## 2 Mechanism behind Thermal-Induced Measurement Errors
### Development and Verification of the Image Error Model for a Specific Camera System
The CCD focal plane is generally fabricated using silica-based materials that have a very small thermal expansion coefficient. The thermal expansion coefficient has a magnitude of approximately \(10^{-7}\)\({}^{\circ}\)C\({}^{-1}\). Considering that the temperature variation range of the focal plane is of the order of \(10\)\({}^{\circ}\)C, the relative distortion in the captured image induced by the deformations of the CCD focal plane is of the order of \(10^{-6}\), which will lead to a maximum pixel drift of 0.001 pixels. Take a 1000 by 1000 resolution as an example. Therefore, the influence of the thermal deformations (\(\delta_{X}^{CCD}\) and \(\delta_{Y}^{CCD}\)) of the CCD focal plane on image acquisition can be neglected. Combined with the above experimental results as well as theoretical analysis, the thermal behaviours that generate considerable image errors are given in Table 2. One part of the behaviours (\(\delta_{Z}^{c}\), \(\delta_{Z}^{m}\), \(\delta_{Z}^{l}\), and \(t_{Z}^{CCD}\)) introduced image distortion errors, and another part (\(t_{X}^{CCD}\) and \(t_{Y}^{CCD}\)) introduced image translation errors. Introducing both parts of the thermal behaviours into Equation (2) and combining this with Equation (1), we established the image distortion error model, expressed as:  
\[\alpha_{e}=\frac{(Z_{C}^{P}+v)(\delta_{Z}^{c}+\delta_{Z}^{m}+\delta_{Z}^{l})- Z_{C}^{P}\delta_{Z}^{CCD}}{Z_{C}^{P}v-v(\delta_{Z}^{c}+\delta_{Z}^{m}+\delta_{Z}^{l})} =\frac{(Z_{C}^{P}+v)(\lambda_{C}^{c}\Delta T_{c}+\lambda_{Z}^{m} \Delta T_{m}+\lambda_{Z}^{l}\Delta T_{l})-Z_{C}^{P}\delta_{Z}^{CCD}\Delta T_{ c}}{Z_{C}^{P}v-v(\lambda_{Z}^{c}\Delta T_{c}+\lambda_{Z}^{m}\Delta T_{m}+ \lambda_{Z}^{l}\Delta T_{l})} \tag{11}\]  
Figure 7: Thermal responses of rotations and translations of the CCD focal plane: (**a**) experimental arrangement for measuring out-plane thermal behaviours, (**b**) experimental arrangement for measuring in-plane thermal behaviours, and (**c**) responses of thermal behaviours to the case temperature. and image translation error model, expressed as:  
\[\left\{ \begin{array}{l}\Delta x_{e}=\frac{t_{X}^{CCD}}{dx}=\frac{k_{X}^{CCD} \Delta T_{c}}{dy}\\ \Delta y_{e}=\frac{t_{Y}^{CCD}}{dy}=\frac{k_{Y}^{CCD}\Delta T_{c}}{dy}\end{array} \right. \tag{12}\]  
In this case, \(\lambda_{Z}^{c}\), \(\lambda_{Z}^{m}\), and \(\lambda_{Z}^{l}\) are the thermal deformation parameters of the camera case, mount, and lens, respectively. \(k_{X}^{CCD}\), \(k_{Y}^{CCD}\), and \(k_{Z}^{CCD}\) are the thermal translation parameters of the CCD focal plane along \(X\), \(Y\), and \(Z\) directions, respectively. \(\Delta T_{c}\), \(\Delta T_{m}\), and \(\Delta T_{l}\) represent the temperature variations in the camera case, mount, and lens, respectively. These can be calculated as:  
\[\left\{ \begin{array}{l}1-R_{1}(\Delta T_{ikb}-\Delta T_{c})=K_{1}\frac{d( \Delta T_{ikb})}{dt}\\ R_{1}(\Delta T_{ikb}-\Delta T_{c})-R_{2}(\Delta T_{c}-\Delta T_{e})-R_{3}( \Delta T_{c}-\Delta T_{m})=K_{2}\frac{d(\Delta T_{c})}{dt}\\ R_{3}(\Delta T_{c}-\Delta T_{m})-R_{4}(\Delta T_{m}-\Delta T_{e})-R_{5}( \Delta T_{m}-\Delta T_{l})=K_{3}\frac{d(\Delta T_{m})}{dt}\\ R_{5}(\Delta T_{m}-\Delta T_{l})-R_{6}(\Delta T_{l}-\Delta T_{e})_{l}=K_{4} \frac{d(\Delta T_{l})}{dt}\end{array} \right. \tag{13}\]  
This was determined in our previous investigation [14]. Equation (13) expresses the relationship between the environmental temperature, temperature of the camera components, and thermal parameters (\(R_{1}\)\(\sim\)\(R_{6}\) and \(K_{1}\)\(\sim\)\(K_{4}\)). The camera system investigated in this study is the same as that used in an earlier study [14]. Therefore, the previously mentioned thermal parameters should also be the same, which are listed in Table 3.  
Next, we carried out an experiment to verify the correctness of these models developed using the investigated camera system.