## 2 Mechanism behind Thermal-Induced Measurement Errors
### Development and Verification of the Image Error Model for a Specific Camera System  
In this subsection, we discuss the thermal responses of the selected camera system, which consists of an IPX-16M3-L CCD camera and Sigma macro 105-mm F2.8 DG lens, to temperature variations. We assess the influence of these thermal responses on image acquisition and develop the specific thermal-induced image error model using the investigated camera system. Figure 5a shows the experimental arrangement for measuring the responses of the rotations (\(A_{X}^{c}\), \(A_{Y}^{c}\), and \(A_{Z}^{c}\)) and translation (\(\delta_{Z}^{c}\)) of the camera case to temperature variations under two types of fixed modes, i.e., a simple fixed mode with a single contact point and an improved fixed mode with multiple contact planes. The thermal rotations and translation were measured using a high-precision laser displacement sensor (LDS, LK-H050, precision of \(\pm 1\)\(\upmu\)m) during self-heating of the camera. Meanwhile, the temperature of the camera case was measured using a thermal sensor. The thermal rotations (\(A_{X}^{c}\), \(A_{Y}^{c}\), and \(A_{Z}^{c}\)) and translation (\(\delta_{Z}^{c}\)) are plotted as a function of the temperature of the camera case in Figure 5b. Compared to the simple fixed mode, the improved fixed mode can effectively eliminate all the thermal rotations and translation of the camera case, and further the adverse influence on image acquisition. Neither of the two types of fixed modes can eliminate the thermal deformations (\(\delta_{Z}^{c}\), \(\delta_{Z}^{m}\), and \(\delta_{Z}^{l}\)) of the camera case, mount, and lens along the optical axis. In our previous investigation [14], the thermal responses (\(\delta_{Z}^{c}\), \(\delta_{Z}^{m}\), and \(\delta_{Z}^{l}\)) of these components of the investigated camera system to temperature variations were measured, as shown in Figure 6.  
Figure 4: Pixel drifts under real experimental conditions. Figure 5: _Cont._Under the improved fixed mode, the responses of rotations and translations of the CCD focal plane to the temperature of the camera case were measured during self-heating of the camera, respectively, as shown in Figure 7a,b. The laser displacement sensor was used to measure the displacement of the CCD focal plane at different points shown in Figure 7a to calculate the out-plane rotations (\(A_{X}^{CCD}\) and \(A_{Y}^{CCD}\)) and out-plane translation (\(t_{Z}^{CCD}\)). To calculate the in-plane rotation (\(A_{Z}^{CCD}\)) and in-plane translations (\(t_{X}^{CCD}\) and \(t_{Y}^{CCD}\)), the CCD focal plane was imaged using a reached thermal-balance camera, and its position was recognised via a matching algorithm, as shown in Figure 7b. The thermal responses of the CCD focal plane were plotted as a function of the temperature of the camera case, as shown in Figure 7c. The thermal translations (\(t_{X}^{CCD}\), \(t_{Y}^{CCD}\), and \(t_{Z}^{CCD}\)) of the CCD focal plane vary linearly with the temperature of the camera case and introduce considerable image errors. The thermal rotations (\(A_{X}^{CCD}\) and \(A_{Z}^{CCD}\)) present randomness with variations in the case temperature, and the mean values of these rotations were extremely small and hardly caused image errors. The rotation (\(A_{Y}^{CCD}\)) presents a significant positive correlation with the case temperature, but the maximum error in the pixel drift caused by the maximum rotation during variations in the case temperature has a magnitude of \(\sim\)0.001 pixel under normal experimental conditions. This is reasonably negligible compared to the measurement resolution of 1/100 of a pixel in photo mechanics measurements. Figure 5: Experimental investigation of thermal behaviours of the camera case under different fixed modes: (**a**) schematic of experimental setup, and (**b**) variation in thermal behaviours with the temperature of the camera case. Figure 6: Thermal responses of deformations of the camera case, mount, and lens.