# Analysis of MTF in TDI-CCD Subpixel Dynamic Super-Resolution Imaging by Beam Splitter
## 2 Sampling MTF of Subpixel Dynamic Super-Resolution Imaging
### MTF of Over-Sampling with TDI-CCD  
Comparing with CCD, the output signal of TDI-CCD pixel is the accumulation from multiple CCD linear arrays for the same scene during the scanning process to improve the response signal-noise rate (SNR) [27]. Actually, there often exists the drift angle caused by the influence of the space-borne camera. Let \(V_{g}\) represent the moving speed of the target (or the push-broom scanning direction of camera), the angle between \(V_{g}\) and TDI integral direction is the drift angle \(\theta\):  
\[\theta=\arctan\biggl{(}\frac{V_{r}}{V_{c}}\biggr{)} \tag{4}\]  
where \(V_{r}\) is the velocity component of \(V_{g}\) in the direction of the CCD linear array; and \(V_{c}\) is the velocity component of \(V_{g}\) in the TDI integral direction (Figure 5b).  
Figure 4: Sampling results of linear CCD pair. (**a**) Sampling of the first linear CCD; (**b**) Sampling of the second linear CCD; (**c**) Sampling results of the first linear CCD and the second linear CCD.  
If the TDI-CCD line scanning speed and the target speed of motion is not synchronized, which may cause the image shift along the drift angle \(\theta\). Therefore, the output response of the pixel is:  
\[p(x,y)=\left[\frac{1}{pM\tan\theta}\cdot rect(\frac{x}{pM\tan \theta})\right]*\left[\frac{1}{pM\theta}\cdot rect(\frac{y}{pM\theta})\right]\] \[=\left\{\begin{array}{ll}\frac{1}{p^{2}M\tan\theta}\cdot d|x|< \frac{pM\tan\theta}{2},|y|<\frac{pM\theta}{2}\\ 0\quad\text{elsewhere}\end{array}\right. \tag{5}\]  
where pixel's center distance of the TDI-CCD is \(p\) (the length is same as the width); \(M\) is the TDI integral stages; \(d=\frac{\Delta V}{V_{i}}\) is the synchronization accuracy; \(\Delta V\) a is the difference between the image scanning speed \(V_{i}\) and the TDI charge transfer rate.  
The Fourier transform of \(p(x,y)\) is:  
\[P(u,v)=\sin c(\pi u\cdot pM\tan\theta)\cdot\sin c(\pi v\cdot pMd) \tag{6}\]  
For oversampling in the imaging of TDI-CCD, the shifted distance is b in the vertical distance, assuming that the shifted distance is a in the horizontal direction (Figure 6).  
A square and a triangle in Figure 6a are regarded as a basic unit, with the dashed box shown in the lower right corner. Using the division method of Wigner-Seitz cell interpolation, the regular square's lattices are obtained and shown in Figure 6b.  
Figure 5: Schematic of time delay integration (TDI)–CCD. (**a**) Operation mode of TDI-CCD; (**b**) The relationship between each velocity.  
Figure 6: Sampling results of TDI-CCD. (**a**) Sampling results of TDI-CCD pairs; (**b**) Sampling results of TDI-CCD pairs after partition.  
The Wigner-Seitz function of TDI-CCD \(w(x,y)\) is equal to \(\frac{4}{p^{2}}\) when \(x\in(-\frac{p-a}{2},\frac{a}{2})\) and \(y\in(-\frac{p-b}{2},\frac{b}{2})\). In other cases, it is 0.  
The modulation transfer function (\(M\!T\!F_{2}\)) of the sampling results from two TDI-CCD is (The derivation details can be found in Appendix A):  
\[\begin{array}{rl}M\!T\!F_{2}=&\sin c(\pi u\cdot pM\tan\theta)\cdot\{\frac{a }{p}\sin c(\pi uu)+(\frac{p-a}{p})\sin c[\pi u(p-a)]\}\\ &\cdot\sin c(\pi v\cdot pMd)\cdot\{\frac{b}{p}\sin c(\pi vb)+(\frac{p-b}{p}) \sin c[\pi v(p-b)]\}\end{array} \tag{7}\]  
For obtaining the condition of limitation, setting \(\frac{\partial(M\!T\!F_{2})}{\partial a}=0\), we can obtain the following:  
\[\cos[\pi u(p-a)]=\cos(\pi ua) \tag{8}\]  
Then setting \(\frac{\partial(M\!T\!F_{2})}{\partial b}=0\), thus we can obtain:  
\[\cos[\pi v(p-b)]=\cos(\pi vb) \tag{9}\]  
As Equations (8) and (9) are applicable to all of \(u\) and \(v\), when \(p-a=a\) and \(p-b=b\), the \(M\!T\!F_{2}(a,b)\) has a maximum value when \(a=\frac{p}{2}\) and \(b=\frac{p}{2}\).  
Similar to analysis in Section 2.2, when there are \(n\) image pairs generated from TDI-CCD with shift distance \(a_{i}\) and \(b_{i}(i=1,2,\cdots,2n-1)\) respectively. From the derivation process in Appendix A, when \(a_{1}=a_{2}=\cdots=a_{2n-1}=p/2n\) and \(b_{1}=b_{2}=\cdots b_{2n-1}=p/2n\), \(M\!T\!F_{mT}\) reach its maximum value.