## 2 Materials and Methods
### Segmentation of Overlapping-Droplet-Image-Processing Method
#### 2.5.2 Overlapping-Droplet-Splitting Method
Image processing techniques are used to extract the contour of the overlapping droplets, as shown in Figure 8a. Convex-packet calculation and concave-point detection. Using the convex-packet algorithm in computational geometry, the smallest convex polygon encircling the entire contour can be obtained, as shown in Figure 8b. Then, by comparing the original contour with its convex envelope, convex-defect regions can be identified. The vertices of these convex-defect regions are the desired concave points. Concave-point screening and candidate concave-point selection. Not all detected concave points are meaningful, and it is necessary to screen significant concave points based on the depth of convex defects, filtering out insignificant concave points by setting appropriate depth thresholds, retaining only the concave points that have a substantial impact on segmentation, and, finally, obtaining the three concave points, i, j, and k, as shown in Figure 8c. Based on the assumption that the deepest concave point usually represents the most significant segmentation location, we iterated through all the concave  
Figure 8: Segmentation process for non-parallel overlapping droplet. Where (**a**) is Binary Image Extraction; (**b**) shows Convex Hull Construction; (**c**) shows Concave Points Filtering & Candidate Selection; (**d**) shows Non-Parallel Segmentation Line Generation. Figure 7: Segmentation of strongly overlapping droplets in a single concave point. Where (**a**) is Binary Image Extraction; (**b**) shows Concave Defect Detection& Key Points Localization; (**c**) shows Key Points Connection & Slope Calculation; (**d**) shows Baseline L Generation & Intersection Detection; (**e**) is Perpendicular Segmentation Line AD Identification. points that have been screened, and used the convex-defect-depth formula to select the deepest concave point as a candidate concave point. The candidate concave point is i.  
\[D_{\text{max}}=\text{max}(D_{i})\,i=1,2,\cdots,n \tag{1}\]  
where \(D_{i}\) denotes the depth of the ith concave point and n is the total number of concave point. The nearest concave-point pairing and segmentation-line generation uses Euclidean distance as a metric to find the concave point with the shortest distance to the candidate concave point among the non-candidate concave points. The Euclidean distance is calculated as follows:  
\[d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \tag{2}\]  
where (\(x_{1}\),\(y_{1}\)) and (\(x_{2}\),\(y_{2}\)) are the coordinates of the two concave points, respectively. Connecting the candidate concave point i and the found nearest concave point j forms the segmentation line of the non-parallel overlapping droplets, as shown in Figure 8d. This improved segmentation method cannot only effectively deal with complex non-parallel overlapping droplets, but can also adapt to a variety of serial and mixed overlapping cases, greatly improving the accuracy and applicability of segmentation. By combining geometric analysis and mathematical modelling, it is possible to identify and segment all kinds of complex overlapping-droplet structures more accurately. For parallel overlapping droplets, the centre-of-mass segmentation method is used. The segmentation method of parallel overlapping droplets is different to that of other types of overlapping droplets due to their unique structural features. In closed (non-porous) parallel overlapping droplets, the overlapping portion is located inside the entire region and the number of overlapping droplets is equal to the number of recessed regions. This property makes the segmentation methods for non-parallel overlapping droplets not applicable in this case. By analysing the image characteristics of parallel overlapping droplets, a concise and effective segmentation method was found. Figure 8 shows this process in detail, where the process is as follows:  
Concave-point screening (Figure 9a-c).