# Infrared Dim and Small Target Detection Based on the Improved Tensor Nuclear Norm
## 6 Results and Analysis
### Evaluation Indicators  
In order to better compare the performance of this algorithm with that of other algorithms, some evaluation metrics commonly used in the field of infrared dim and small target detection were used in this study to objectively evaluate the performance of this algorithm in terms of background suppression, target energy enhancement, and noise cancellation. Two evaluation metrics, the background suppression factor (BSF) and the signal-to-noise ratio (SNR), were introduced in the paper. The BSF is mainly used to evaluate the effect of the background image predicted by the algorithm by identifying the difference between the image and the original image. The calculation formula is as follows:  
\[BSF=\sigma_{in}/\sigma_{out} \tag{23}\]  
where \(\sigma_{in}\) and \(\sigma_{out}\) are the mean squared differences between the input image and the difference image, respectively. The signal-to-noise ratio can be used to evaluate the contrast between the target and the noise energy obtained by the algorithm to verify the detection and background estimation effects of the algorithm, as follows:  
\[SNR=10\times\text{log}_{10}^{(A_{E}-A_{B})/\sigma_{B}} \tag{24}\]  
where is the mean gray value of target region \(A_{E}\), \(A_{B}\) is the mean gray value of the background region, and \(\sigma_{B}\) is the standard deviation of the gray value of the background region. The calculation results for the nine classical algorithms and the algorithm described in this paper for the three evaluation metrics in six scenes are shown in Tables 1 and 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Scene 1**} & \multicolumn{2}{c}{**Scene 2**} & \multicolumn{2}{c}{**Scene 3**} \\ \cline{2-7}  & **Bsf** & **SNR** & **Bsf** & **SNR** & **Bsf** & **SNR** \\ \hline Anisotropy & 24.79 & 2.17 & 16.87 & 2.15 & 29.45 & 2.94 \\ Top\_hat & 67.26 & 8.84 & 39.13 & 6.14 & 76.07 & 6.92 \\ mpcm & 424.01 & 13.94 & 75.83 & 11.64 & 316.09 & 14.07 \\ GST & 640.39 & 15.30 & 124.10 & 10.46 & 217.61 & 10.21 \\ VATF & 92.30 & 10.54 & 35.83 & 6.24 & 78.51 & 6.26 \\ FKWR & 187.01 & 11.39 & 46.23 & 7.06 & 73.58 & 7.24 \\ SRWS & 1.43 & 11.95 & 1.04 & 13.92 & 1.91 & 14.51 \\ Aadcdd & 428.21 & 14.49 & 113.59 & 9.05 & 246.90 & 11.21 \\ PSTNN & 307.94 & 15.07 & 87.53 & 9.62 & 186.54 & 10.65 \\ Proposed & **744.35** & **16.92** & **469.04** & **16.03** & **827.34** & **14.54** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Scene 1–3 Bsf and SNR.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Scene 4**} & \multicolumn{2}{c}{**Scene 5**} & \multicolumn{2}{c}{**Scene 6**} \\ \cline{2-7}  & **Bsf** & **SNR** & **Bsf** & **SNR** & **Bsf** & **SNR** \\ \hline Anisotropy & 31.93 & 5.92 & 23.65 & 3.17 & 29.73 & 5.93 \\ Top\_hat & 80.27 & 8.86 & 57.09 & 8.93 & 80.31 & 5.92 \\ mpcm & 365.00 & 15.77 & 334.98 & 14.35 & 261.97 & 9.75 \\ GST & 473.14 & 14.38 & 407.67 & 12.63 & 489.30 & 8.76 \\ VATF & 53.78 & 6.85 & 132.99 & 13.46 & 369.76 & 7.92 \\ FKWR & 103.34 & 10.44 & 173.07 & 12.27 & 87.48 & 1.20 \\ SRWS & 1.68 & 15.23 & 1.04 & 15.17 & 1.70 & 16.68 \\ Aadcdd & **574.57** & 15.28 & 403.66 & 12.87 & 347.99 & 13.71 \\ PSTNN & 223.49 & 12.59 & 221.20 & 14.92 & 408.40 & 8.68 \\ Proposed & 513.87 & **16.22** & **424.06** & **16.13** & **1753.70** & **18.04** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Scene 4–6 Bsf and SNR.  
To further investigate the performance of these algorithms, ROC curves were plotted for six scenes, where the horizontal axis is the detection rate (\(P_{d}\)) and the vertical axis is the false alarm rate (\(P_{f}\)), as shown in Equation (25). NTDT is the number of real targets detected; NFDT is the number of false alarm targets detected; NT is the total number of real targets present in the image; NP is the total number of targets detected in the image.  
\[\left\{\begin{array}{l}P_{d}=\frac{NTDT}{N}\times 100\%\\ P_{f}=\frac{NDT}{NP}\times 100\%\end{array}\right. \tag{25}\]