## 2 Materials and Methods
### Segmentation of Overlapping-Droplet-Image-Processing Method
#### 2.5.2 Overlapping-Droplet-Splitting Method  
In this study, a fine and efficient segmentation process for overlapping droplets is designed, which consists of six key steps: pre-segmentation, concave-point detection and concave-point segmentation, further separation of complex overlapping droplets, and ellipse fitting. In order to achieve the best segmentation results, three specialised segmentation methods have been developed for each of the three typical types of overlapping droplets mentioned above (single-concave-point strong overlap, parallel overlap, and non-parallel overlap). This differentiated segmentation strategy takes into account the characteristics and structure of each type of overlapping droplet, ensuring the accuracy and adaptability of the segmentation process. 1. Pre-segmentation  
Pre-segmentation of the extracted overlapping droplets is achieved by separating the overlapping droplets with low degree of overlap using morphological weak corrosion [26]. Figure 6 shows the pre-segmentation process. 2. Concave-point detection and segmentation  
After the pre-segmentation process, the detection of concave points for strongly overlapping droplets begins, and in this paper, a convex-defect-based method is used to detect the concave points [27]. Firstly, the boundary of the contour is determined by calculating the convex packet of the contour; then, the convexity defect function is used to identify the concave region between the convex packet and the contour. For each depressed region, significant dents are screened according to a preset depth threshold, and their coordinates and depression depth are recorded. This method effectively identifies significant concave points in the contour and provides an accurate characterisation for further shape analysis. For the single-concave-point strongly overlapping droplets, the vertical connective segmentation method is used to, firstly, extract a binarised image of the strongly overlapping droplets, as shown in Figure 7a. Subsequently, concave defects and concave points are detected on this binarised image, and the results are shown in Figure 7b. In this example, only one concave defect is detected in the overlapping target, and its critical points are labelled as concave point A, as well as the start point of the concave defect, B1, and the end point, B2. Next, the points B1 and B2 are connected, and the slope of this connecting line is calculated, as shown in Figure 7c. Using this slope, a straight line is drawn across the concave point A, which intersects the droplet-edge contour at points C1 and C2, and this line is labelled L [28], as shown in Figure 7d. In order to determine the best segmentation line, all points on the edge contour on the other side of the straight line, L, are traversed. The goal is to find a particular point, D, such that the line connecting AD is perpendicular to the straight line, L, as shown in Figure 7e. The point D is found and the points A and D are connected, and this connecting line (represented by a solid black line in the figure) is the optimal demarcation line for the overlapping droplets. Through this precise geometric analysis method, the complex overlapping droplets can be effectively segmented, laying the foundation for subsequent image analysis and processing. Figure 6: Pre-segmentation of overlapping images. For the non-parallel overlapping droplets, the minimum-distance segmentation method is used, and for the non-parallel overlapping droplets with the number of concave points exceeding 1, which occupies a considerable proportion of the overlapping-droplet types, the segmentation method is significantly different from the previous two cases. Non-parallel overlapping droplets can be further subdivided into serial-overlapping and mixed-overlapping droplets, within which the segmentation of mixed-overlapping droplets is mainly based on the segmentation method of serial-overlapping droplets. The following is the optimised segmentation process:  
Contour analysis.