# Optical Element Surface Defect Size Recognition Based on Decision Regression Tree
## 1 Introduction  
The optical integration level of sophisticated equipment is getting higher and higher in recent years. For instance, laser inertial confinement fusion equipment, a communication satellite, and optical quantum computer have integrated a large quantity of fine optical elements [1; 2; 3]. The existence of optical elements surface defects not only affects the performance of this sophisticated equipment but also causes the collapse of the whole equipment system in some extreme situations. Hence, it is necessary to carry out the quality evaluation of optical elements at an early age. Defect size recognition is an important quantitative analysis basis of optical elements' surface quality evaluation, and its accuracy directly affects the final quality evaluation result. At present, the defect size is mainly deduced from the defect image gray distribution collected by the optical imaging system. This process is also called the reverse recognition of defect size [4]. Conventional optical imaging systems for defect detection contains a dark-field imaging system [5; 6], bright-field imaging system [7; 8], etc. Since dark-field imaging system only captures the scattered light inducedby the defect, the defect region will be brighter than the background in the image, which makes the defect to be detected at higher contrast and realizes super-resolution detection to a certain extent. Thus, a dark-field imaging system is more appropriate for detecting some tiny defects. When the size of the tiny defect is larger than the diffraction limit of the dark-field imaging system, the gray distribution of the defect has two peaks, and the defect size can be derived from the distance between two peaks. But as the defect size is close to the diffraction limit of the imaging system, its gray distribution becomes one peak, and the defect size cannot be recognized by the distance between two peaks anymore. To solve this problem, the commonly used image process method is image threshold segmentation, such as the Otsu method [9; 10]. By setting or calculating an appropriate gray threshold, the defect size can be approximately extracted from the image. But its recognition accuracy is usually low, and it needs some additional calibration technique to ensure its accuracy. It is very inconvenient because some known size defects need to be prepared for calibration in advance. Thus, the current image process method for defect size recognition needs to be reconsidered. Recently, with the development of the artificial intelligence field, machine learning is widely applied in the image process [11; 12; 13], and it has superior performance in defect detection, classification, etc. Yet, it has been rarely reported in defect size recognition. The main reason is that machine learning needs more known size defect images than the calibration procedure mentioned above to learn the relationship between the characteristic of the defect gray distribution and the defect size, which means a large quantity of known size defect samples needs to be made. This is extremely difficult to be achieved because the actual defect machining technique and machining accuracy cannot reach this condition, especially when the defect size is close to the diffraction limit of the imaging system. Hence, the scarce of the training data set limits the usage of machine learning. In this paper, we propose a defect size recognition method based on the machine learning of the decision regression tree (DRT) [14; 15; 16], whose training data is from the simulation model of the dark-field imaging system. Here, we take scratch as an example, using the DRT algorithm to recognize the width of the scratch. Firstly, an electromagnetic simulation model of microscopic scattering dark-field imaging (MSDI) system is built based on the finite-difference time-domain (FDTD) method [17; 18] to gain the far-field simulation light intensity distribution of the scratch.