including
Following steps:
S1, in conjunction with to fitting data, definition based on L1 without constraint cost problem；
S2, new constraint cost problem solving will be converted to without constraint cost problem in S1；
S3, by the constraint cost problem in S2 split into it is new without constraint cost problem solving；
S4, multiple alternately minimum sons that solve will be converted to without constraint cost problem in S3 using alternately minimum iterative strategy and asked
Topic solves each alternately minimum solution subproblem, solves ellipse fitting parameter using division bregman alternative manner.
On the basis of above technical scheme, it is preferred that being defined in S1 based on L1 includes following without constraint cost problem
Step:
S101, to general conic section F (K, δ)=ax2+bxy+cy2+ dx+ey+f=0 enables K=(x2,xy,y2,x,
Y, 1), δ=(a, b, c, d, e, f)T；
S102, in order to ensure conic section be ellipse, it is desirable that the discriminate of general conic section meets b2- 4ac < 0,
Primal constraints minimization problem is established according to least absolute deviation are as follows:
S103, when discriminate be b2When -4ac=-1, meet oval condition b2- 4ac < 0, equivalent form 4ac-b2=
1, matrix form are as follows: δTδ=1 C is further converted to primal constraints minimization problem
In formula, C is constant matrices,
S104, pass through method of Lagrange multipliers, formula (2) be converted to and is solved as follows without constraint cost problem:
In formula, λ is LaGrange parameter, which is a constant greater than zero.
It is further preferred that being converted to new constraint cost problem specific steps without constraint cost problem in S2 are as follows: introduce
Formula (3) in S104 is converted to following new constraint cost problem by auxiliary variable s=K δ:
It is further preferred that in S3 by the constraint cost problem in S2 split into it is new without constraint cost problem specific steps
Are as follows: be introduced into penalty term the formula (4) in S2 is split into it is following new without constraint cost problem:
In formula, μ is the punishment parameter greater than zero, and t is bregman variable.
It is further preferred that multiple alternately minimum solution subproblems are respectively as follows: in S4
Subproblem about elliptic parameter δ:
Subproblem about auxiliary variable s:
Subproblem about bregman variable t:
In formula (6), formula (7) and formula (8), k indicates the number of iteration, for marking the variable of different the number of iterations.
It is further preferred that being obtained using partial differential method direct solution formula (6):
It is further preferred that solving formula (7) using two dimension shrink operator, obtain: sk+1=shrink (K δk+1+tk,1/
μ) (10)；
In formula, two-dimentional shrink operator is defined as:
It is further preferred that the Bregman alternative manner using standard solves formula (8), obtain: tk+1=tk+Kδk+1-sk+1
(12)。
It is further preferred that in S4 using division bregman alternative manner solve ellipse fitting parameter specifically include it is following
Step:
S401, initialization elliptic parameter δ0=0, t0=0, s0=(KT)-1；
S402, initialization maximum number of iterations kMax=300, primary iteration number k=0；
S403, setting iteration termination condition:
S404, elliptic parameter δ is updated using formula (9)；
S405, auxiliary variable s is updated using formula (10)；
S406, bregman variable t is updated using formula (12)；
S407, it is less than maximum number of iterations when the elliptic parameter δ in S404 meets current iteration number, andOr current iteration number determines that the elliptic parameter δ that currently updates is when being equal to maximum number of iterations
Optimal solution；
S408, iteration obtain optimal elliptic parameter δ=(a, b, c, d, e, f)TAfterwards, by each member in elliptic parameter δ
Element is normalized divided by f, the expression formula η after being normalized, whereinA, B, C', D, E, F are respectively elliptic parameter δ normalization
The corresponding normalized value of each element afterwards；
S409, the oval center of circle is set as (Cx,Cy), oval major semiaxis is Rx, oval semi-minor axis is Ry, ELLIPTIC REVOLUTION angle is θ,
Ellipse fitting parameter is calculated according to the following formula:
In formula, XcAnd YcRepresent x-axis and y-axis.
A kind of high-precision ellipse fitting method of the invention has the advantages that compared with the existing technology
(1) absolute value error and minimum are introduced into elliptic parameter and estimates frame, obtain the algebraic distance based on L1
With minimum cost function, the problem of traditional L2 cost function is to noise-sensitive is overcome；
(2) present invention can obtain high for noise free data, Gauusian noise jammer data, non-Gaussian noise interference data
The elliptic parameter fitting precision of precision.
Detailed description of the invention
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technical description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
Some embodiments of invention for those of ordinary skill in the art without creative efforts, can be with
It obtains other drawings based on these drawings.
Fig. 1 is a kind of flow chart of high-precision ellipse fitting method of the present invention；
Fig. 2 is the flow chart of S1 in a kind of high-precision ellipse fitting method of the present invention；
Fig. 3 is that S4 uses the flow chart of division bregman alternative manner in a kind of high-precision ellipse fitting method of the present invention；
Fig. 4 is the noiseless ellipse data of emulation；
Fig. 5 is Fig. 4 iteration 300 times, ellipse fitting result images；
Fig. 6 be emulation comprising mean value be 0, standard deviation be 1 Gaussian noise oval data；
Fig. 7 is Fig. 6 iteration 6 times, ellipse fitting result images；
Fig. 8 is the oval data comprising non-gaussian noise spot at random of emulation；
Fig. 9 is Fig. 8 iteration 26 times, ellipse fitting result images.
Specific embodiment
Below in conjunction with embodiment of the present invention, the technical solution in embodiment of the present invention is carried out clearly and completely
Description, it is clear that described embodiment is only some embodiments of the invention, rather than whole embodiments.Base
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts all
Other embodiments shall fall within the protection scope of the present invention.
As shown in Figure 1, a kind of high-precision ellipse fitting method, comprising the following steps:
S1, in conjunction with to fitting data, definition based on L1 without constraint cost problem；As shown in Fig. 2, specifically including following
Step:
S101, to general conic section F (K, δ)=ax2+bxy+cy2+ dx+ey+f=0 enables K=(x2,xy,y2,x,
Y, 1), δ=(a, b, c, d, e, f)T；Wherein, the expression formula of general conic section F (K, δ) is the standard of general conic section
Expression formula, a, b, c, d, e, f are each term coefficient of general conic section F (K, δ), the expression formula of general conic section F (K, δ)
For the common knowledge of this field；
S102, in order to ensure conic section be ellipse, it is desirable that the discriminate of general conic section meets b2- 4ac < 0,
Primal constraints minimization problem is established according to least absolute deviation are as follows:In formula, | | | |1Represent L1
Norm,Indicate variate-value when being minimized objective function, s.t. is the contracting of subject to inside mathematics
Write, be the controlled meaning, mathematics habit be " so that ... meet ..., belong to the common knowledge of this field；
S103, when discriminate be b2When -4ac=-1, meet oval condition b2- 4ac < 0, equivalent form 4ac-b2=
1, matrix form are as follows: δTδ=1 C is further converted to primal constraints minimization problem
In formula, C is constant matrices,
S104, pass through method of Lagrange multipliers, formula (2) be converted to and is solved as follows without constraint cost problem:
In formula, λ is LaGrange parameter, which is a constant greater than zero, δTIndicate the transposition of δ.
In the present embodiment, formula (3) is non-differentiability containing L1 absolute value term and can not be split, to solve this problem,
S2-S4 is using division bregman alternative manner, the basic resolving ideas of this method are as follows: auxiliary variable s=K δ is introduced, it will be in S1
Unconstrained minimization problem has been converted to constraint cost problem, and being introduced into penalty term will have constraint cost problem to split into newly in S2
Without constraint cost problem, multiple alternately minimums will be converted to without constraint cost problem in S3 using alternately minimum iterative strategy and asked
Subproblem is solved, the solution that constraint condition updates three subproblems is executed by division bregman iteration, obtains elliptic parameter δ most
Excellent solution, then ellipse fitting parameter is obtained by normalized.
S2, new constraint cost problem solving will be converted to without constraint cost problem in S1；Specific steps are as follows: introduce auxiliary
Variable s=K δ is helped, the formula (3) in S104 is converted into following new constraint cost problem:
S3, by the constraint cost problem in S2 split into it is new without constraint cost problem；Specific steps are as follows: introduce penalty term
Formula (4) in S2 is split into following new without constraint cost problem:
In formula, μ is the punishment parameter greater than zero, and t is bregman variable, and penalty term is constant matrices, in the present embodiment
In, since influence of the parameter to calculated result of penalty term is little, the parameter of penalty term is without limitation.
S4, multiple alternately minimum sons that solve will be converted to without constraint cost problem in S3 using alternately minimum iterative strategy and asked
Topic solves each alternately minimum solution subproblem, solves ellipse fitting parameter using division bregman alternative manner.
Wherein, the minimum iterative strategy of alternating, which refers to, fixes other variables, solves one of variable.In the present embodiment
In, when solving some variable, the subitem containing the variable is put into an equation in wushu (5), and as the alternating of the variable is most
Small solution subproblem solves subproblem by multiple alternately minimums that alternately minimum iterative strategy obtains and is respectively as follows:
Subproblem about elliptic parameter δ:
Subproblem about auxiliary variable s:
Subproblem about bregman variable t:
In formula (6), formula (7) and formula (8), k indicates the number of iteration, for marking the variable of different the number of iterations.
It is typical quadratic equation minimization problem, therefore, in the present embodiment since (6) can be micro- when solving formula (6)
Middle use asks partial differential method direct solution formula (6), the final result of solution are as follows:Due to
The method for seeking partial differential belongs to existing mathematic calculation, and therefore, the present embodiment omits the detailed process for seeking partial differential, directly
Provide the final result solved using partial differential.
When solving formula (7), directlys adopt two-dimentional shrink operator and solve formula (7), two-dimentional shrink operator is defined as:Formula (7) can be calculated according to the definition of two-dimentional shrink operator
Further expression formula are as follows: sk+1=shrink (K δk+1+tk, 1/ μ) and (10), in the present embodiment, K δk+1+tkIn corresponding (11)
X, the r in 1/ μ corresponding (11) can find out the final result of formula (10) according to the computation rule of formula (11).Due to formula
(10) calculating process belongs to existing mathematic calculation, therefore, the solution procedure of formula (10) is omitted in the present embodiment.
When solving formula (8), using division Bregman alternative manner newer (8), obtain: tk+1=tk+Kδk+1-sk+1
(12).Since division Bregman alternative manner belongs to the prior art, it is not repeated herein.
In the present embodiment, LaGrange parameter λ=1, punishment parameter μ=1, K are takenTThe transposition of representing matrix K, K-1It indicates
The inverse matrix of matrix K.As shown in figure 3, in S4 using division bregman alternative manner solve ellipse fitting parameter specifically include with
Lower step:
S401, initialization elliptic parameter δ0=0, t0=0, s0=(KT)-1；
S402, initialization maximum number of iterations kMax=300, primary iteration number k=0；
S403, setting iteration termination condition:
S404, elliptic parameter δ is updated using formula (9)；
S405, auxiliary variable s is updated using formula (10)；
S406, bregman variable t is updated using formula (12)；
S407, it is less than maximum number of iterations when the elliptic parameter δ in S404 meets current iteration number, andOr current iteration number determines that the elliptic parameter δ that currently updates is when being equal to maximum number of iterations
Optimal solution；
S408, iteration obtain optimal elliptic parameter δ=(a, b, c, d, e, f)TAfterwards, by each member in elliptic parameter δ
Element is normalized divided by f, the expression formula η after being normalized, whereinA, B, C', D, E, F are respectively elliptic parameter δ normalization
The corresponding normalized value of each element afterwards；Wherein, f is an element of elliptic parameter δ；
S409, the oval center of circle is set as (Cx,Cy), oval major semiaxis is Rx, oval semi-minor axis is Ry, ELLIPTIC REVOLUTION angle is θ,
Ellipse fitting parameter is calculated according to the following formula:
In formula, XcAnd YcRepresent x-axis and y-axis.
Step S401-S407 is the optimal solution preocess for iteratively solving elliptic parameter δ, and the optimal solution of elliptic parameter δ is
Meet the elliptic parameter δ of absolute value error minimum.
S401-S407 obtains optimal elliptic parameter δ=(a, b, c, d, e, f) through the above stepsTAfterwards, that is, it is aware of ellipse
Equation of a circle, according to elliptic parameter δ=(a, b, c, d, e, f)T, elliptical five ellipse fitting parameters, step S408- can be solved
S409 is the solution procedure of ellipse fitting parameter.
The ellipse fitting method recorded using the present embodiment, has obtained relevant experimental data, hereinafter, simple introduce utilizes this
The experimental data that method obtains.
Fig. 4 is the noiseless ellipse data of emulation, the oval center of circle (Cx,Cy)=(151,257), major semiaxis Rx=17, it is short by half
Axis Ry=13, rotation angle θ=0.872665 radian；Fig. 5 is Fig. 4 iteration 300 times, ellipse fitting result images, the oval center of circle (Cx,
Cy)=(151.000176,257.000378), major semiaxis Rx=17.017225, semi-minor axis Ry=12.987198, rotation angle θ=
0.865209 radian；Comparison diagram 4 and Fig. 5 it is found that using the present embodiment record ellipse fitting method, after iteration 300 times, with
The noiseless ellipse data of emulation have no too big difference, and almost can be approximated to be is the same ellipse, thus, it will be seen that
The ellipse fitting method precision that the present embodiment is recorded is high；
Fig. 6 be emulation comprising mean value be 0, standard deviation be 1 Gaussian noise oval data, the oval center of circle (Cx,Cy)=
(64,48), major semiaxis Rx=32, semi-minor axis Ry=30, rotation angle θ=2.356194 radians；
Fig. 7 is Fig. 6 iteration 6 times, ellipse fitting result images, the oval center of circle (Cx,Cy)=(64.076388,
47.953661), major semiaxis Rx=32.291132, semi-minor axis Ry=30.073432, rotation angle θ=2.334187 radians；Comparison
Even Fig. 6 and Fig. 7 utilizes the ellipse fitting method of the present embodiment record it is found that in the oval data with Gaussian noise
Also it is oval that high-precision can be fitted；
Fig. 8 is the oval data comprising non-gaussian noise spot at random of emulation, the oval center of circle (Cx,Cy)=(151,257),
Major semiaxis Rx=17, semi-minor axis Ry=13, rotation angle θ=0.872665 radian；
Fig. 9 is Fig. 8 iteration 26 times, ellipse fitting result images, the oval center of circle (Cx,Cy)=(151.031648,
257.018581), major semiaxis Rx=16.948239, semi-minor axis Ry=12.945079, rotation angle θ=0.818174 radian.Comparison
Fig. 8 and Fig. 9 utilizes the ellipse fitting method of the present embodiment record it is found that in the oval data of non-gaussian noise spot at random
It is oval that high-precision can be fitted.
In conjunction with Fig. 4 to Fig. 9 it is found that ellipse fitting method described herein can be dry for noise free data, Gaussian noise
Disturb data, non-Gaussian noise interference data obtain high-precision elliptic parameter fitting precision.
The foregoing is merely better embodiments of the invention, are not intended to limit the invention, all of the invention
Within spirit and principle, any modification, equivalent replacement, improvement and so on be should all be included in the protection scope of the present invention.
Claims (9)
1. a kind of high-precision ellipse fitting method, it is characterised in that: the following steps are included:
S1, in conjunction with to fitting data, definition based on L1 without constraint cost problem；
S2, new constraint cost problem solving will be converted to without constraint cost problem in S1；
S3, by the constraint cost problem in S2 split into it is new without constraint cost problem solving；
S4, multiple alternately minimum solution subproblems will be converted to without constraint cost problem in S3 using the minimum iterative strategy of alternating,
Each alternately minimum solution subproblem is solved, division bregman alternative manner is used to solve ellipse fitting parameter.
2. a kind of high-precision ellipse fitting method as described in claim 1, it is characterised in that: definition is based on L1's in the S1
Without constraint cost problem the following steps are included:
S101, to general conic section F (K, δ)=ax2+bxy+cy2+ dx+ey+f=0 enables K=(x2,xy,y2, x, y, 1), δ
=(a, b, c, d, e, f)T；
S102, in order to ensure conic section be ellipse, it is desirable that the discriminate of general conic section meets b2- 4ac < 0, according to most
A small multiplication establishes primal constraints minimization problem are as follows:
S103, when discriminate be b2When -4ac=-1, meet oval condition b2- 4ac < 0, equivalent form 4ac-b2=1,
Matrix form are as follows: δTδ=1 C is further converted to primal constraints minimization problem
In formula, C is constant matrices,
S104, pass through method of Lagrange multipliers, formula (2) be converted to and is solved as follows without constraint cost problem:
In formula, λ is LaGrange parameter, which is a constant greater than zero.
3. a kind of high-precision ellipse fitting method as claimed in claim 2, it is characterised in that: asked in the S2 without constraint cost
Topic be converted to new constraint cost problem specific steps are as follows: introduce auxiliary variable s=K δ, by the formula (3) in S104 be converted to as
New constraint cost problem down:
4. a kind of high-precision ellipse fitting method as claimed in claim 3, it is characterised in that: by the constraint in S2 in the S3
Cost problem split into it is new without constraint cost problem specific steps are as follows: be introduced into penalty term the formula (4) in S2 is split into it is as follows
It is new without constraint cost problem:
In formula, μ is the punishment parameter greater than zero, and t is bregman variable.
5. a kind of high-precision ellipse fitting method as claimed in claim 4, it is characterised in that: multiple alternately minimum in the S4
Subproblem is solved to be respectively as follows:
Subproblem about elliptic parameter δ:
Subproblem about auxiliary variable s:
Subproblem about bregman variable t:
In formula (6), formula (7) and formula (8), k indicates the number of iteration, for marking the variable of different the number of iterations.
6. a kind of high-precision ellipse fitting method as claimed in claim 5, it is characterised in that: directly asked using partial differential method
Solution formula (6), obtains:
7. a kind of high-precision ellipse fitting method as claimed in claim 5, it is characterised in that: asked using two dimension shrink operator
Solution formula (7), obtains: sk+1=shrink (K δk+1+tk,1/μ) (10)；
In formula, two-dimentional shrink operator is defined as:
8. a kind of high-precision ellipse fitting method as claimed in claim 5, it is characterised in that: changed using the Bregman of standard
Formula (8) are solved for method, are obtained: tk+1=tk+Kδk+1-sk+1 (12)。
9. such as a kind of described in any item high-precision ellipse fitting methods of claim 5 to 8, it is characterised in that: make in the S4
With division bregman alternative manner solve ellipse fitting parameter specifically includes the following steps:
S401, initialization elliptic parameter δ0=0, t0=0, s0=(KT)-1；
S402, initialization maximum number of iterations kMax=300, primary iteration number k=0；
S403, setting iteration termination condition:
S404, elliptic parameter δ is updated using formula (9)；
S405, auxiliary variable s is updated using formula (10)；
S406, bregman variable t is updated using formula (12)；
S407, it is less than maximum number of iterations when the elliptic parameter δ in S404 meets current iteration number, andOr current iteration number determines that the elliptic parameter δ that currently updates is when being equal to maximum number of iterations
Optimal solution；
S408, iteration obtain optimal elliptic parameter δ=(a, b, c, d, e, f)TAfterwards, by each element in elliptic parameter δ divided by
F is normalized, the expression formula η after being normalized, whereinA, B, C', D, E, F are respectively elliptic parameter δ normalization
The corresponding normalized value of each element afterwards；
S409, the oval center of circle is set as (Cx,Cy), oval major semiaxis is Rx, oval semi-minor axis is Ry, ELLIPTIC REVOLUTION angle be θ, according to
Lower formula calculates ellipse fitting parameter:
In formula, XcAnd YcRepresent x-axis and y-axis.
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| JUNLI LIANG ETC.: ""Robust Ellipse Fitting Based on Sparse Combination of Data Points"", 《IEEE TRANSACTIONS ON IMAGE PROCESSING》 * |
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| CN112464780A (en) * | 2020-11-24 | 2021-03-09 | 宁波大学 | Ellipse object feature extraction method based on maximum entropy criterion |
| CN112464780B (en) * | 2020-11-24 | 2024-01-09 | 扬州汇隆科技有限公司 | Elliptic object feature extraction method based on maximum entropy criterion |
| CN112967256A (en) * | 2021-03-09 | 2021-06-15 | 扬州大学 | Tunnel ovalization detection method based on spatial distribution |
| CN112967256B (en) * | 2021-03-09 | 2023-11-24 | 扬州大学 | Tunnel ovalization detection method based on spatial distribution |
| CN113640726A (en) * | 2021-10-19 | 2021-11-12 | 青岛杰瑞自动化有限公司 | Multi-azimuth ellipse fitting calibration method and system for double-shaft magnetometer |
| CN113640726B (en) * | 2021-10-19 | 2021-12-21 | 青岛杰瑞自动化有限公司 | Multi-azimuth ellipse fitting calibration method and system for double-shaft magnetometer |
| WO2023226065A1 (en) * | 2022-05-25 | 2023-11-30 | 北京航空航天大学杭州创新研究院 | Ellipse positioning data processing method and apparatus, and device and computer-readable medium |
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