## 2 Previous Studies in Auto-Vision of Inspection System  
The two types of measurements are diametral and radial roundness [33]. Diametral measurements are made with two-point measuring tools such as calipers or micrometers, whereas radial measurements require precision spindle instruments that are both expensive and time-consuming. These studies introduced a method for evaluating roundness and cylindricity tolerances, addressing difficulties encountered by technicians and engineers, focusing on optimizing manual procedures, and improving skills [34]. Furthermore, roundness profiles with pseudo and actual diameters were evaluated, and relationship models with neural network regression based on coordinate measuring machines were developed [35]. The \"roundness tolerance band\" is defined by the International Organization for Standardization (ISO) as the area between two concentric circles at the same cross-section where the difference in their radii equals the roundness tolerance value. This value emphasizes the geometric roundness tolerance standard by representing the allowable range of deviation from perfect roundness in a circular object. It specifies the acceptable margin of error for the roundness of a part [13; 36; 37]. This study addressed four ISO 14405-1:2016 modifiers, providing simple algorithms for evaluating circularity errors and investigating the impact of measurement system strategies on these new specifications [38]. In engineering processes, the surface structure can be defined by three primary features: form errors, roughness, and waviness. Errors encompass three vital criteria essential for the assembly and connection of different parts, namely straightness, flatness, and circularity, commonly referred to as roundness. Out-of-round (OOR) indicates the disparity between the actual and measured radius at a particular point. Roundness measurement is usually based on four globally recognized reference circles discussed in ISO-GPA and ASME Y14.5(1994), such as Maximum Inscription Radial Circle (MIC), Minimum Circumscribing Circle (MCC), Minimum Zone for Circularity (MZC), and least squares circle (LSC), as addressed by [10; 39; 40; 41]. The comparison of the benchmark algorithms for calculating round errors based on MCC, MIC, and MZC was shown by [42]. Likewise, this research focused on two robust cylinder fitting alignment algorithms that combined PCA, regression, and algebraic methods, as well as a PCD cylinder alignment algorithm that can handle high-percentage deviations and improve shape recognition [43; 44]. This study also conducted an iterative procedure for robust circle fitting that employs [PERSON]'s method to obtain the center and radius, as well as geometric distances to the circle, to identify and remove outliers, thereby reducing the corrupted effect on circle parameter estimates [45]. A new roundness measurement algorithm and least squares circle fit arithmetic were employed in a rapid quality detection system for electrochemically fabricated micro-holes. The findings demonstrated reliable results and high precision, meeting the requirements for quality inspection of electrochemical machining [46]. The optimal wire spark erosion machining (WSET) turning condition minimized roundness deviation and maximized material erosion rate. It produced a suitable surface for a variety of applications, with a roundness deviation of 15.51 m and a material erosion rate of 9.35 mm\({}^{3}\)/min. These findings will be useful in the advanced fabrication of miniature products from difficult-to-machine materials such as titanium [47]. The hybrid approach evaluates circularity error using the least squares method (LSM) and probabilistic global search Lausanne (PGSL) techniques, which have been proven to be efficient and suitable computer-aided circularity measuring instruments [48]. A circular profile surface has a maximum point that is connected by a wide circle diameter, as shown in Figure 1a. The center of the circle is represented by C, and its radius can be calculated for all points.