## 2 Previous Studies in Auto-Vision of Inspection System
### The Minimum Zone Circles  
The MZC method determines roundness error by using two reference circles, which provide a quantitative measure by comparing the actual contour with two reference circles: the Maximum Inscribed Circle (MIC) and the Minimum Circumscribed Circle (MCC). In this technique, a circle is drawn around the circularity profile, while another circle is drawn inside the roundness profile to fit it as described by [51]. In the measurement of roundness, two circles, namely the circumscribed circle and the inscribed circle, play a crucial role. By identifying the center of these circles, which forms the minimal zone circle, the center of the measured shape is determined. This center is used as a reference point to assess the roundness profile. The roundness error is then computed by measuring the radial distance between the minimum inscribed circle and the maximum circumcised circle. The difference in radii between these circles indicates the magnitude of the roundness error present in the measured object. The roundness error is established using the MZC technique by measuring the radial separation between two circles, specifically the minimum inscribed circle and the maximum circumscribed circle. The difference in radii between these two circles indicates the roundness error of the object being measured.  
\[\left\{\begin{array}{l}R_{1}-R_{2}\\ \sqrt{\left(xi-xc\right)^{2}+\left(yi-yc\right)^{2}}\leq R_{1}\\ \sqrt{\left(xi-xc\right)^{2}+\left(yi-yc\right)^{2}}\geq R_{2},i=1,2,3,\ldots n \\ R_{1}-R_{2}\geq 0\\ R_{L}\leq R\leq R_{LI};\\ x_{L}\leq x_{c}\leq x_{U};\\ y_{L}\leq y_{c}\leq y_{U}\end{array}\right\} \tag{1}\]  
where the \(R_{1}\) and \(R_{2}\) are the radii of \(c1\) and \(c2\), respectively, and thus share a center at \((xc,yc)\). The minimum radial separation is defined as the roundness error. It is mathematically formulated as  
\[e_{MZC}=R_{max}-R_{min} \tag{2}\]  
In recent years, there has been a growing interest in implementing computer vision systems in industrial workplaces to detect defects in products. These systems use various image processing techniques to analyze the features and surface of the model, providing valuable information to the experts. Advanced industrial systems necessarily involve ever-improving product performance and improved product quality during the production process [52; 53; 54]. Defects, such as scratches, imperfections, or holes on the surface of the product, on the other hand, have an adverse influence mainly on the product's aesthetics and user comfort but also on its performance [55; 56; 57; 58]. Therefore, defect detection is an efficient strategy for minimizing the environmental effects of product defects [59; 60; 61]. Machine vision in industrial processes  
Figure 1: The fundamentals of calculating roundness using three different algorithms: (**a**) MIC, (**b**) MCC, and (**c**) MZC [50].  
can help with a wide range of industrial tasks [62; 63; 64]. A typical industrial visual inspection system is categorized into three components: optical illumination, image acquisition, image processing, and defect detection [29; 65]. The development of an automated smart system-based interpreter for detecting and measuring the surface feature of Example 1, Part 21, of ISO 14649 standard using machine vision inspection (3 SMVI) is proposed in this study [66]. The 3 SMVI system will be developed in the milling machine using a camera system and lighting system [50; 66]. The 3 SMVI system is widely adopted as a standard tool for measuring and inspecting surfaces. This article explores the increasing use of camera systems to validate the accuracy of surface features in design and measurement. It also discusses the challenges of measuring roundness in circles using vision and validates the system based on IoT.