## 4 Conclusions  
To address the problem that the two-level discrete coupled sampling effect contained in a fiber-optic imaging system bundle as a relay imaging element could lead to the limitations of the conventional imaging quality evaluation model, a numerical analysis method of the modulation transfer function based on the coupling of hexagonally arranged fiber-optic imaging bundle and area array CCD image elements is proposed. The explicit numerical relationships between the number of fibers and input signal parameters and the coupled-MTF are established, and mathematical simulations were performed. The results show that at the Nyquist frequency, when the fiber parameters, the deviation of the input cosine target signal peak from the initial position of the fiber center, and the deviation of the image element coupling are fixed, then the oscillation convergence of the coupled-MTF increases with an increase in the number of fibers and converges to a stable value that equals the number of fibers; then, the number of fibers increases again, and the oscillation period and the final convergence value of the coupled-MTF remain unchanged. In order to analyze the mechanism of the deviation between the spatial frequency of the input cosine target signal and the Nyquist frequency on the coupled-MTF, a comparative study of the relationship between the convergences of the coupled-MTF with different input spatial frequencies was conducted. This study shows that when the deviation between the spatial frequency of the input cosine target and the Nyquist frequency gradually increases from 0.1% to 1%, the coupled-MTF shows an obvious trend of faster convergence and smaller oscillation periods, and the final convergence value gradually increases and converges to a stable value of 0.5 when the amount of frequency deviation is 1%. Thereafter, when the frequency deviation increases from 1% to 2%, the coupled-MTF oscillation period, convergence rate, and final convergence stable value remain unchanged. This paper further investigates the effect of the initial position deviation on the coupled-MTF and shows that if the input signal is a cosine, the coupled-MTF oscillates periodically with the increase in the initial position deviation, and the oscillation period is two fiber diameters. When the initial position deviation is 6.4, which is 1 fiber diameter, it drops to the minimum value of 0. It reaches the maximum value of 1 when the initial position deviation is 0/12.8, which is 0, or 2 fiber diameters. The transmission model of the hexagonally aligned fiber-optic imaging bundles established in this paper is more in line with the actual situation. This study produces important guidelines for the selection of the number of fibers in a hexagonally arranged fiber-optic imaging bundle, the input spatial frequency and the initial position deviation, and also for other discrete transmission components.  
Conceptualization, W.L. and C.W.; methodology, W.L.; software, W.L. and Y.H.; validation, C.H. and C.W.; formal analysis, W.L.; investigation, W.L. and Y.H.; resources, C.W.; writing--original draft preparation, W.L.; writing--review and editing, W.L.; funding acquisition, C.W. and H.Z. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (grant number 61805235) and Natural Science Foundation of Jilin Province (grant number 20200201203C). The authors declare no conflict of interest.