# Improved Direct Linear Transformation for Parameter Decoupling in Camera Calibration
## 2 Parameter Coupling Analysis
### Error Coupling Analysis  
The two equations in Equation (2) have the same form. We thus analyze the first equation; all conclusions can apply to the second equation. We assume that \(u_{0}\) has error \(\triangle u_{0}\) and that \(t_{x}\) has error \(\triangle t_{x}\). From Equation (2), we have:  
\[ \begin{split}\Delta u^{\prime}_{i}&=u_{i}-(u_{0}+ \Delta u_{0})-a_{x}\frac{r_{11}X_{i}+r_{12}Y_{i}+r_{13}Z_{i}+(t_{x}+\Delta t_{x })}{r_{13}X_{i}+r_{32}Y_{i}+r_{33}Z_{i}+t_{z}}\\ &=\Delta u_{i}-(\Delta u_{0}+\frac{d_{x}d_{y}}{r_{31}X_{i}+r_{32} Y_{i}+r_{33}Z_{i}+t_{z}})\end{split}  \tag{4}\]  
when \(\Delta t_{x}=-(r_{31}X_{i}+r_{32}Y_{i}+r_{33}Z_{i}+t_{z})/a_{x}\Delta u_{0}\), we have \(\Delta u^{\prime}_{i}=\Delta u_{i}\). The image re-projection error will not change. However, because the space point coordinate values are different, other image point re-projection errors will be changed. We assume that:  
\[ \begin{split}&\Delta t_{x}=-\frac{r_{13}X_{1}+r_{32}Y_{1}+r_{33} Z_{1}+t_{z}}{a_{x}}\Delta u_{0}\\ &=-\frac{r_{31}X_{i}+r_{32}Y_{i}+r_{33}Z_{i}+t_{z}}{a_{x}}\Delta u _{0}-\frac{r_{31}dX_{i}+r_{32}dY_{1}+r_{33}dZ_{i}}{a_{x}}\Delta u_{0}\end{split}  \tag{5}\]  
where \(dX_{i}=X_{1}-X_{i}\), \(dY_{i}=Y_{1}-Y_{i}\), \(dZ_{i}=Z_{1}-Z_{i}\), and image re-projection errors are given by:  
\[ \begin{split}\Delta u^{\prime}_{i}&=\Delta u_{i}+ \Delta u^{\prime\prime}_{i}\text{ with }\Delta u^{\prime\prime}_{i}=\frac{r_{31}dX_{i}+r_{32}dY_{i}+r_{33}dZ_{i}}{r_{3 1}X_{i}+r_{32}Y_{i}+r_{33}Z_{i}+t_{z}}\Delta u_{0}\end{split}  \tag{6}\]  
Based on Equation (6), if \(dZ_{i}\) is very small and \(t_{z}\) is very large, the change in image re-projection errors will be very small. This is because coupling \(\triangle u_{0}\) and \(\triangle t_{x}\) reduces the image re-projection error. In order to reduce the coupling effect, better results can be obtained if the 3D template is large and close to the camera. In fact, the template size is limited, and the calibration distance is limited by the field of view and the lens parameter. With respect to focal distance and translation vector, we assume \(a_{x}\) has error \(\triangle a_{x}\) and \(t_{z}\) has error \(\triangle t_{z}\). From Equation (2), we have:  
\[ \begin{split}\Delta u^{\prime}_{i}&=u_{i}-u_{0}- \frac{(a_{x}+\Delta a_{x})C_{i}}{B_{i}+\Delta t_{z}}\end{split}  \tag{7}\]  
where \(B_{i}=r_{31}X_{i}+r_{32}Y_{i}+r_{33}Z_{i}\) and \(C_{i}=r_{11}X_{i}+r_{12}Y_{i}+r_{13}Z_{i}\). We assume that \(\Delta a_{x}/\Delta t_{z}=a_{x}/B_{1}\), and we obtain \(\Delta u^{\prime}_{1}=\Delta u_{1}\). Other image re-projection errors are:  
\[ \begin{split}\Delta u^{\prime}_{i}&=\Delta u_{i}+ \Delta u^{\prime\prime}_{i}\text{ with }\Delta u^{\prime\prime}_{i}=-\frac{a_{x}C_{i}\Delta t_{z}}{B_{1}(B_{i}+ \Delta t_{z})}\approx-\frac{a_{x}C_{i}}{B_{i}}\times\frac{\Delta t_{z}}{B_{1}} \end{split}  \tag{8}\]From Equation (1), we have \(a_{x}C_{i}/B_{i}\approx(u_{i}-u_{0})\). We assume that the Euler angles of the world coordinate system relative to the camera coordinate system are zero. By setting the first space point \(Z_{1}=~{}1\), we have:  
\[\Delta{u^{\prime\prime}}_{i}=-\frac{u_{i}-u_{0}}{t_{z}}\Delta t_{z} \tag{9}\]  
Based on Equation (9), if \((u_{i}-u_{0})\) is small and \(t_{z}\) is large, the change in the image re-projection error will be very small. This is because coupling \(\triangle a_{x}\) and \(\triangle t_{z}\) reduces the image re-projection error. To reduce the coupling effect, better results can be obtained if the 3D template is closer to the camera. Similar to the distortion, the camera focal distance and translation vector coupling error are related to the image point location. Therefore, under the influence of distortion, the camera focal distance and translation vector coupling error may increase the calibration error. The coupling error is present in the geometric model. The camera model is also called a pinhole model; that is, the space point, image point and camera origin are located on the same line.