# Analysis and Design of Infrared Search and Track System with Afocal Zoom Telescope
## 2 Homogeneous Coordinate Transformation in IRST
### Imaging Model under Static Condition  
In the static case, the object coordinates in the target coordinate system are  
\[\mathbf{X}_{0}=\begin{bmatrix}x_{0}\\ y_{0}\\ z_{0}\\ 1\end{bmatrix} \tag{1}\]  
The transform matrix from target coordinates to platform coordinates is  
\[\mathbf{T}_{\mathbf{s}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&R\\ 0&0&0&1\end{bmatrix} \tag{2}\]  
where \(R\) denotes the distance between the target coordinates and the platform coordinates.  
The transform matrix from the platform to the telescope is  
\[\mathbf{T}_{\text{TM1}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&l_{1}\\ 0&0&0&1\end{bmatrix} \tag{3}\]where \(l_{i}\), \(i\) = 1, 2, 3, 4, is the distance between different coordinates. The action matrix of the afocal telescope is calculated as follows:  
\[\mathbf{T_{tele}}=\begin{bmatrix}m&0&0&0\\ 0&m&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \tag{4}\]  
where the angular magnification of the telescope is denoted by m. Numerically, in an IRST system, \(m=\frac{\theta_{mt}}{\theta_{in}}\), as \(\theta_{out}\) represents the emergence angle of refraction or reflection, and \(\theta_{in}\) declares the angle of incidence.  
The transform matrix from the telescope to the FSM is  
\[\mathbf{T_{TM2}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&l_{2}\\ 0&0&0&1\end{bmatrix} \tag{5}\]  
The action matrix of FSM is calculated as follows:  
\[\mathbf{T_{FSM}}=1-2\mathbf{N_{FSM}}\mathbf{N_{FSM}}\mathbf{N_{FSM}} \tag{6}\]  
where \(\mathbf{N_{FSM}}\) is the normal vector of the FSM in the platform coordinates.  
The translation matrix from the FSM to the FM is calculated as follows:  
\[\mathbf{T_{TM3}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&l_{3}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \tag{7}\]  
The action matrix of FM is calculated as follows:  
\[\mathbf{T_{FM}}=1-2\mathbf{N_{FM}}\mathbf{N_{FM}}\mathbf{N_{FM}}=\begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix} \tag{8}\]  
The transform matrix from the FM to the object lens is  
\[\mathbf{T_{TM4}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&l_{4}\\ 0&0&0&1\end{bmatrix} \tag{9}\]  
where \(\mathbf{N_{FM}}\) is the normal vector of the folding mirror in the platform coordinate system.  
The action matrix of the imaging objective is calculated as follows:  
\[\mathbf{T_{obj}}=\begin{bmatrix}-\frac{f^{\prime}_{obj}}{R}&0&0&0\\ 0&-\frac{f^{\prime}_{obj}}{R-f^{\prime}_{obj}}&0\\ 0&0&-\frac{f^{\prime}_{obj}}{R-f^{\prime}_{obj}}&0\\ 0&0&0&1\end{bmatrix} \tag{10}\]  
where \(f^{\prime}_{obj}\) represents the objective focal length.  
The action matrix from the imaging objective coordinate system to the image plane coordinate system is calculated as follows:  
\[\mathbf{T_{img}}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&f^{\prime}_{obj}\\ 0&0&0&1\end{bmatrix} \tag{11}\]  
The image location in the image plane coordinate system is calculated as follows:  
\[\mathbf{X_{i}}=\begin{bmatrix}x_{i}\\ y_{i}\\ z_{i}\\ 1\end{bmatrix} \tag{12}\]  
Therefore, under the static condition, the imaging model can be written as  
\[\mathbf{X_{i}}=\mathbf{T_{img}}\mathbf{T_{obj}}\mathbf{T_{TM4}}\mathbf{T_{FM}}\mathbf{T_{TM3}}\mathbf{T_{ FSM}}\mathbf{T_{TM2}}\mathbf{T_{tele}}\mathbf{T_{TM1}}\mathbf{T_{p}}\mathbf{X_{0}} \tag{13}\]  
Thus, we acquire the position of the image in the image coordinate system.