## 2 Model of Dynamic Imaging  
### Dynamic Frame-by-Frame Imaging Models  
In order to expand the ground imaging range, the SiC mirror rotates periodically in front of the CMOS camera scans, at a certain velocity, in the dynamic imaging mode [22]. In this process, the dynamic frame-by-frame imaging schematic is established, as shown in Figure 1.  
When the satellite flies to the observation area along the orbit, the mirror rotates with a certain angular speed at the initial position, and the initial sub-frame image is exposed at \(T_{1}\). The exposure time interval is calculated by satellite, according to the single frame width, and then, the single-frame images along the vertical track direction are obtained according to the time interval T2, T3... Tn. Finally, the extended-amplitude image can be gained after splicing all the frame images according to the time sequence. However, the ground area, covered by CMOS pixels, changes from a square to an approximate trapezoidal image, with different sizes, as the imaging attitude angle increases in the dynamic imaging mode. At the same time, the anisotropic motion between pixels leads to serious distortion of the image on the CMOS focal plane. In addition, the internal rigid structure, when subjected to dynamic disturbance, produces serious random vibration. It makes the CMOS focal plane deviate from the target image by a certain angle, resulting in the degradation of image quality. For the sake of effectively analyzing the distorted output image and the degraded image caused by random vibration in the dynamic frame-by-frame imaging process, the imaging simulation link is established, as shown in Figure 2.  
The ground target image is obtained by one exposure of the camera, as shown in Figure 2, where the pixel's coordinate function is \(g(x,y)\). According to the invariance of the field of view, distortion image on the camera's focal plane is obtained, as shown in Figure 2, where the pixel's coordinate function is \(p(x,y)\). The generalized function \(\xi\{\}\) is expressed as the transformation operation between the ground target and the distorted image on the camera focal plane, so the mapping relationship can be expressed as:  
\[p(x,y)=\xi\{g(x,y)\} \tag{1}\]  
The degraded image can be obtained by the convolution of the distorted image function and the line spread function \(h(x,y)\), which is caused by satellite vibration, as shown in Figure 2.  
Figure 1: Schematic diagram of dynamic frame-by-frame imaging.  
Figure 2: Simulation link for dynamic imaging.  
\[o(x,y)=h(x,y)*\xi\{g(x,y)\} \tag{2}\]  
In the formula, \(o(x,y)\) is the degraded image interfered by random vibration. With the transformation of the Fourier transform, the frequency domain calculation formula is expressed as:  
\[O\big{(}f_{x},f_{y}\big{)}=\text{MTF}\big{(}f_{x},f_{y}\big{)}\cdot F\{\xi[g(x, y)]\} \tag{3}\]  
where, \(O\big{(}f_{x},f_{y}\big{)}\) is the output-image spectrum; \(F\{\xi[g(x,y)]\}\) is the input image spectrum; \(\text{MTF}\big{(}f_{x},f_{y}\big{)}\) is called the modulation transfer function, which is the Fourier transform of the line spread function \(h(x,y)\) caused by random vibration.  
\[\text{MTF}\big{(}f_{x},f_{y}\big{)}=\big{|}\iint h(x,y)\exp\big{[}-j2\pi\big{(} f_{x}x+f_{y}y\big{)}\big{]}\text{d}x\text{d}y\big{|} \tag{4}\]  
It represents that the different frequency components are modulated by the filtering system. The distortion degraded image is obtained by the inverse Fourier transform of the output-image spectrum \(O\big{(}f_{x},f_{y}\big{)}\), as shown in Figure 2.