## 3 Use Case: (Com)2 INet for Levee Piping Hazard Inspection via Remote Sensing
For each date point \(x_{j}(j=1,\ldots,m)\) and each Gaussian component \(i(i=1,\ldots,k)\), compute the posterior probability (responsibility) that \(x_{j}\) belongs to the \(i-\)th Gaussian:\(\gamma_{ji}=\frac{\pi_{i}\mathbb{N}(x_{j}|\chi_{i},\underline{\Sigma})}{ \sum_{j=1}^{k}\mathbb{N}(x_{j}|\chi_{i},\underline{\Sigma})}\) where \(\mathbb{N}(x_{j}|\chi_{i},\underline{\Sigma}_{i})\) is the Gaussian probability density function:\(\mathbb{N}(x_{j}|\chi_{i},\underline{\Sigma})=\frac{1}{(2\pi)^{d/2}|\underline{ \Sigma}|^{1/2}}\). M-step (Maximization):
7. Update the parameters of the Gaussian components based on the responsibility \(\gamma_{ji}\): a. Update the mixing coefficients:\(\pi_{i}=\frac{1}{m}\sum_{j=1}^{m}\gamma_{ji}\) b. Update the means:\(\chi_{i}=\frac{\sum_{j=1}^{m}\gamma_{ji}x_{j}}{\sum_{j=1}^{m}\gamma_{ji}}\) c. Update the covariance matrices:\(\underline{\Sigma}_{i}=\frac{\sum_{j=1}^{m}\gamma_{ji}(x_{j}-\chi_{i})(x_{j}- \chi_{i})^{T}}{\sum_{j=1}^{m}\gamma_{ji}}\) Check for Convergence:
8. Compute the log-likelihood of the data under the current model: \(logL=\sum_{j=1}^{m}log\big{(}\sum_{i=1}^{k}\pi_{i}\cdot\mathbb{N}(x_{j}|\chi_ {i},\underline{\Sigma}_{i})\big{)}\).. If the change in log-likelihood is less than the threshold \(\epsilon\) stop the iteration. 9. Output:
10. The parameters of the Gaussian components \((\chi_{i},\underline{\Sigma}_{i},\pi_{i})\) and the cluster assignments based on the maximum posterior probability: Cluster assignment for \(x_{j}=argmax_{i}\gamma_{ji}\). ```  
**Algorithm 4** GMM-based levee piping hazard inspection. **Performance Evaluation**. _Evaluation Indicator_. Mean Intersection over Union (mIoU) and total runtime of per frame are adopted in this paper for evaluating accuracy of target extraction and algorithm efficiency. The mIoU quantitatively measures the accuracy of predicted segmentation results compared to the label truth. It evaluates the overlap between the predicted segmentation region and the true region for each class and computes the average across all classes. As a robust and interpretable metric, mIoU is particularly suitable for assessing the performance of clustering and segmentation algorithms, including those applied to UAV thermal infrared imagery for levee piping hazard detection. For the specific target, the IoU and the total runtime are computed as below. \[IoU=\frac{True\ Positive}{True\ Positive+False\  
Experimental DataThe experimental data in this manuscript were obtained using remote sensing images captured by the DJI H20T sensor mounted on the DJI M300 UAV. The images were taken from a nadir perspective with a flight speed of 8 m per second, primarily acquiring thermal infrared remote sensing images. The data collection took place in the Pajiang River Basin, Fogang County, Qingyuan City, Guangdong Province, on 20 April 2023. The UAV operated at a flight altitude of 30 m, and the thermal infrared camera had a resolution of \(640\times 512\) with a Noise Equivalent Temperature Difference (NEDT) \(\leq 50 mK@f/1.0\). During this data acquisition, eight frames of thermal infrared remote sensing images containing piping hazards were collected, as shown in Figure 6. These thermal infrared remote sensing images with piping hazards were used in this manuscript to test and evaluate the performance of the proposed SC-based SACM algorithm, in terms of the accuracy of target extraction and computational time of benchmarks. To be specific, the study site is located in Qingyuan city, Guangdong province, China. Qingyuan city is characterized by complex terrain with undulating mountains. It has a subtropical monsoon climate, where summers are hot and winters are warm. The flood season occurs from April to September. During this period, precipitation is heavily influenced by monsoons and typhoons, making the area prone to flooding disasters. Moreover, the data was collected from the levee at the confluence of the Pajiang River and the Pa'ershui River. Due to its geographical location, this area is frequently affected by floods.