## 3 Results  
### Objective Evaluation Metrics  
(1) Model-based approach evaluation metrics  
To assess the effectiveness of the algorithm, we adopted four widely used metrics: the Receiver Operating Characteristic (ROC) curve, Signal-to-Clutter Ratio Gain (SCRG), Background Suppression Factor (BSF), and Contrast Gain (CG). The ROC curve, representing the Receiver Operating Characteristic curve, provides a comprehensive evaluation of detection performance. The \(y\)-axis of the curve represents the True Positive Rate (TPR), while the \(x\)-axis represents the False Positive Rate (FPR). The formulas for these metrics are as follows:  
\[TPR=\frac{number\ of\ detected\ target}{actual\ number\ of\ target} \tag{40}\]  
\[FPR=\frac{number\ of\ detected\ background\ pixels}{total\ number\ of\ image\ pixels} \tag{41}\]  
Detection performance can be quantified using the Area Under the Curve (AUC). A higher AUC value indicates better detection performance. The Signal-to-Clutter Ratio (SCR) measures the difficulty of detecting targets in infrared images. \[SCR=\frac{\left|\mu_{t}-\mu_{b}\right|}{\delta_{b}} \tag{42}\]  
Here, \(\mu_{t}\) represents the average grayscale of the target region, \(\mu_{b}\) denotes the average pixel value of the local neighborhood region, and \(\delta_{b}\) is the standard deviation. \(SCR_{in}\) and  
Figure 4: Small target detection process: The input image sequence is reorganized to retain only the ommatidial imaging regions. A weighting operator is then computed, followed by the construction of the proposed model, which is subsequently optimized using the ADMM framework. The term “soft” refers to the soft-thresholding function, which is used during the iterative updates of T.  
\(SCR_{out}\) refer to the SCR values of the input source and output detection images, respectively. The Signal-to-Clutter Ratio Gain (SCRG) is defined as SCR Gain. \[SCRG=\frac{SCR_{out}}{SCR_{in}} \tag{43}\]  
The effectiveness of background suppression in the detection algorithm is quantified by the Background Suppression Factor (BSF), expressed as follows:  
\[BSF=\frac{\delta_{in}}{\delta_{out}} \tag{44}\]  
Here, \(\delta_{in}\) and \(\delta_{out}\) represent the standard deviations of the local region before and after suppression, respectively. In summary, higher SCRG and BSF values indicate better performance of the algorithm in target enhancement and background suppression. However, in methods based on sparse and low-rank recovery, if background noise is minimized, the standard deviation becomes almost negligible, resulting in a calculated value of Inf. To address this issue, we utilize the Contrast Gain (CG) metric, defined as follows:  
\[CG=\frac{CON_{out}}{CON_{in}} \tag{45}\]  
Here, \(CON_{in}\) and \(CON_{out}\) represent the contrast of the input and output infrared images, respectively. Additionally, the calculation of CON is as follows:  
\[CON=|\mu_{t}-\mu_{b}| \tag{46}\]  
Furthermore, in the measurements of SCRG, BSF, and CG, the neighborhood size is defined as the area of the effective imaging region of a single ommatidium. (2) Deep learning-based approach evaluation metrics  
To fairly evaluate and compare the performance of both traditional model-based and deep learning-based small target detection methods, we adopt two commonly used metrics: precision and recall. These metrics are model-agnostic and can be consistently applied across different types of detection outputs (saliency maps, confidence maps, bounding boxes), making them well suited for cross-method evaluation. Recall is defined as the proportion of true targets that are successfully detected.