# Study of the Error Caused by Camera Movement for the Stereo-Vision System
## 3 Camera Movement Errors
### Identical Translations  
Figure 2 shows the relationship between a pair of camera coordinate systems before and after identical translations. In this case, the global world coordinate system experiences rigid body translations. The translation amount is denoted by \(\Delta\mathbf{T}_{w}=\left[\Delta T_{wx},\ \Delta T_{wy},\ \Delta T_{wz}\right]^{T}\). Let the world coordinate of any point, before and after identical translations, be \(\ X_{w}(X_{w},Y_{w},\ Z_{w})\) and \({}^{\prime}_{w}(X^{\prime}_{w},Y^{\prime}_{w},\ Z^{\prime}_{w})\), respectively. \(\Delta\mathbf{T}_{w}\) is considered to be the error introduced in the initial 3D coordinate \(\ X_{w}\). Then, the 3D coordinate error \(\Delta\mathbf{X}_{w}\) is described as follows.  
\[\Delta\mathbf{X}_{w}=\left[\begin{array}{c}X^{\prime}_{w}\\ Y^{\prime}_{w}\\ Z^{\prime}_{w}\end{array}\right]-\left[\begin{array}{c}X_{w}\\ Y_{w}\\ Z_{w}\end{array}\right]=\left[\begin{array}{c}\Delta T_{wx}\\ \Delta T_{wy}\\ \Delta T_{wz}\end{array}\right], \tag{4}\]From Equation (4), the 3D coordinate error \(\Delta\mathbf{X}_{w}\) is equal to the identical translation \(\Delta T_{w}\).  
Suppose that any two points on the measured object are defined as \(\mathbf{X}_{w1}(X_{w1},\,Y_{w1},\,\,Z_{w1})\) and \(\mathbf{X}_{w2}(X_{w2},\,Y_{w2},\,\,Z_{w2})\) in the initial world coordinate system. After camera movements, \(\mathbf{X}^{\prime}_{w1}(X^{\prime}_{w1},\,Y^{\prime}_{w1},\,Z^{\prime}_{w1})\) and \(\mathbf{X}^{\prime}_{w2}(X^{\prime}_{w2},\,Y^{\prime}_{w2},\,Z^{\prime}_{w2})\) are the corresponding points in the new world coordinate system. The distances between the two selected points before and after the identical translations are expressed as  
\[\left\{\begin{array}{l}L_{w}=\sqrt{\left(X_{w1}-X_{w2}\right)^{2}+\left(Y_{ w1}-Y_{w2}\right)^{2}+\left(Z_{w1}-Z_{w2}\right)^{2}}\\ L^{\prime}_{w}=\sqrt{\left(X^{\prime}_{w1}-X^{\prime}_{w2}\right)^{2}+\left(Y ^{\prime}_{w1}-Y^{\prime}_{w2}\right)^{2}+\left(Z^{\prime}_{w1}-Z^{\prime}_{ w2}\right)^{2}}\end{array}\right., \tag{5}\]  
For the two pairs of points, viz., \(\mathbf{X}_{w1}\) and \(\mathbf{X}^{\prime}_{w1},\,\mathbf{X}_{w2}\) and \(\mathbf{X}^{\prime}_{w2}\), the following relations from Equation (4) can be obtained as  
\[\left\{\begin{array}{l}X^{\prime}_{w1}=X_{w1}+\Delta T_{wx}\\ Y^{\prime}_{w1}=Y_{w1}+\Delta T_{wy}\\ Z^{\prime}_{w1}=Z_{w1}+\Delta T_{wz}\end{array}\right.,\left\{\begin{array}{ l}X^{\prime}_{w2}=X_{w2}+\Delta T_{wx}\\ Y^{\prime}_{w2}=Y_{w2}+\Delta T_{wy}\\ Z^{\prime}_{w2}=Z_{w2}+\Delta T_{wz}\end{array}\right. \tag{6}\]  
Combining Equations (5) and (6) yields the following expression  
\[L_{w}=L^{\prime}_{w} \tag{7}\]  
It was concluded that the distance between any two points remains unchanged after identical translations.  
Displacement and strain form an important group of parameters concerning the deformation measurement. Displacement represents the change of position of an object in a certain period of time. Strain refers to the ratio of the change in length of a small line segment to the original length. From Equations (4) and (7), the error in measuring the displacement should be equal to the amount of identical translations, while the strain measurement is not affected.