## 2 Sensor Calibration
### Camera Parameter Calibration
#### 2.1.3 Inclined Camera Imaging Model  
To optimize the imaging optical path of the sensor according to the Scheimpflug principle, it can be treated equivalently to the image plane of an oblique camera. Based on this principle, an oblique camera model is established, as shown in Figure 3. The actual oblique image plane \(\Pi_{\mathrm{S}}\) can be obtained by rotating an ideal non-tilted image plane \(\Pi_{\mathrm{C}}\) around the principal point \(o\). This rotation relationship \(\mathbf{R_{\mathrm{\tau}}}\) can be decomposed into consecutive rotations around the \(x_{\mathrm{c}}\)-axis and the \(y_{\mathrm{c}}\)-axis of the image physical coordinate system \(o-x_{\mathrm{c}}y_{\mathrm{c}}\). The rotation angles \(\tau_{x}\) and \(\tau_{y}\), are expressed as follows:  
\[\mathbf{R_{\mathrm{\tau}}}= \begin{bmatrix}\cos\tau_{y}&0&\sin\tau_{y}\\ 0&1&0\\ -\sin\tau_{y}&0&\cos\tau_{y}\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&\cos\tau_{x}&-\sin\tau_{x}\\ 0&\sin\tau_{x}&\cos\tau_{x}\end{bmatrix}  \tag{5}\]  
After tilting the image plane, \(P_{\mathrm{W}}\) will be projected as point \(P_{\mathrm{S}}\) on the tilted image plane, establishing the oblique camera coordinate system \(O_{\mathrm{S}}-X_{\mathrm{S}}Y_{\mathrm{S}}Z_{\mathrm{S}}\) and the oblique image physical coordinate system \(o-x_{\mathrm{s}}y_{\mathrm{s}}\). Based on the collinear relationship of points \(P_{\mathrm{W}}\), \(P_{\mathrm{C}}\), and \(P_{\mathrm{S}}\), as well as the rotation matrix \(\mathbf{R_{\mathrm{\tau}}}\), the transformation relationship between the coordinates \(P_{\mathrm{S}}(X_{\mathrm{S}},Y_{\mathrm{S}},Z_{\mathrm{S}})\) in \(O_{\mathrm{S}}-X_{\mathrm{S}}Y_{\mathrm{S}}Z_{\mathrm{S}}\) and the coordinates \(P_{\mathrm{C}}(X_{\mathrm{C}},Y_{\mathrm{C}},Z_{\mathrm{C}})\) in \(O_{\mathrm{C}}-X_{\mathrm{C}}Y_{\mathrm{C}}Z_{\mathrm{C}}\) can be obtained [12], as follows:\[X_{\text{S}}^{\prime}=\frac{X_{\text{S}}}{Z_{\text{S}}},\;Y_{\text{S}}^{\prime}= \frac{Y_{\text{S}}}{Z_{\text{S}}},\;X_{\text{C}}^{\prime}=\frac{X_{\text{C}}}{Z_ {\text{C}}},\;Y_{\text{C}}^{\prime}=\frac{Y_{\text{C}}}{Z_{\text{C}}},\]  
\[\lambda_{1} \begin{bmatrix}X_{\text{S}}^{\prime}\\ Y_{\text{S}}^{\prime}\\ 1\end{bmatrix} =\boldsymbol{\Lambda}\boldsymbol{R}_{\tau}^{T} \begin{bmatrix}X_{ \text{C}}^{\prime}\\ Y_{\text{C}}^{\prime}\\ 1\end{bmatrix}  \tag{6}\]  
\[\boldsymbol{\Lambda}= \begin{bmatrix}\cos\tau_{x}\text{cos}\tau_{y}&0&\sin \tau_{y}\\ 0&\cos\tau_{x}\text{cos}\tau_{y}&-\sin\tau_{x}\text{cos}\tau_{y}\\ 0&0&1\end{bmatrix}  \tag{7}\]  
where \(X_{\text{S}}^{\prime}\), \(Y_{\text{S}}^{\prime}\), \(X_{\text{C}}^{\prime}\), and \(Y_{\text{C}}^{\prime}\) are normalized coordinates; \(\lambda_{1}\) is the scale factor from Equation (6), determined by the normalization of the coordinates on both sides of the equation; and \(\boldsymbol{\Lambda}\) is the derived Scheimpflug matrix [12]. Further analysis shows that the principle of transforming from the oblique camera coordinates \((X_{\text{S}},Y_{\text{S}},Z_{\text{S}})\) to the image pixel coordinates \((u,v)\) on \(\Pi_{\text{S}}\) is consistent with the traditional camera imaging model. Therefore, without considering distortion, the ideal transformation relationship between the pixel points in the oblique camera imaging model and their corresponding three-dimensional world points is as follows:  
\[\lambda_{2} \begin{bmatrix}u\\ v\\ 1\end{bmatrix} =\boldsymbol{K}\boldsymbol{\Lambda}\boldsymbol{R}_{\tau}^{T}( \boldsymbol{R} \begin{bmatrix}X_{\text{W}}\\ Y_{\text{W}}\\ Z_{\text{W}}\end{bmatrix} +\boldsymbol{T}) \tag{8}\]  
where \(\lambda_{2}\) is the scale factor, determined by the normalization of the coordinates on both sides of the equation. The above-established camera imaging model ignores lens distortion. However, in practical applications, due to unavoidable defects in the design, manufacturing, and installation processes of the lens, image deformation caused by lens distortion is significantly present.