## 1 Introduction  
In recent years, fiber-optic imaging technology has seen increasing adoption in traditional photoelectric imaging systems [1]. The progress in the preparation process has made the fiber-optic imaging bundle more flexible than traditional imaging components, owing to significant advantages [2], especially in fields such as industrial use, medical scopes [3], military night vision [4], and space target surveillance. The hexagonal arrangement of the fiber-optic imaging bundle is of great interest to researchers because of its good stability, high transmission rate, high resolution [5], and other advantages.  
The addition of a hexagonally aligned fiber-optic imaging bundle to an optoelectronic imaging system using a CCD as a detector can significantly improve the overall imaging performance of the system. The image quality of conventional optoelectronic imaging systems is evaluated based on the assumption that the optical system is a spatially invariant linear system, but hexagonally aligned fiber-optic imaging bundles, CCD devices, etc., do not have the characteristic \"space invariance\". [PERSON] [6] extended the definition of the local isoplantic condition for discrete imaging systems, which allows the modulation transfer function (MTF) to be used for the image quality evaluation of discrete imaging systems in the frequency range satisfying the Nyquist criterion. [PERSON] et al. [7] derived the MTF based on the Fourier transform of the spectrum of the image being the spectrum of the object, to make an average transfer function curve for hexagonal gazing focal plane arrays with rectangular and hexagonal arrangements. [PERSON] [8] presented a method for measuring the modulation transfer function of line-array fiber-optic image bundles and obtained good agreement with theoretically computed results. [PERSON] and team at the University of California, San Diego, designed an optical system with two-glass symmetric monocentric lenses coupled with fiber bundles [9] and developed an ultra-wide-angle camera using the structure of concentric lens-coupled fiber-optic panels with a camera field of view of 126\({}^{\circ}\)\(\times\) 16\({}^{\circ}\), a transfer function larger than 0.4 in the full field of view at 200 lp/mm, and a billion pixels [10]. [PERSON]. [11] studied the methods to mitigate moire artifacts and local obscuration of fiber bundles. [PERSON] [12] proposed a restoration method to remove honeycomb patterns and improve resolution for fiber bundle images. In China, research on fiber-optic image transfer has been ongoing. [PERSON] [13] conducted a study on the equivalent evaluation of the MTF for a fiber-optic panel image transmission system and confirmed that the image transmission process of a fiber-optic panel is an integral sampling process. [PERSON] of the Changchun Institute of Optics studied the MTF of linear fiber bundles [14] and established a one-dimensional coupled discrete sampling model for the MTF [15]. Most of the existing research on the coupled-MTF of fiber-optic imaging bundles is based on the one-dimensional arrangement of physical models, and [PERSON] studied the coupled-MTF characteristics between square-arranged fiber bundles and area array CCD [16]. However, fiber-optic imaging bundles are arranged in hexagonal structures. In this paper, we establish the numerical model of the coupled-MTF between hexagonally aligned fiber-optic imaging bundles and rectangular image CCD, based on the special structure of hexagonally aligned fiber-optic imaging bundles, using the assumption of generalized null invariance and the definition of MTF, and analyze the relevant characteristics of the coupled-MTF. The numerical model of the coupled-MTF of a hexagonally aligned fiber-optic imaging bundle and a rectangular image element CCD was established, and the relevant characteristics of the coupled-MTF were analyzed.