发布时间: 2018-10-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.170680 2018 | Volume 23 | Number 10 计算机图形学

 收稿日期: 2018-01-08; 修回日期: 2018-04-08 第一作者简介: 吴建霖, 1993年生, 男, 硕士研究生, 主要研究方向为视觉测量与智能检测。E-mail:1596938843@qq.com;蒋理兴, 男, 教授, 主要研究方向为测量仪器与计量检定。E-mail:lixing_jiang@126.com;王安成, 男, 讲师, 主要研究方向为组合导航和精确测量。E-mail:acwang_xd@163.com;谷友艺, 男, 硕士研究生, 主要研究方向为精密仪器测量。E-mail:1943624866@qq.com;于彭, 男, 硕士研究生, 主要研究方向为结构光测量。E-mail:1264506260@qq.com. 中图法分类号: TP391 文献标识码: A 文章编号: 1006-8961(2018)10-1549-09

# 关键词

Eccentricity error compensation for circular targets
Wu Jianlin, Jiang Lixing, Wang Ancheng, Gu Youyi, Yu Peng
Strategic Support Force Information Engineering University, Zhengzhou 450001, China

# Abstract

Objective Currently, circular target is widely used in a multitude of vision measurement systems in which center positioning accuracy determines the accuracy of the measuring system.The projection of a circular feature is generally an ellipse and not a true circle because the main optical axis of a camera is not parallel with the feature surface.When the angle between the main optical axis of a camera and the feature surface is large, the eccentricity error will affect the accuracy of the measurement system extremely.Most of the research on eccentricity errors in the last two decades has examined quite a few methods and conducted experiments to derive the mechanism of eccentricity and attempted to correct the errors.Incidentally, the past studies used geometric parameters to calculate the eccentricity errors, which increased the complexity of the process.This paper introduces a new method for correcting the eccentricity error with the help of the three concentric circle targets. Method Most algorithms for eccentricity errors usually involve several geometric parameters, and calibration and bundle adjustment are always required to obtain these parameters.These algorithms increase the computational complexity and reduce the rate of convergence.Our method designs three concentric circles as the target, which has common center coordinates in the object plane and different centers in the image plane, which are on a line.The moment invariants of Zernike moment are used for the edge detection of the pixel level to obtain the precise positioning of the sub-pixel level edge.The center of the ellipse is determined with the least-square ellipse fitting.To achieve better results, the images of the concentric circle targets should include at least 20 pixels to ensure sufficient effective edge points.The ellipse center is easily calculated with the sub-pixel level edge, then we can use the three groups of ellipse centers to calculate the eccentricity error model.The three concentric circles in the error equation have the same six parameters.Thus, the corresponding parameters can be set into blocks as new variables, which in turn can be reduced to three unknown parameters.The error equations can be sufficiently solved with the help of the three concentric circles.Through the formulas derived in this study, the eccentricity errors can be solved completely.Obtaining geometrical parameters should be avoided and the nonlinear model should be solved. Result A possible solution for the correction of this systematic eccentricity error is proposed in this paper.The method can effectively improve the positioning accuracy of the circular target center.Simulation experiment results by using MATLAB show that the eccentricity errors can be compensated from the pixel level to the 10-11-pixel level when the targets are photos taken in different angles, distances, and sizes of the targets.This study designs a target that has three concentric circles and diameters of 6, 12, and 18 cm.To calculate the true center of the circles, a circle with a size of 2 mm is designed in the central area of the target.During image processing, we use the improved gray barycenter localization algorithm to calculate the center of the small circle.By comparison, its radius is extremely small, and the simulation experiment shows that its eccentric error is only 0.02 pixels, which can be ignored compared with the three concentric circles used in the experiment and regarded as the true value in the experiment.Experiment results show that the measurement errors can be controlled at 0.1 mm.Relative to the concentric circles method, the accuracy is two times than that of before and the eccentricity error is decreased by approximately 80%. Conclusion This paper presents a new eccentricity compensation method for calculating eccentricity error by using three concentric circle targets to add constraint.Unlike in previous eccentricity error correction methods, additional parameters should be estimated to correct the eccentricity error.Consequently, the computation complexity increases and convergence speed decreases.Prior knowledge about the geometric parameters of the measurement system (target and camera) are not needed; rather, the proportional relationship of the circles and ellipse center coordinates are the only information required.The experiment results show the efficiency of the proposed method for eccentricity error compensation.The algorithm can improve the location efficiency of circular targets.Consequently, the algorithm can enhance the effect on depth image matching that is based on non-coding markers, the precision of the automatic camera calibration method that is based on circular markers, and the robustness of the navigation and positional system.

# Key words

eccentricity error; three concentric circles; ellipse fitting; image measurement errors; visual location

# 1.1 圆形标志偏心差模型

 $\left\{ \begin{array}{l} {P^2} = \\ \frac{{\left( {{{u'}_{{{B'}_1}}} - {{u'}_{{{B'}_3}}}} \right)\left( {r_1^2 - r_2^2} \right)r_3^2 - \left( {{{u'}_{{{B'}_1}}} - {{u'}_{{{B'}_2}}}} \right)\left( {r_1^2 - r_3^2} \right)r_2^2}}{{\left( {{{u'}_{{{B'}_1}}} - {{u'}_{{{B'}_3}}}} \right)\left( {r_1^2 - r_2^2} \right) - \left( {{{u'}_{{{B'}_1}}} - {{u'}_{{{B'}_2}}}} \right)\left( {r_1^2 - r_3^2} \right)}}\\ K = \frac{{{{u'}_{{{B'}_1}}} - {{u'}_{{{B'}_2}}}}}{{\frac{1}{{\frac{{{P^2}}}{{r_1^2}} - 1}} - \frac{1}{{\frac{{{P^2}}}{{r_2^2}} - 1}}}}\\ L = \frac{{ - \left( {{{v'}_{{{B'}_1}}} - {{v'}_{{{B'}_2}}}} \right)}}{{\frac{1}{{\frac{{{P^2}}}{{r_1^2}} - 1}} - \frac{1}{{\frac{{{P^2}}}{{r_2^2}} - 1}}}} \end{array} \right.$ (13)

 $\left\{ \begin{array}{l} {u_c} = {{u'}_{{{B'}_1}}} - \frac{1}{2}\left( {\frac{K}{{\frac{P}{{{r_1}}} - 1}} - \frac{K}{{\frac{P}{{{r_1}}} + 1}}} \right)\\ {v_c} = {{v'}_{{{B'}_1}}} - \frac{1}{2}\left( {\frac{K}{{\frac{P}{{{r_1}}} + 1}} - \frac{K}{{\frac{P}{{{r_1}}} - 1}}} \right) \end{array} \right.$ (14)

# 2 仿真实验及分析

 $\left( {\begin{array}{*{20}{c}} X\\ Y\\ Z \end{array}} \right) = \lambda \mathit{\boldsymbol{R}}\left( {\begin{array}{*{20}{c}} x\\ y\\ { - f} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {{X_s}}\\ {{Y_s}}\\ {{Z_s}} \end{array}} \right)$ (15)

 $\left\{ \begin{array}{l} X = {X_0} + r\cos \left( \theta \right)\\ Y = {Y_0} + r\sin \left( \theta \right)\\ Z = 0 \end{array} \right.$ (16)

 $\left\{ \begin{array}{l} x = - f\frac{{{a_1}\left( {X - {X_s}} \right) + {b_1}\left( {Y - {Y_s}} \right) + {c_1}\left( {Z - {Z_s}} \right)}}{{{a_3}\left( {X - {X_s}} \right) + {b_3}\left( {Y - {Y_s}} \right) + {c_3}\left( {Z - {Z_s}} \right)}}\\ y = - f\frac{{{a_2}\left( {X - {X_s}} \right) + {b_2}\left( {Y - {Y_s}} \right) + {c_2}\left( {Z - {Z_s}} \right)}}{{{a_3}\left( {X - {X_s}} \right) + {b_3}\left( {Y - {Y_s}} \right) + {c_3}\left( {Z - {Z_s}} \right)}} \end{array} \right.$ (17)

$\theta$分为64份，计算64个点坐标，利用所有像点拟合椭圆，并计算椭圆中心点坐标${u_{\rm{e}}}\left( {{x_{\rm{e}}}, {y_{\rm{e}}}} \right)$，并计算标志中心实际对应的像点坐标，计算椭圆偏心差大小为$d = \left\| {{u_{\rm{e}}}-{u_{\rm{c}}}} \right\| = \sqrt {{{\left( {{x_{\rm{e}}}-{x_{\rm{c}}}} \right)}^2} + {{\left( {{y_{\rm{e}}}-{y_{\rm{c}}}} \right)}^2}}$

Table 1 Effects of eccentricity errors compensation under different factors

 /像素 序号 不同夹角补偿前/后 不同距离补偿前/后 不同标志大小补偿前/后 1 0.407/0.162×10-11 0.407/0.162×10-11 0.407/0.162×10-11 2 0.448/0.013×10-11 0.412/0.075×10-11 0.586/0.081×10-11 3 0.505/0.042×10-11 0.417/0.162×10-11 0.797/0.168×10-11 4 0.581/0.163×10-11 0.422/0.086×10-11 1.042/0.065×10-11 5 0.682/0.010×10-11 0.428/0.167×10-11 1.318/0.084×10-11 6 0.817/0.408×10-11 0.434/0.085×10-11 1.628/0.121×10-11 7 0.996/0.485×10-11 0.441/0.072×10-11 1.969/0.122×10-11 8 1.234/2.508×10-11 0.448/0.162×10-11 2.344/0.018×10-11 9 1.556/1.752×10-11 0.455/0.323×10-11 2.751/0.121×10-11 10 1.993/1.942×10-11 0.462/0.247×10-11 3.191/0.087×10-11 11 2.595/0.333×10-11 0.470/0.487×10-11 3.663/0.127×10-11

1) 相机采集同心圆标志图像，使用标定参数进行校正。

2) 图形进行预处理，提取椭圆图像部分获取亚像素级边缘。

3) 边缘点进行最小二乘拟合，剔除误差大于阈值的拟合点，进行二次拟合。对拟合椭圆求取中心坐标。

4) 将得到的3组椭圆中心坐标代入式(13)(14)，得到补偿结果。

# 3 实物实验结果与分析

Table 2 Effects of eccentricity errors compensation with object targets

 /像素 序号 大圆 中圆 小圆 文献[9] 三同心圆 1 13.111 0 5.607 4 1.380 3 0.475 3 0.251 3 2 11.018 9 4.661 5 1.091 3 0.482 0 0.221 7 3 9.795 5 4.151 4 1.056 4 0.473 3 0.259 7 4 10.278 4 4.472 5 1.104 9 0.470 7 0.263 5 5 12.339 8 5.209 8 1.162 4 0.568 5 0.295 5 6 11.182 0 4.783 1 1.087 0 0.481 6 0.247 0 7 6.778 7 2.839 4 0.567 4 0.301 8 0.149 1 8 10.582 3 4.730 4 1.092 1 0.508 0 0.211 6 9 14.145 7 6.063 3 1.414 7 0.415 9 0.141 8

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