# Iterative Camera Calibration Method Based on Concentric Circle Grids
## 1 Introduction  
Camera calibration is a crucial technology in the fields of vision measurement [1; 2; 3; 4; 5], robot navigation [6; 7; 8; 9; 10; 11], and 3D reconstruction [12; 13; 14; 15; 16]. To acquire precise measurement information, the accurate estimation of camera parameters is essential. Camera calibration parameters include intrinsic parameters and extrinsic parameters. Intrinsic parameters include main point coordinates, effective focal length, and distortion parameters, which represent the characteristics of the camera. Extrinsic parameters include the translation vector and rotation matrix, which represent the relative position relationship between the world coordinate system and the camera coordinate system. Schemes for calibration can be classified into two categories: self-calibration [17; 18] and target-based calibration. Self-calibration methods obtain the camera parameters by establishing multivariate equations of corresponding feature points from different viewpoints. These methods are flexible and do not require calibration targets. However, they are only effective in well-textured scenes. Moreover, in scenarios where the camera motion constraint is absent, the self-calibration method is has been proven to be inadequate for engineering applications. Additionally, this approach can hardly achieve the precise sub-pixel extraction of feature points in calibration images. On the contrary, the target-based calibration method can yield reliable calibration results and can be applied to various scenes. Calibration targets can be divided into the following three categories: 1D targets [19; 20; 21], 2D targets [22; 23; 24], and 3D targets [25; 26]. The 1D target consists of several collinear points. Due to the ease of capturing feature points on the target using multiple cameras simultaneously, it is commonly used in multi-camera calibration. However, the 1D target lacks strong spatial geometric constraints, resulting in low calibration accuracy. Traditional 3D targets typically consist of two or three orthogonal planes, which aid in achieving higher calibration accuracy. However, designing such targets may be challenging and expensive. Especially in the case of some large measurement scenes, this calibration method is more restricted by space. The sphere is used as a simple 3D target due to its strong contour continuity, which helps compensate for distortion caused by large viewing angles [27]. It is important to note that the size and distance of the sphere significantly affect calibration results. Additionally, there are only a few feature points involved in the calibration process. Compared to 1D and 3D targets, designing a 2D target is relatively easier. This is because calibration patterns can be printed on a planar board, allowing for the easy manufacturing of high-precision targets. [PERSON]'s chessboard calibration method [28] is considered one of the most representative 2D target-based calibration methods. This approach realizes the conversion between the world coordinate system and the image coordinate system by extracting Harris corners from chessboard images. The initial values of the camera parameters are obtained using a linear model. Subsequently, the reprojection error function is established, and the Levenberg-Marquardt algorithm is utilized for optimization. However, the detection accuracy of chessboard corner points is easily affected by image noise, distortion, and resolution. On the other hand, circular feature points offer the advantages of easy recognition and noise resistance, making them commonly employed in camera calibration and measurement [29; 30; 31; 32; 33; 34]. In perspective projection, a circle is imaged as an ellipse. The projection of the space of the circle center does not coincide with the ellipse center. To address this issue, many methods have been investigated to accurately locate the imaged circle center. [PERSON] et al. [35] introduced a novel constructive geometric method for estimating the imaged center and optimizing it using homological constraints.