## 1 Introduction  
The stability and reliability of automobile brake systems play a vital role in ensuring the safety of drivers and passengers, as well as the safety of cars in the process of high-speed driving and rapid stops within short distances. The automobile brake piston component is an essential constituent of the braking system. In the process of manufacturing and processing parts, due to errors in the processing system itself the positioning error of the part; the fixture error; and other factors, errors in the processed parts are unavoidable. As a kind of position tolerance, the coaxiality error has a great influence on the assembly of shaft parts. To ensure the stability and reliability of the brake system and extend its service life, strict assembly tolerance requirements are necessary. Therefore, it is very important to measure the coaxiality of automobile brake piston components. The rapid and accurate measurement of coaxiality error has always been a difficulty in engineering practice. The coaxiality measurement methods of traditional shaft parts can be divided into contact and non-contact. At present, common tools for contact detection include coaxiality measuring instruments, coordinate measuring machines, and bearing gauges. Common non-contact detection methods include machine vision detection, laser collimators, and laser displacement sensors. [PERSON] et al. [1] proposed a subpixel edge detection algorithm based on polynomial interpolation, and obtained image edges through a series of image preprocessing algorithms. Then, the subpixel edge of the image was obtained via quadratic polynomial interpolation of the edge points. Finally, an improvedleast squares method was employed to fit the circular edges and calculate the coaxiality error of the parts, thus enhancing the speed and accuracy of coaxiality detection during the assembly process of aerospace small parts. [PERSON] et al. [2] proposed a method utilizing three capacitive sensors to measure the coaxiality between the suspended coil and the outer magnetic yoke. They also established a mathematical relationship and theoretical model between the output of capacitive sensors and the coaxiality error, providing a new method for the alignment of geometric coaxiality in the field of precision instrument research. [PERSON] et al. [3] proposed a coaxiality error measurement method based on 3D point clouds. Firstly, they utilized a laser scanner to obtain the 3D point cloud model of the measured part, and a circular method of rotary cutting was put forward to obtain equally spaced optimal slices of the crankshaft journal. Then, they utilized an optimized Pratt circle fitting method to obtain the centers of the slice. Finally, the reference axis was calculated via the least squares method, and the coaxiality error was calculated. [PERSON] et al. [4] proposed a non-contact optical measurement approach. According to the data points of a cross-section obtained using laser displacement sensors, data points of the gear top were separated to fit the center of the section. Then, the coordinates of the central point of the cross-section were calculated using the least squares method. Following this, the reference axis was fitted using the particle swarm optimization algorithm, and the coaxiality error of the bevel gear relative to the spline was calculated according to the criterion of minimum containment area. They achieved the rapid detection of coaxiality for compound gear shafts. [PERSON] et al. [5] proposed a rapid method for measuring the coaxiality of small-diameter series holes based on laser displacement sensors. This measurement method did not require the rotation angle information of the rod. After rotating randomly several times, the coaxiality error of the series holes was calculated using the measured values from a laser displacement sensor. [PERSON] et al. [6] proposed a concentricity measurement method based on the principle of laser ranging.