## 2 Theory and Derivation
### Selection of CCD Image Elements and Sampling Direction  
The hexagonally arranged fiber-optic bundles are arranged in a cascaded, cross-stacked manner, which results in a stable fiber bundle structure with less deformation on the end face and a high fill rate. The literature shows that at approximately 0.907 [17], the fill factor of a hexagonally aligned fiber bundle is higher than that of a squarely aligned fiber bundle with the same fiber material. That is, both the imaging quality and the optical image transmission rate of a hexagonally aligned fiber bundle are better under the same conditions. Figure 3a shows the cell structure of the hexagonal fiber bundle in which the fiber core radius is r, and the fiber outer radius is R. In the right-angled \(\Delta\) ABC, BC = 4\(R\) and AB = 2\(R\), so we can obtain \(AC=2\sqrt{3}R\). To achieve equal spacing sampling, a rectangular image element CCD with an image element size of \(\sqrt{3}R\) is proposed. In general, \(r\) is less than or equal to \(\sqrt{3}R\). Figure 1: The input end has the same arrangement of fiber imaging bundles as the output end. Figure 2: Working principle of hexagonally arranged fiber imaging bundles. Assuming \(r=\frac{\sqrt{3}}{2}R\), i.e., the single fiber core diameter divided by fiber diameter, the fiber core layer area over fiber area is \(\sqrt{3}:2\). Considering the effect of the filling factor, the light transmission area of the imaging bundle can be obtained as a factor of about 68% of the total area. [PERSON] [18] classified the particular orientations of the hexagonally arranged fiber-optic imaging bundles, as shown in Figure 3b, by classifying the directions marked as solid lines in the figure as one category and the directions marked as dashed lines \(II\) as another category, i.e., by adding an angular coordinate system to the beam, where the directions \(0\), \(\frac{\pi}{3}\), and \(\frac{2\pi}{3}\) are one category, and the directions \(\frac{\pi}{6}\), \(\frac{\pi}{2}\), and \(\frac{5\pi}{6}\) are another category. [PERSON] et al. [19,20] confirmed the validity of using the function for sampling, i.e., when the input signal is a cosine target, the direction of 2D discrete sampling is not restricted, and any direction is possible. Combining the above two points, we studied the modulation transfer function of a hexagonally arranged fiber-optic imaging bundle coupled with a rectangular image element CCD based on the direction of \(I\) in Figure 3b. The numerical model of a hexagonally arranged fiber-optic imaging bundle is simplified as follows:  
1. Fiber-optic imaging bundles are manufactured using multi-component glass fiber, and unlike communication fiber, multi-component glass fiber has a large light imaging area. In practice, the thickness of the single fiber cladding of the optical fiber bundle is controlled to be approximately 1/10 of the diameter of the fiber. In general, the light imaging area of various types of imaging bundles accounts for 60-90% of the total area of the image bundle. The attenuation of the 1000 m communication fiber at a wavelength of 1300 nm is approximately 1 dB, while the attenuation of image transmission of the multi-component glass fibers in the broad spectrum (400 nm-800 nm) beam is approximately 1 dB/m, and for a few tens of millimeters of the hexagonal fiber imaging bundle, the attenuation is much smaller than 1 dB, which is negligible;
2. The multi-component glass fiber introduces a glass cladding structure that substantially reduces the severe crosstalk effect between adjacent fibers. Light-absorbing materials are introduced into the structure of the optical fiber imaging element to attenuate the inter-fiber crosstalk and absorb stray light. In this study, it was assumed that there was no crosstalk between fibers and that each monofilament fiber transmitted image information independently and efficiently;
3. Broken filaments are undesirable, but likely to exist in fiber-optic bundles.