# Gear Integrated Error Determination Using the Gaussian Template Convolution-Facet Method  
[PERSON]  
1 Beijing Engineering Research Center of Precision Measurement Technology and Instruments,  
Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China;  
[EMAIL_ADDRESS]  
[PERSON]  
1 Beijing Engineering Research Center of Precision Measurement Technology and Instruments,  
Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China;  
[EMAIL_ADDRESS]  
[PERSON]  
2 School of Construction Machinery, Shandong Jiaotong University, Jinan 250357, China; [EMAIL_ADDRESS]  
[PERSON]  
1 Beijing Engineering Research Center of Precision Measurement Technology and Instruments,  
Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China;  
[EMAIL_ADDRESS]  
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: [[https://www.mdpi.com/journals/pdf/10012](https://www.mdpi.com/journals/pdf/10012)]([https://www.mdpi.com/journals/pdf/10012](https://www.mdpi.com/journals/pdf/10012))  
have applied sub-pixel detection algorithms to gear measurement, resulting in improved measurement accuracy. [PERSON], [PERSON] et al. [11] used the Zernike moment sub-pixel algorithm and morphological filtering method to extract and refine gear contours. A spur gear with a tip diameter of 49.751 mm and 23 teeth was selected as the measurement object, achieving a relative error of approximately 5% in the accumulated total deviation of the measured tooth pitch. [PERSON], [PERSON] et al. [12] measured the outer diameters of 12 spur gears with a nominal diameter of 62.014 mm using an interpolation-based sub-pixel operator provided by the Halcon algorithm library, with an error range between 2 um and 30 um. [PERSON], [PERSON] et al. [13] proposed a sub-pixel edge localization algorithm based on Gaussian integral surface fitting, which meets the profile deviation measurement requirements of five-level precision involute spur cylindrical gears with a tooth number of 60 and a modulus of 2.0 mm.  
It is evident that while certain sub-pixel algorithms have been employed to enhance measurement accuracy, they are only able to accommodate gears of medium modulus, and there is a lack of research assessing the accuracy of fine modulus (modulus less than or equal to 1.0 mm) gears. Additionally, the current methods are limited to measuring individual errors, rather than gear integrated errors, leading to insufficient gear error information derived from images.  
In this study, a gear integrated error measurement method based on GTC\(-\)Facet is proposed. This method utilizes the Canny operator to determine the gear edge transition area, and employs the GTC method to acquire the Hessian matrix. By matrix manipulation, the gray distribution Facet model is promptly obtained, enhancing the positioning speed and accuracy of the gear profile. Additionally, it integrates the theory of gear integrated error, enabling the instant acquisition of gear integrated error curves. This approach not only facilitates the decomposition of individual errors such as pitch deviation and profile deviation in gear analysis measurement, but also enables the decomposition of tangential composite deviation in gear functional measurement, significantly improving gear error analysis capabilities.