## 2 Mechanism behind Thermal-Induced Measurement Errors  
In this section, the underlying physical mechanism behind thermal-induced measurement errors is discussed in detail. It is the theoretical principle for establishing temperature compensation methods. First, we determined the thermal behaviours of camera components that may lead to thermal-induced errors and established a physical model to illustrate the relationship between these thermal behaviours and the image error based on optical imaging relationships. Then, the thermal responses of each  
The thermal behaviours (\(\delta^{c}_{Z}\), \(A^{c}_{X}\), \(A^{c}_{Y}\), \(A^{c}_{Z}\), \(F^{c}_{Y}\), \(\delta^{m}_{Z}\), and \(\delta^{l}_{Z}\)) of the camera case, mount, and lens cause the variation in the space position and orientation of the camera coordinate system (\(O_{C}\)-\(X_{C}Y_{C}Z_{C}\)), which changes into (\(O_{C}\)-\(X_{C}Y_{C}Z_{C}\)), relative to the world coordinate system (\(O_{W}\)-\(X_{W}Y_{W}Z_{W}\)). The thermal behaviours (\(A^{CCD}_{X}\), \(A^{CCD}_{Y}\), \(A^{CCD}_{Z}\), \(t^{CCD}_{X}\), \(t^{CCD}_{Y}\), \(t^{CCD}_{Z}\), \(\delta^{CCD}_{X}\), \(\delta^{CCD}_{Y}\), \(\delta^{m}_{Z}\), and \(\delta^{l}_{Z}\)) of the CCD focal plane, mount, and lens lead to the variation in the position and orientation of the changed physics image coordinate system (\(O_{s^{\prime}}\)-\(X_{s^{\prime}}Y_{s^{\prime}}Z_{s^{\prime}}\)), which changes into (\(O_{s}\)-\(X_{s^{\prime}}\)-\(Y_{s^{\prime}}\)-\(Z_{s^{\prime}}\)) relative to the changed camera coordinate system (\(O_{C}\)-\(X_{C}Y_{C}Z_{C}\)). Based on the quantitative geometric analysis of the spatial position and orientation variation of these coordinate systems induced by the previously mentioned thermal behaviours, the coordinates (\(X^{O}_{s^{\prime\prime}}\), \(Y^{O}_{s^{\prime\prime}}\), \(Z^{O}_{s^{\prime\prime}}\)) and (\(X^{p}_{s^{\prime\prime}}\), \(Y^{p}_{s^{\prime\prime}}\), \(Z^{p}_{s^{\prime\prime}}\)) of the changed equivalent optical center \(O^{{}^{\prime}}\) and space point \(P\) are acquired in the changed physics image coordinate system (\(O_{s^{\prime}}\)-\(X_{s^{\prime}}\)-\(Y_{s^{\prime}}\)-\(Z_{s^{\prime\prime}}\)). Furthermore, point \(p^{\prime\prime}\) at which line \(\overline{PO^{\prime}}\) intersects plane \(X_{s}\)-\(O_{s^{\prime}}\)-\(Y_{s^{\prime\prime}}\) is the image point corresponding to the space point \(P\) after the thermal behaviours occur. Equation (2) expresses the relationship between the world coordinates (\(X^{p}_{W}\), \(Y^{p}_{W}\), \(Z^{p}_{W}\)) of the space point \(P\) and the pixel coordinates (\(x^{p^{\prime\prime}}_{s^{\prime\prime}}\), \(y^{p^{\prime\prime}}_{s^{\prime\prime}}\)) of the corresponding image point. \[\left[ \begin{array}{c}x^{p^{\prime\prime}}_{s^{\prime\prime}}\\ y^{p^{\prime}}_{s^{\prime\prime}}\\ 1\end{array} \right]=\frac{1}{Z^{p}_{s^{\prime\prime}}-Z^{O}_{s^{\prime\prime}} }\left[ \begin{array}{ccc}\frac{1}{dx^{\prime}}&0&C_{x}\\ 0&\frac{1}{dy^{\prime}}&C_{y}\\ 0&0&1\end{array} \right]\left[ \begin{array}{ccc}Z^{p}_{s^{\prime\prime}},&0&- X^{p}_{s^{\prime\prime}}\\ 0&Z^{p}_{s^{\prime\prime}},&-Y^{p}_{s^{\prime\prime}}\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\textbf{R}^{*}\textbf{R}^{*} \textbf{R}\left[\begin{array}{c}X^{P}_{W}\\ Y^{P}_{W}\\ Z^{P}_{W}\end{array}\right]+\textbf{R}^{{}^{\prime}}\textbf{T}+\textbf{T}^{{}^{ \prime}}\\ Z^{P}_{W}\end{array} \right] \tag{2}\]  
where  
\[\left[ \begin{array}{c}X^{p}_{s^{\prime\prime}}\\ Y^{p}_{s^{\prime\prime}}\\ Z^{p}_{s^{\prime\prime}}\end{array} \right]=\textbf{R}^{*}\textbf{R}^{*} \textbf{R}\left[ \begin{array}{c}X^{P}_{W}\\ Y^{P}_{W}\\ Z^{P}_{W}\end{array} \right]+\textbf{R}^{*}\left[ \begin{array}{c}0\\ 0\\ v^{\prime}\end{array} \right]+\textbf{R}^{*}\textbf{R}^{{}^{\prime}}\textbf{T}+ \textbf{R}^{*}\textbf{T}^{{}^{\prime}}+\textbf{T}^{{}^{\prime}} \tag{3}\]  
and  
\[\left[ \begin{array}{c}X^{O^{\prime}}_{s^{\prime\prime}}\\ Y^{O^{\prime}}_{s^{\prime\prime}}\\ Z^{O^{\prime}}_{s^{\prime\prime}}\end{array} \right]=\textbf{R}^{*}\left[