目的 圆形标志目前正广泛地应用于各类视觉测量系统，其圆心定位精度决定了测量系统的测量精度。当相机主光轴与标志表面不平行时，圆被映射为椭圆，圆心位置计算产生偏差。光轴与标志表面夹角较大或标志较大等情况下，偏心差较大而严重影响系统测量精度。本文提出了一种基于三同心圆圆形标志的投影偏心差补偿算法。方法 算法基于三同心圆的圆形标志设计，根据3组椭圆拟合中心坐标解算偏心差模型进行计算补偿。结果 针对圆形标志偏心差问题，同心圆补偿算法取得良好效果，有效提升了圆形标志定位精度。仿真结果表明，在拍摄角度、拍摄距离、圆形标志大小不同的情况下，偏心差在像素量级，补偿后偏心差在10-11pixel量级。实物实验结果表明，若设计有直径分别为6cm，12cm，18cm的三同心圆标志，经解算补偿结果较以往两同心圆算法精度提高一倍，偏心差值减小80%，测量误差在0.1mm左右。结论 本文提出了一种新的偏心差补偿算法，利用三同心圆标志增加约束解算偏心差。与以往偏心差补偿算法相比，此方法精度更高，且无需预先平差解算相机与目标的距离、拍摄角等参数，仅需要知道标志圆形半径比例及椭圆中心坐标即可计算补偿，具有很高的实用性，可用于改善基于非编码标志点的深度像匹配、基于圆形标志点的全自动相机标定方法、视觉导航定位等应用中。
Study on the eccentricity error compensation for circular targets
Wu Jianlin,Jiang Lixing,Wang Ancheng,Gu Youyi,Yu Peng(Strategic Support Force Information Engineering University)
Objective Nowadays, circular target is widely used in a multitude of vision measurement systems, which center positioning accuracy determines the accuracy of the measuring system. The projection of a circular feature is generally an ellipse, not a true circle, since that the main optical axis of camera is not parallel with the feature surface. When the angle between the main optical axis of camera and the feature surface is large, the eccentricity error will extremely affect the accuracy of measurement system. Much of the research in the eccentricity errors in the last two decades has examined quite a few methods and experiments to derive the mechanism of the eccentricity and tried to correct the errors. Unfortunately, they 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 a three concentric circles target. Method Most algorithms of the eccentricity errors usually involve some geometric parameters, and these parameters always need calibration and Bundle Adjustment to obtain. 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, what’s more, they are on a line. The moment invariants of Zernike moment are used for the edge detection of the pixel level in order to get the precise positioning of sub-pixel level edge. Through the least-squares ellipse fitting, the center of ellipse is determined. For better results, the images of the concentric circles targets should include at least twenty pixels to ensure that there are enough effective edge points. The ellipse center is easily calculated with the sub-pixel level edge, then we can use the three groups of ellipse center to calculate the eccentricity error model. Since the three concentric circles in the error equation has six same parameters, corresponding parameters can be set into blocks as new variables, which will reduce to three unknown parameters. It’s enough to solve the error equations with the help of three concentric circles. Through the formulas which are derived in this paper, the eccentricity errors can be solved completely. It’s excellent to avoid obtaining the geometrical parameters and solve the nonlinear model. Result The possible solution for a correction of this systematic eccentricity error is proposed in this paper, and the method works well in improving the positioning accuracy of the circular target center. The simulation experiment results with MATLAB show that the eccentricity errors can be compensated from the level of pixel to the level of 10-11 pixel when the targets are toke photos in different angles, distances and sizes of the targets. This paper designs a target which has three concentric circles and diameters of 6cm,12cm,18cm. To calculate the true center of the circles, a circle which size is 2mm is designed in the central area of the target. In the process of 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 its eccentric error is only 0.02 pixels, it can be ignored to compared with the three concentric circles used in the experiment which is suitable to be regard as the true value in the experiment. The experiment results show that the measurement errors could be controlled in 0.1mm. Compared with the concentric circles method, its accuracy increases up to two times than before and decreases the eccentricity error about 80%. Conclusion This paper presents a new eccentricity compensation method to calculate eccentricity error by using three concentric circles target to add constraint. Compared with previous eccentricity error correction methods, additional parameters need to be estimated for correcting the eccentricity error. In consequence, it will increase the computation complexity and reduce convergence speed. It’s no need to know the geometric parameters of the measurement system regarding target and camera in advance, just only need the proportional relationship of circles and ellipse center coordinates. The experiment results show the efficiency of the proposed method for eccentricity error compensation. Since the algorithm can improve the location efficiency of circular targets, it will improve the effect in depth image matching which is based on non-coding markers, the precision of the automatic camera calibration method which is based on circular markers, and the robustness in the navigation and positional system.