# Significance of Camera Pixel Error in the Calibration Process of a Robotic Vision System
## 2 Experimental Setup and Measurement Uncertainties
### Camera to Object Calibration  
Consider a 3D point \(P\), represented by homogeneous coordinates \((x,y,z,1)^{T}\), and its corresponding 2D image projection point \(p\) represented by homogeneous coordinates \((\alpha,\beta,1)^{T}\). Assuming a pinhole camera model the mapping between 3D point and the corresponding 2D image point can be defined as:  
\[s[\alpha,\beta,1]^{T} =M\cdot[x,y,z,1]^{T} \tag{13}\] \[=\begin{bmatrix}m_{11}&m_{12}&m_{13}&m_{14}\\ m_{21}&m_{22}&m_{23}&m_{24}\\ m_{31}&m_{32}&m_{33}&m_{34}\end{bmatrix}\cdot[x,y,z,1]^{T}\] (14) \[=K[R_{Q}|T_{Q}]\cdot[x,y,z,1]^{T}, \tag{15}\]  
where \(s\) is a scaling factor, \(M\) is the camera projection matrix of dimension \(4\times 3\), \(K\) is \(3\times 3\) camera intrinsic matrix, and \([R_{Q}|T_{Q}]\) is a \(3\times 4\) extrinsic camera matrix, with the first three columns \(R_{Q}\) representing rotation and the last column \(T_{Q}\) representing translation, relating object coordinate frame to the camera coordinate frame.  
Finally the 2D image point coordinates \(\alpha\) and \(\beta\) are calculated as:  
\[\alpha =\frac{m_{11}x+m_{12}y+m_{13}z+m_{14}}{m_{31}x+m_{32}y+m_{33}z+m_{34}} \tag{16}\] \[\beta =\frac{m_{21}x+m_{22}y+m_{23}z+m_{24}}{m_{31}x+m_{32}y+m_{33}z+m_{ 34}}. \tag{17}\]  
To find the matrices \(K\), \(R_{Q}\) and vector \(T_{Q}\), a camera calibration is required [7]. For that process a known 3D points in the object coordinate frame (e.g., checkerboard pattern) and their corresponding 2D image points are often used.  
While performing this type of calibration, various factors can influence its accuracy, examples include: camera resolution, sensor noise, inaccurate feature point, instability of camera mounting device, accuracy of the numerical technique used etc. One of the most feasible way to estimate the uncertainty associated with this type of calibration is to evaluate the re-projection errors for each image. A re-projection error is known as a quantitative measure of accuracy, which can be evaluated by estimating the distance between a feature point detected in a calibration object (checkerboard), and a corresponding world point projected into the same image. Ideally the value of re-projection error should be close to zero.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Kuka Link** & \(\mathbf{\alpha}^{0}\pm\mathbf{\Delta}\mathbf{\alpha}^{0}\) & \(\mathbf{a}\pm\mathbf{\Delta}\mathbf{a}\) **(mm)** & \(\mathbf{\theta}^{0}\pm\mathbf{\Delta}\mathbf{\theta}^{0}\) & \(\mathbf{d}\pm\mathbf{\Delta}\mathbf{d}\) **(mm)** \\ \hline
1 & \(90-0.013\) & \(25+0.009\) & \(\theta_{1}+0\) & \(0\) \\
2 & \(0+0.005\) & \(455+0.447\) & \(\theta_{2}+0.005\) & \(0+0.010\) \\
3 & \(90+0.008\) & \(35+0.172\) & \(\theta_{3}-89.990\) & \(0\) \\
4 & \(-90+0.041\) & \(0-0.104\) & \(\theta_{4}-0.083\) & \(-420-0.341\) \\
5 & \(90-0.001\) & \(0+0.081\) & \(\theta_{5}+0.038\) & \(0+0.056\) \\
6 & \(0\) & \(0\) & \(\theta_{6}+0\) & \(-80+0\) \\ \hline \end{tabular}
\end{table}
Table 2: Parameter values of KUKA robot with offset (\(\Delta\)).