# Iterative Camera Calibration Method Based on Concentric Circle Grids
## 3 Experimental Results and Analysis
### Synthetic Experiments  
We first tested the effect of the deflection angle between the object plane and the image plane on circle center location accuracy. The camera settings were \(u_{0}=1045\) pixels, \(v_{0}=1010\) pixels, \(f_{u}=1507\) pixels, and \(f_{v}=1507\) pixels, and the radii of the simulation concentric circle feature point were \(10\)mm and \(20\)mm. The pitch angle \(\theta_{x}\), yaw angle \(\theta_{y}\) and rotation angle \(\theta_{z}\) represented the object plane deflections within the range of \(-1\) rad to \(1\) rad along the \(x\)-axis, \(y\)-axis, and \(z\)-axis of the camera coordinate system, and the step was \(0.1\) rad. The translation vector \(\mathbf{T}\) was set to \([0,0,1500]^{\mathrm{T}}\) and the rotation matrix \(\mathbf{R}\) was calculated using \(\theta_{x}\), \(\theta_{y}\), and \(\theta_{z}\). After selecting \(10\) points on each circle, based on the camera model, the projection points were calculated, which were used to fit the projection conics. Therefore, the boundary points and the fitted ellipses in the simulation images were obtained. The program was implemented in Python. The residual error between the calculated projection and the real projection of the circle center was used as the evaluation criteria to evaluate the accuracy of the location method. Figure 6a, Figure 7a, and Figure 8 show the residual error in the \(u\) and \(v\) directions by using the conventional method [34] and the proposed method.  
Figure 6: Effect of pitch angle on location accuracy. **(a)** The conventional and the proposed method; **(b)** enlarged view of the proposed method.  
Figure 7: Effect of yaw angle on location accuracy. **(a)** The conventional and the proposed method; **(b)** enlarge view of the proposed method.  
As indicated in Figures 6a and 7a, the conventional method is greatly affected by the deflection angles of the object plane around the \(x\)-axis, \(y\)-axis, and the maximum residual error can reach 0.39 pixels and 0.45 pixels, respectively. Figures 6b and 7b present the residual error of the proposed method. They are independent of the deflection angle. In Figure 8, the influence of the deflection angle around the \(z\)-axis on the two methods can be ignored.  
Subsequently, the rotation angles were all set to 1 rad. The simulated image was subjected to the addition of Gaussian noise with a mean of zero and a standard deviation that \(\sigma\) varies between 0 and 2 pixels. The final result was obtained by taking the average error from 100 independent experiments conducted for each noise level.  
As we can see from Figure 9, the location accuracy of the two methods decreases with the increase in image noise. Compared with the conventional method, the proposed method can ensure the center location accuracy under a large rotation angle and high image noise, and the residual error of the proposed method is always less than 0.26 pixels.  
Then, a concentric circle planar calibration target was generated. The camera parameters were kept unchanged, and the lens distortion was set at \(k_{1}=-0.1,k_{2}=0.08\)  
Figure 8: Effect of rotation angle on location accuracy.  
Figure 9: Effect of Gaussian noise on location accuracy.  
\(P_{1}=-0.0005\), \(P_{2}=-0.0005\) as a slight distortion and at \(k_{1}=-0.4,k_{2}=0.3\), \(P_{1}=-0.002\), \(P_{2}=-0.002\) as a heavy distortion. We simulated the calibration images under the same poses and Gaussian noise. On this basis, the Zhang method, the conventional method, and the proposed method were compared. The calibration results are listed in Tables 1 and 2. The proposed method does not exhibit a noticeable improvement effect in the presence of slight lens distortion. As the lens distortion becomes heavier, the result achieved using the proposed approach becomes more accurate.