# Design of an Imaging Optical System for Large-Sized Stepped Shaft Diameter Detection
## 1 Introduction  
Large-sized stepped shafts are commonly used in large-scale mechanical equipment as positioning and installation components [1]. The online automatic detection of their diameter parameter has emerged as a crucial technology in measurement equipment research. The imaging optical system plays a vital role in detecting the diameter of these shafts and directly impacts the measurement accuracy. Currently, the widely used projection method for diameter measurement is effective for axis-like components due to its relatively simple technical implementation and the ease of designing collimating optical systems [2]. However, this method has limitations when it comes to larger diameter sizes, primarily due to the limited projection area and the small effective size of the linear array CCD used. Hence, there is a need to propose a diameter projection measurement method suitable for larger-sized diameters. Currently, optical imaging methods for measuring diameter using the projection method include parallel light imaging, collimated light imaging, and dual-CCD imaging. The parallel light imaging method is only suitable for measuring diameters in the range of 30 mm to 100 mm and is not applicable for measuring large-sized stepped shafts. The collimated light imaging method is suitable for measuring objects of various shapes, but for measuring the diameter of large-sized stepped shafts, it may require a longer focal length and a larger collimation angle to obtain clear imaging effects. In contrast, the dual-CCD imaging method utilizes two symmetric CCD sensors to capture the object's images and measure the diameter based on these images. It has higher accuracyand real-time performance and allows for adjustment of the detection range by adjusting the distance between the two CCDs. For large-sized stepped shafts, the dual-CCD imaging method can better handle larger objects. However, further improvement is needed in terms of imaging quality. Therefore, this paper proposes the addition of a dual-telecentric optical system based on the dual-CCD imaging method. At the beginning of the 21 st century, researchers utilized projection methods for diameter measurements [3], which were improved and upgraded by subsequent studies. In recent years, many scholars have expanded upon the projection method to explore dual-telecentric optical imaging systems [4; 5]. For instance, [PERSON] designed an optical system with dual fields of view, featuring two magnification factors, which were \(-0.275\) and \(-0.1375\), respectively, primarily addressing the complications and accuracy issues associated with changing lenses of different magnifications [6]. However, this system requires 15 spherical lenses, making its structure overly complex compared to standard dual-telecentric systems. [PERSON] focused on the application requirements of machine vision in detecting the positional accuracy of pin needles, designing a dual-telecentric optical system with a depth of field of 0.5 mm, maximum distortion less than 0.1%, and telecentricly controlled within 0.007\({}^{\circ}\), correcting various aberrations to enhance detection accuracy [7]. Yet, at a spatial frequency of 208 lp/mm, the lens contrast is low, with an MTF value just over 0.2, indicating poor imaging quality. [PERSON] proposed a dual-telecentric optical system with ultra-wide depth of field and low distortion, comprising seven spherical lenses with distortion less than 0.1% and telecentricity at a maximum of 0.013\({}^{\circ}\). This system achieves industrial measurement requirements in terms of image quality with a good cost-performance ratio, even with fewer lenses [8]. However, its object-side field of view is only 60 mm, making it suitable only for measuring small diameters and not capable of measuring large-sized step shafts. [PERSON] introduced a method using laser scanners to measure diameters within the range of 200 mm-900 mm. However, the error margin between 2 mm and 6 mm renders it unsuitable for high-precision measurements [9].