# Iterative Camera Calibration Method Based on Concentric Circle Grids
## 1 Introduction
The method iteratively finds the solutions without any parameter information. Nevertheless, it is important to verify the efficiency and stability of this method. Based on the characteristics of tangent lines and the Hough transform, [PERSON] et al. [36] utilized four tangent lines of the concentric circle to determine a straight line, where the projected center is located on this line. By finding several straight lines, the intersection point of these lines can be identified as the projected center. In [37], the imaged circle center and the vanishing line were recovered by exploring the properties of the common self-polar triangle of concentric circles. But this method requires calculating the inversion and eigenvectors of the matrix. [PERSON] et al. [38] investigated the geometric and algebraic constraints associated with the projected concentric circle. They introduced a rank-1 constraint to determine the center by solving a quartic equation. However, this approach may lead to numerical instability. And [PERSON] et al. [39] formulated the solution of the imaged circle center into a first-order polynomial eigenvalue problem by considering the pole-polar relationship based on the concentric circle. Compared with the previous work of [PERSON], this technique reduces computation complexity. To minimize the perspective projection error, [PERSON] et al. [40] confirmed the position of the imaged circle center by computing the eigenvectors of the concentric circle projection matrix. The aforementioned research address the problem of center deviation of circular feature points in the linear perspective projection. However, due to lens distortion, the reprojection error function established in calibration is nonlinear [41; 42]. Thus, it is crucial to consider the influence of lens distortion on calibration accuracy. [PERSON] [43] proposed a calibration method that takes into account radial distortion, which is rapid and not affected by the local minimum problem. While the radial component of lens distortion is dominant, it is also coupled with the tangential component, and thus both should be considered. [PERSON] [44] corrected the images and calculated the lens distortion separately from the intrinsic and extrinsic parameters of the camera. However, this algorithm is more complex. [PERSON] et al. [41] discussed the impact of lens distortion and presented a robust geometric camera calibration method. The objective of compensation is to reduce the distance between the model and the actual position by adjusting all camera parameters. [PERSON] et al. [41] analyzed lens distortion in camera imaging and designed a template that consists of a line through the center of the concentric circle. Orthogonal vanishing points were utilized to solve the camera's intrinsic parameters. Additionally, they optimizedthe positions of the distortion points to approach the ideal points, thereby enhancing the accuracy of camera calibration. In the circle grids target-based calibration method, two sources affect the accuracy of camera calibration, namely the high-precision location of the circle center and the optimization of camera parameters. To address the above limitations, this paper proposes an improved camera calibration method that combines a concentric circle center location algorithm and an iterative compensation algorithm for the calibration parameters. The imaged concentric circle center obtained by cross-ratio invariance is not affected by the perspective projection, which ensures the location accuracy of the feature point. In addition, to achieve higher calibration accuracy, an iterative compensation framework is developed to refine the calibration results. The remainder of this paper is organized as follows. Section 2 presents the concentric circle center location method, which is based on the invariance of the cross-ratio, and introduces a novel iterative approach for camera calibration that takes into account the influence of lens distortion.