# Iterative Camera Calibration Method Based on Concentric Circle Grids
## 2 Materials and Methods
### Concentric Circle Center Location Method Based on Cross-Ratio Invariance
Therefore, the centers of the ellipses projected by concentric circles and the imaged centers are collinear points. Figure 2 shows a concentric circle feature point and the projection on the image plane.\(\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2}\) represent the centers of two ellipses, and \(\mathbf{P}_{u}\) is the projection of the concen  
tric circle center \(\mathbf{P}_{w}\). A line \(l_{u}\) is established connecting \(\mathbf{P}_{u},\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2}\), and this line intersects both ellipses at four different points \(\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2},\mathbf{P}_{u}^{3}\), and \(\mathbf{P}_{u}^{4}\). Based on the characteristics of projective geometry, there is a line \(l_{w}\) that passes through \(\mathbf{P}_{w}\) and intersects the concentric circle at four points \(\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2},\mathbf{P}_{u}^{3}\), and \(\mathbf{P}_{u}^{1}\),\(\mathbf{P}_{u}^{\infty}\) is the vanishing point on the line \(l_{w}\). Additionally, \(\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2},\mathbf{P}_{u}^{3},\mathbf{P}_{u}^{4},\mathbf{P}_{u}^{\infty}\) demonstrate the perspective projection relationship with \(\mathbf{P}_{u}^{1},\mathbf{P}_{u}^{2},\mathbf{P}_{u}^{3},\mathbf{P}_{u}^{4},\mathbf{P}_{u}^{\infty}\). With the properties of the harmonic conjugated points, the cross-ratio equations can be developed as follows:  
\[\left(\mathbf{P}_{w^{\prime}}^{1}\mathbf{P}_{w^{\prime}}^{2};\mathbf{P}_{w},\mathbf{P}_{w}^{ \infty}\right)=\left(\mathbf{P}_{w^{\prime}}^{3}\mathbf{P}_{w^{\prime}}^{4};\mathbf{P}_{w },\mathbf{P}_{w}^{\infty}\right)=-1. \tag{9}\]  
Because the cross-ratio remains unchanged under projective transformation, we obtain:  
\[B ={h_{12}}^{2}+{h_{22}}^{2}-{h_{32}}^{2}r^{2}\]0  
In Equation (10), \(\mathbf{P}_{u^{\prime}}^{1},\mathbf{P}_{u}^{2},\mathbf{P}_{u}^{3},\mathbf{P}_{u}^{4}\) can be obtained by solving equations of the line and ellipses, and \(\mathbf{P}_{u}\) is the imaged center. We choose the point that lies inside the concentric circle as the imaged center. In the perspective projection, when the image plane and the circle target plane are not parallel, a circle is imaged as an ellipse. There is a deviation between the ellipse center and the projection of the circle center, which is called the eccentricity error. The eccentricity error is only related to the angle between the image plane and the target plane. Figure 3 shows the mathematical model of the concentric circle; \(r_{in},r_{out}\) are the radius of the concentric circle; \(err=\left\|\mathbf{P}_{u}-\mathbf{P}_{u}^{1}\right\|\) is the eccentricity error of the concentric circle. \[B ={h_{12}}^{2}+{h_{22}}^{2}-{h_{32}}^{2}r^{2}\]1 Mathematical model of the concentric circle. \[B ={h_{12}}^{2}+{h_{22}}^{2}-{h_{32}}^{2}r^{2}\]2 Concentric circle feature point. The process of locating the imaged center of a concentric circle can be summarized in the following steps:  
1. Apply the sub-pixel edge detection algorithm to fit the projection conics and estimate the coefficient matrices. Traditional edge detection methods include the Roberts operator, Laplace operator, Prewitt operator, and Canny operator. These methods can only locate to the pixel level and cannot meet the requirements of accurate location. To meet the high precision location requirements of the camera calibration system, we use a Canny-Zernike algorithm to detect the edge points. Firstly, the Canny operator is used to preliminarily locate the edge points. Then, the Zernike moment algorithm is applied in sub-pixel edge extraction which improves the location accuracy. The centers and coefficient matrices of ellipses are obtained using the direct least square method. 2. Identify the intersection points between the line that traverses the ellipses' centers and the two ellipses. Because the centers of the ellipses projected by concentric circles and the imaged centers are collinear points, we connect the two ellipses centers to establish a straight line. By combining the line equation and the projection conic equations, we obtain four intersection points. 3.