# Spectral Information Divergence-Driven Diffusion Networks for Hyperspectral Target Detection
## 4 Results
### Evaluation Metrics  
To comprehensively evaluate HTD performance, this paper uses the 2-D and 3-D Receiver Operating Characteristic Curves (ROC) and the target-background separation map as objective evaluation metrics. The ROC curve illustrates the relationship between the detection rate \(P_{D}(\tau)\) and the false alarm rate \(P_{F}(\tau)\) as the classification threshold \(\tau\) varies. The corresponding formula is provided in Equations (25) and (26). \[P_{D}(\tau)=\frac{TP}{TP+FN} \tag{25}\]  
\[P_{F}(\tau)=\frac{FP}{FP+TN} \tag{26}\]  
The detection rate \(P_{D}(\tau)\) is defined as the proportion of positive cases correctly detected at a given threshold \(\tau\), as described in Equation (25). The false alarm rate \(P_{F}(\tau)\) is the proportion of incorrectly detected positive cases relative to all negative cases at threshold \(\tau\), as shown in Equation (26). An ideal ROC curve approaches the upper left corner (\(0,1\)), indicating a high detection rate and a low false alarm rate. By varying the threshold \(\tau\), 2D ROC curves \((\tau,P_{D}(\tau))\) and \((\tau,P_{F}(\tau))\) can be plotted. A faster rise in the 2D ROC curves \((\tau,P_{D}(\tau))\) indicates a higher detection rate, and a faster decline in the 2D ROC curves \((\tau,P_{D}(\tau))\) indicates a lower false alarm rate. The Area Under the Curve (AUC) represents the area enclosed by the ROC curve and \(P_{D}(\tau)\), as defined in Equation (27). A larger AUC value indicates a higher detection rate for the model. \[\text{AUC}=\int_{0}^{1}P_{D}(\tau)dP_{F}(\tau) \tag{27}\]  
To further evaluate the target detection performance of the model, [PERSON] et al. [57] proposed a 3D ROC curve detection method. The 3D ROC curve includes five key metrics: \(\text{AUC}_{(P_{F},P_{D})},\text{AUC}_{(\tau,P_{D}(\tau))}\text{AUC}_{(\tau, P_{F}(\tau))},\text{AUC}_{BS},\text{AUC}_{SNPR}\). Calculated using Equations (28) and (29):  
\[\text{AUC}_{BS}=\text{AUC}_{(P_{F},P_{D})}-\text{AUC}_{(\tau,P_{F}(\tau))} \tag{28}\]  
\[\text{AUC}_{SNPR}=\frac{\text{AUC}_{(\tau,P_{D})}}{\text{AUC}_{(\tau,P_{F}( \tau))}} \tag{29}\]  
The value of \(\text{AUC}_{(P_{D},P_{F})}\) is interpreted the same way as AUC: a larger value indicates better detection algorithm performance, with an optimal value of \(1\). \(\text{AUC}_{(P_{D},P_{F})}\) represents the overall performance of the algorithm and is the most important metric. However, \(\text{AUC}_{(\tau,P_{F}(\tau))}\) is interpreted differently: a smaller value signifies better performance, with the optimal value being \(0\). a smaller \(\text{AUC}_{(\tau,P_{F}(\tau))}\) value reflects a lower false alarm rate and better background suppression. \(\text{AUC}_{BS}\) is a composite indicator calculated by weighting and summing the two AUC metrics. Consequently, a larger \(\text{AUC}_{BS}\) value indicates better detection algorithm performance, with the optimal value being \(1\). \(\text{AUC}_{SNPR}\) is inspired by the concept of Signal-to-Noise Ratio (SNR), where \(\text{AUC}_{(P_{F},P_{D})}\) represents the signal and \(\text{AUC}_{(\tau,P_{F}(\tau))}\) represents the noise. A larger \(\text{AUC}_{SNPR}\) value indicates better performance of the detection algorithm, with the optimal value being \(+\infty\). The 3D ROC curve provides a three-dimensional visualization, allowing for intuitive assessment of the impact of threshold \(\tau\) on the detection rate \(P_{D}(\tau)\) and the false alarm rate \(P_{F}(\tau)\). The target-background separation map, using the Box-Whisker method, analyzes the separation between target and background. There are two main criteria for evaluation. The first criterion is whether there is any overlap between the target box and the background box. If overlap exists, it indicates that the target and background are mixed during thedetection process, making it difficult to distinguish them. The second criterion is based on the area of the background box. The upper and lower bounds of the box represent the 75 th and 25 th percentiles of the data, respectively.