## 2 THz Imaging Character of Parabolic Antennas  
The observation geometry of the spatial target for the space-based terahertz radar is depicted in Figure 2. The LOS vector links the \(O_{Target}\) and \(O_{Radar}\) points. In the case of a three-axis stabilized space target, its orientation remains constant with respect to the coordinate system of the target, and the projection onto the imaging plane is determined by the corresponding LOS angles of the radar. The radar LOS can be represented as:  
\[LOS=\left[\cos\theta\cos\varphi,\ \ \ \cos\theta\sin\varphi,\ \ \ \sin\theta \right]^{\mathrm{T}} \tag{1}\]  
where the superscript T denotes transposition, \(\theta\in[-90^{\circ},90^{\circ})\), and \(\varphi\in[-180^{\circ},180^{\circ})\), which represent the elevation angles and azimuth angles, respectively. The orientation of a parabolic antenna can be expressed as follows:  
\[k=\left[\cos\beta\cos\alpha,\ \ \ \cos\beta\sin\alpha,\ \ \ \sin\beta \right]^{\mathrm{T}} \tag{2}\]  
where \(k\) is usually called the normal vector and \(\alpha\in[-180^{\circ},180^{\circ})\) and \(\beta\in[-90^{\circ},90^{\circ})\) represent the yaw angle and the pitch angle, respectively. The R-axis and the D-axis of the ISAR plane are determined by the LOS vector and derivation LOS vector, respectively [27]. Since the projection relationship is not the key part of this article, we overlook its fussy details. It is important to recognize that all these angles can vary arbitrarily, leading to a complex issue. In reality, it is more common to focus on just a single rotational axis in order to achieve the highest possible imaging quality. After further analysis, we can see that the elevation and azimuth angles of the target relative to the radar's LOS are, respectively, denoted as  
\[\vartheta=\theta-\beta \tag{3}\]  
\[\psi=\varphi-\alpha \tag{4}\]  
For the sake of analysis convenience, in the imaging setup of this paper, it is assumed that the elevation angle \(\vartheta\) remains constant and only the rotation accumulation of the azimuth angle \(\psi\) is considered to change. Without loss of generality, let \(\alpha=0\), namely, the relative azimuth angle \(\psi\) is equal to the \(\varphi\) angle. Figure 1: The overall framework diagram of the proposed method. We assume that the 2D ISAR imaging plane of the space target is composed of the R-axis and the D-axis, where the R-axis of the imaging plane is determined by the LOS vector, while the D-axis is defined based on the differentiation of the LOS vector. Given a point \(M=[u,v,w]^{T}\), which represents a 3D vector, its corresponding projection onto the 2D R-D imaging plane is referred to as \(N=[r,d]^{T}\). This projection relationship can be expressed as:  
\[N=P\cdot M \tag{5}\]  
where \(P\) is the projection matrix, represented as  
\[P= \begin{bmatrix}\cos\theta\cos\phi&\cos\theta\sin\varphi&\sin\theta\\ P_{21}&P_{22}&\cos\theta\theta^{\prime}\end{bmatrix}  \tag{6}\]  
and \(P_{21}\) and \(P_{22}\) are denoted as  
\[ \begin{cases}P_{21}=-\sin\theta\cos\varphi\theta^{\prime}-\cos\theta\sin \varphi\varphi^{\prime}\\ P_{22}=-\sin\theta\sin\varphi\theta^{\prime}+\cos\theta\cos\varphi\varphi^{ \prime}\end{cases}  \tag{7}\]  
where \(\theta^{{}^{\prime}}\) and \(\varphi^{{}^{\prime}}\) represent the instantaneous velocities of the elevation and azimuth angles, respectively. Additionally, if the scatter point \(M\) is rotated along the \(O_{Target}-UVW\) coordinate with angle \([\ \theta_{\mu}\ \theta_{\
u}\ \theta_{\omega}\ ]^{\mathrm{T}}\), the rotated point \(\hat{M}\) is  
\[\hat{M}=R_{W}\cdot R_{V}\cdot R_{U}\cdot M \tag{8}\]  
where \(R_{U}\,R_{V}\,\) and \(R_{W}\) are the rotation matrices along the \(U-\), \(V-\), and \(W-\)axes, respectively. Specifically,  
\[R_{U}= \begin{bmatrix}1&0&0\\ 0&\cos\theta_{\mu}&-\sin\theta_{\mu}\\ 0&\sin\theta_{\mu}&\cos\theta_{\mu}\end{bmatrix}  \tag{9}\]  
\[R_{V}= \begin{bmatrix}\cos\theta_{\
u}&0&-\sin\theta_{\
u}\\ 0&1&0\\ \sin\theta_{\
u}&0&\cos\theta_{\
u}\end{bmatrix}  \tag{10}\]  
\[R_{W}=