## 4 Simulation Experiment
### Simulation Results  
As mentioned above, the simulation method is available for calculating the MTF for arbitrary image motions, which is illustrated theoretically. Therefore, the simulation method works in the case of a simple form of image motion, such as linear motion and sinusoidal motion. The simulation method can be verified in the case of simple form of image motion, such as linear motion and sinusoidal motion. For the data shown in Figure 5b, the image plane is \(76~{}\upmu\mathrm{m}\times 76~{}\upmu\mathrm{m}\), with \(128\times 128\) sampling grids and \(256\times 256\) displaying grids. The maximum sampling frequency is \(16.89\) lp/mm. Firstly, the simulation method is applied to the linear image motion. According to the [PERSON] sampling theorem, linear motion can be sampled twice in the exposure. The MTF for the linear motion derived by other literature can be expressed as follows [11]:  
\[\mathrm{MTF}(\mathrm{f})=\frac{\sin(2\pi\mathrm{f}\mathrm{v}\mathrm{t}\mathrm{ e}_{\mathrm{e}})}{2\pi\mathrm{f}\mathrm{v}\mathrm{t}\mathrm{e}_{\mathrm{e}}} =\sin\mathrm{c}(2\pi\mathrm{f}\mathrm{d}), \tag{26}\]  
where \(\mathrm{d}\) is the non-circular blur radius of the linear motion. Figure 6 shows the MTFs calculated by the sinc function and the MTFs calculated by the simulation method. To evaluate the simulation results, the average relative error is proposed as in Equation (27). \[\mathrm{error_{relative}}=\frac{1}{\mathrm{k}}\sum_{n=1}^{\mathrm{k}}\left| \frac{\mathrm{MTF}_{n}-\mathrm{MTF}_{n}^{\prime}}{\mathrm{MTF}_{n}^{\prime}} \right|\times 100\%, \tag{27}\]  
where \(\mathrm{MTF}_{n}\) represents the simulated result, \(\mathrm{MTF}_{n}^{\prime}\) is the theoretical result and \(\mathrm{k}\) stands for the quantity of the frequency points in the simulation. The average relative error for the linear motion calculated by Equation (27) is \(3.23\%\). According to the existing research, the MTF for sinusoidal motion is presented in the form of the Bessel function of the first kind. The expression can be described as follows [12]:  
\[\mathrm{MTF}= \mathrm{J}_{0}(2\pi\mathrm{f}\mathrm{D}), \tag{28}\]  
Figure 5: Data acquired by ZEMAX. PSF: point spread function. Figure 6: The simulated MTF and theoretical MTF for linear motion. where D stands for the amplitude of the sinusoidal motion. The Nyquist sampling theorem indicates that the sinusoidal signal needs to be sampled at least twice in a cycle. However, studies show that sampling twice in a cycle is not capable of restoring the original sinusoidal signal. In this case, we simulated the sinusoidal vibration signal by sampling 12 times in one period. The comparisons between the simulation results and the theoretical results are presented in Figure 7. The average relative error for the sinusoidal motion is 5.14%. We used the simulation method to calculate the MTF for the mixed image motion of linear motion and sinusoidal motion. The PSF for the complex motion acquired by ZEMAX and MATLAB is shown in Figure 8.  
As mentioned above in Section 1, the former method used the MTF calculation formula for linear motion to calculate the MTF for the complex image motion in the case of short exposure time [17]. The calculation formula is expressed as Equation (26). In order to compare the calculation method presented in this paper with the former method, the simulation experiments were carried out with different exposure times. In the simulation experiment, the velocity of the linear motion was 12 pixels per second, and the amplitude of sinusoidal vibration was eight pixels. The frequency of sinusoidal vibration was set as 1 Hz, and the exposure time range was from 0.5 s to 4 s. The curves of the MTF calculated by the simulation method and the calculation method in different exposure times are presented in Figure 9. Figure 9 also shows the calculated MTFs by the former methods using Equation (26).