## 2 Modulation Transfer Function (MTF) Calculation Method
### The MTF for the Mixed Image Motion of Linear Motion and Sinusoidal Motion  
To accurately calculate the MTF, the exact expression of the image motion needs to be determined. In this section, the mix of common forms of motions--linear motion and sinusoidal motion--is analyzed. The expression of the image motion can be described as follows:  
\[\mathrm{x}(t)=\mathrm{vt}+\mathrm{D}\mathrm{cos}(\omega t), \tag{8}\]  
where \(\mathrm{v}\) is the velocity of the linear motion, \(\mathrm{D}\) stands for the amplitude, and \(\omega\) stands for the circular frequency of the sinusoidal motion. The MTF for the mixed image motion of linear motion and sinusoidal motion becomes the following:  
\[\mathrm{M}(f,t)=\sqrt{\left(\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{ \mathrm{e}}}\sin 2\pi\mathrm{f}[\mathrm{v}t+\mathrm{D}\mathrm{cos}(\omega t)] \mathrm{dt}\right)^{2}+\left\{\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{ \mathrm{e}}}\cos 2\pi\mathrm{f}[\mathrm{v}t+\mathrm{D}\mathrm{cos}(\omega t)] \mathrm{dt}\right\}^{2}}. \tag{9}\]  
It can be seen that the former part of the Equation (9) has the same form as the latter. Therefore, the formula is analyzed by expanding the former part. The former part can be expressed as follows:  
\[ \begin{array}{l}\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}}\sin 2 \pi\mathrm{f}[\mathrm{v}t+\mathrm{D}\mathrm{cos}(\omega t)]\mathrm{dt}=\\ \frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}}\left\{\cos(2\pi \mathrm{f}\mathrm{v}t)\sin[2\pi\mathrm{f}\mathrm{D}\mathrm{cos}(\omega t)] \right\}\mathrm{dt}+\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}} \left[\sin(2\pi\mathrm{f}\mathrm{v}t)\cos[2\pi\mathrm{f}\mathrm{D}\mathrm{cos }(\omega t)]\right]\mathrm{dt}\end{array} . \tag{10}\]  
The integral of the trigonometric functions in Equation (10) is impossible to calculate. The trigonometric function can be expressed as an infinite sum of integral order Bessel functions of the first kind [18]:  
\[ \begin{array}{l}\sin(\mathrm{x}\cos\varphi)=-2\sum\limits_{n=1}^{Y=}(-1)^{n }\mathrm{J}_{2n-1}(\mathrm{x})\cos[(2n-1)\varphi]\\ \cos(\mathrm{x}\cos\varphi)=\mathrm{J}_{0}(\mathrm{x})+2\sum\limits_{n=1}^{Y=}( -1)^{n}\mathrm{J}_{2n}(\mathrm{x})\cos(2n\varphi)\end{array} . \tag{11}\]  
In that case, we used an infinite sum of the integral order Bessel function of the first kind to simplify Equation (10). Thus, the former part of Equation (10) can be described as follows:  
\[ \begin{array}{l}\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}}\cos (2\pi\mathrm{f}\mathrm{v}t)\sin[2\pi\mathrm{f}\mathrm{D}\mathrm{cos}(\omega t )]\mathrm{dt}=\\ \frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}}\cos(2\pi\mathrm{f} \mathrm{v}t)\left\{-2\sum\limits_{n=1}^{\infty}(-1)^{n}\mathrm{J}_{2n-1}(2\pi \mathrm{f}\mathrm{D})\cos[(2n-1)\omega t]\right\}\mathrm{dt}\end{array} . \tag{12}\]  
As the sum of the infinite series is uniform convergent, the operation order of the integrals and sums can be exchanged. Equation (12) can be expressed as follows:  
\[ \begin{array}{l}\frac{1}{t_{\mathrm{e}}}\int\limits_{0}^{t_{\mathrm{e}}}\cos (2\pi\mathrm{f}\mathrm{v}t)\sin[2\pi\mathrm{f}\mathrm{D}\mathrm{cos}(\omega t )]\mathrm{dt}=\\ -\frac{1}{t_{\mathrm{e}}}\sum\limits_{n=1}^{\infty}(-1)^{n}\mathrm{J}_{2n-1}(2 \pi\mathrm{f}\mathrm{D})\left[\frac{\sin((2n-1)\omega+2\pi\mathrm{f}\mathrm{v }t)_{\mathrm{e}}}{(2n-1)\omega+2\pi\mathrm{f}\mathrm{v}}+\frac{\sin((2n-1) \omega-2\pi\mathrm{f}\mathrm{v}t)_{\mathrm{e}}}{(2n-1)\omega-2\pi\mathrm{f} \mathrm{v}}\right].\end{array}  \tag{13}\]  
Based on this method, the MTF calculation formula can be further organized as follows:  
\[\mathrm{M}(f,t)=\sqrt{\left\{\left[\frac{\mathrm{J}_{0}(2\pi\mathrm{f} \mathrm{D})\sin(2\pi\mathrm{f}\mathrm{v}t_{\mathrm{e}})}{2\pi\mathrm{f} \mathrm{v}t_{\mathrm{e}}}+\mathrm{A}+\mathrm{B}\right]^{2}+\left[\frac{\mathrm{J }_{0}(2\pi\mathrm{f}\mathrm{D})\mathrm{sin}^{2}(\pi\mathrm{f}\mathrm{v}t_{ \mathrm{e}})}{\pi\mathrm{f}\mathrm{v}t_where A, B, C, D represent the expressions of the sums of the infinite series.