## 2 Theory and Derivation
### Numerical Modeling of the Coupled-MTF  
In a classical system of unity magnification [17], the input signal is set to be a target with a cosine distribution of optical intensity, whose signal distribution function at the input end of the fiber bundle is as follows:  
\[I_{in}(x)=1+C_{i}(f,\delta)\cos[2\pi f(x+\delta)] \tag{1}\]  
where \(f\) is the spatial frequency of the input signal, \(C_{i}(f,\delta)\) is the input signal modulation system, and \(\delta\) is the cosine wave peak and the fiber center initial position difference. The imaging process of the fiber-optic imaging beam can be seen as first sampled by the input end plane, then uniformly illuminated by the fiber bundle, and then transformed into the outgoing end plane image. Then, the light intensity of the output signal of the first \(i\) row and \(j\) column of the fiber can be expressed as follows:  
\[ \begin{split} I_{out,j,}(x)=2\int\limits_{x_{j}-r}^{x_{j}+r}\{1+ C_{i}(f,\delta)\cos[2\pi f(x+\delta)]\}\Big{[}r^{2}-\big{(}x-x_{j}\big{)}^{2} \Big{]}^{\frac{1}{2}}dx\\ =2\int\limits_{2 jR-r}^{2 jR+r}\{1+C_{i}(f,\delta)\cos[2\pi f(x+ \delta)]\}\Big{[}r^{2}-(x-2 jR)^{2}\Big{]}^{\frac{1}{2}}dx\end{split}  \tag{2}\]  
Two simplified models of the coupling of the hexagonally aligned fiber-optic image transfer beam image element with the area array CCD image element are shown in Figures 4 and 5. For these two coupled discrete sampling cases, the following numerical model is derived. Figure 4: Diagram of the first kind of pixel-coupled error. The first type of coupling error is shown in Figure 4, where the coupling error between the hexagonally arranged fiber-optic imaging beam image elements and the area array CCD image elements only causes a single CCD image element to receive the output optical signal from three adjacent fiber-optic image elements. The coupling error of the image element in the arc vector direction is set to \(\Delta i\), and the coupling error of the image element in the meridian direction is set to \(\Delta j\). The light-receiving area represented by the bow ABCD in the figure is as below:  
\[ \begin{array}{c}S_{ABCD}=\frac{1}{2}(R-\Delta i)\left[\left(\frac{\sqrt{3}}{ 2}R-\Delta j\right)+\sqrt{r^{2}-(R-\Delta i)^{2}}\right]+\\ \frac{1}{2}\left(\frac{\sqrt{3}}{2}R-\Delta j\right)\left[(R-\Delta i)+\sqrt{r ^{2}-\left(\frac{\sqrt{3}}{2}R-\Delta j\right)^{2}}\right]+\\ \frac{r^{2}}{2}\left(\frac{3}{2}\pi-\arccos\frac{\frac{\sqrt{3}}{2}R-\Delta j}{ r}-\arccos\frac{R-\Delta i}{r}\right)\end{array}  \tag{3}\]  
The bow EFGH represents a light-receiving area calculated as follows:  
\[ \begin{array}{c}S_{i,j}=\frac{1}{2}(R-\Delta i)\left[\left(\frac{\sqrt{3}}{ 2}R-\Delta j\right)+\sqrt{r^{2}-(R-\Delta i)^{2}}\right]\\ +\frac{1}{2}\left(\frac{\sqrt{3}}{2}R-\Delta j\right)\left[(R-\Delta i)+\sqrt{ r^{2}-\left(\frac{\sqrt{3}}{2}R-\Delta j\right)^{2}}\right]\\ +\frac{r^{2}}{2}\left(\frac{3}{2}\pi-\arccos\frac{\frac{\sqrt{3}}{2}R-\Delta j }{r}-\arccos\frac{R-\Delta i}{r}\right)\end{array}  \tag{4}\]  
\[ \begin{array}{c}S_{i+1,j+1}=r^{2}\arccos\left(\frac{\frac{\sqrt{3}}{2}R-\Delta j }{r}\right)-\left(\frac{\sqrt{3}}{2}R-\Delta j\right)\sqrt{r^{2}-\left(\frac{ \sqrt{3}}{2}R-\Delta j\right)^{2}}\end{array}  \tag{5}\]  
\[S_{i,j+1}=\pi r^{2}-S_{i,j}-S_{i+1,j+1} \tag{6}\]  
For the case of the first type of coupling error, the expression of the output signal intensity of the first row and column of the image element in the face array CCD is as follows:  
\[I_{coupling,i,j}=\frac{S_{i,j}}{\pi r^{2}}I_{out,i,j}+\frac{S_{i,j+1}}{\pi r^{2 }}I_{out,i,j+1}+\frac{S_{i+1,j+1}}{\pi r^{2}}I_{out,i+1,j+1} \tag{7}\]  
Figure 5: Diagram of the second kind of pixel-coupled error. where, \(I_{out,i,j}\) is the output signal strength of the first \(i\) row and \(j\) column of fibers, \(I_{out,i,j+1}\) is the output signal strength of the first \(i\) row and \(j+1\) column of fibers, and \(I_{out,i+1,j+1}\) is the output signal strength of the first \(i+1\) row and \(j+1\) column of fibers.