## 1 Introduction
However, segmentation algorithms relying on concave-point matching are more sensitive to noise, especially in the case of strongly overlapping droplets with only a single concave point, and existing concave-point methods are prone to misclassification. To address the above problems, this study proposes a multi-stage overlapping-droplet segmentation method that combines an optimised XGBoost classification model with an improved concave-point-matching strategy, aiming to effectively solve the difficulties of boundary blurring and insufficient feature differentiation faced by existing segmentation methods when dealing with irregular overlapping droplets. Specifically, by integrating the advantages of machine learning and geometric analysis, the method not only efficiently handles severe overlapping cases, including strongly overlapping scenes that produce only a single dimple, but also avoids the need for morphological operations (e.g., dilation and erosion) by optimising the feature classification and enhancing the dimple-matching strategy, thus maximising the preservation of critical detail information in the image. This study experimentally validates the effectiveness of the method, demonstrates the potential of combining machine learning with geometric analysis in dealing with overlapping-droplet segmentation under complex conditions, and provides a new technological path for research and applications in related fields. The main research contributions are as follows:1. An overlapping fog droplet image preprocessing method is designed to effectively reduce interfering information through high-pass filtering, grey-scale transformation, binarisation, automatic thresholding and diffuse water filling. 2. Determination of overlapping fog-droplet categories, based on the number of depressions and morphological features detected in overlapping fog droplets, the types of overlapping fog droplets can be classified into three main categories: strong overlapping of single depressions, parallel overlapping, and non-parallel overlapping. This classification provides a clear guiding framework for subsequent segmentation and identification. 3. XGBoost-based type-recognition model. A Bayesian-optimised XGBoost-based triple-classification model is constructed, which takes the fog droplet's contour parameters as the feature input, and significantly improves the classification accuracy and robustness through hyperparameter tuning. Experimental results show that the model achieves 98% classification accuracy for more than 3000 overlapping fog drops, which effectively meets the subsequent classification requirements. 4. Differentiated segmentation strategy and iterative segmentation for overlapping droplets of different overlapping types. This study innovatively proposes a differentiated segmentation strategy and an iterative segmentation mechanism that significantly improve segmentation accuracy and efficiency. This targeted segmentation strategy effectively solves the segmentation problem in complex overlapping scenarios. In order to further optimise the segmentation effect, this study introduces a cyclic iterative segmentation mechanism: the initial segmentation results are analysed twice, and the residual overlapping regions are identified and segmented iteratively until independent droplets are obtained or an inseparable state is reached. 5. Conduct overlapping-droplet segmentation tests for verification and analysis. The experimental results show that the method can efficiently extract overlapped and contacted droplets from a single WSP image in an average of 2.66 s, which is a significant advantage over existing segmentation methods. The method not only solves the problem of low segmentation accuracy when dealing with overlapping or large overlapping droplets, but also improves the unseparation phenomenon generated when segmenting strongly overlapping droplets, and effectively restores the edges of the droplets in the overlapping region. The rest of the article is organised as follows.