# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 5 Experiments
### Video Background Modeling
#### 5.2.2 Quantitative Experiments  
Since the datasets in Section 5.2.1 do not include the ground truth background for each frame of the video, we perform the algorithms on the ChangeDetection.net(CDNet) dataset2014 [50] which consists of visual surveillance and smart environments. However, only part of frames are accompanied with ground truth background and thus we only take out some of frames of the video for testing. That is, we select the frames from 900 to 1000 of the dataset _highway_, _pedestrians_ and _PETS2006_ in the baseline category. Hence the size of the video will be \(h\times w\times 101\), where \(h\times w\) is the resolution of each frame, and 101 is the number of frames. We compare our algorithm with RPCA [4], Gamma [26], SNN [49] and TRPCA [20]. The parameters setting is conducted from Section 5.2.1.  
To evaluate the performance of the experiment results, we use the three metrics for the quantitative measure, that is, Recall, Precision and F-Measure. They are defined as:  
\[\text{Recall}=\frac{\text{TP}}{\text{TP+FN}},\]  
\[\text{Precision}=\frac{\text{TP}}{\text{TP+FP}},\]  
\[\text{F-Measure}=2\frac{\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}},\]  
where TP, FP and FN are true positives numbers, false positives numbers and false negatives numbers respectively. We primarily use F-Measure for overall evaluation since it combines the metrics Recall and Precision and also makes a balance between Recall and Precision which can be viewed as a good indicators for comparison.  
From Table 2, we can see that the proposed algorithm achieves better segmentation performance among the compare methods with respect to the F-Measure. Some of the frames of the segmentation results are shown in Figure 7, it can be observed that among the three dataset in CDNet our method can usually get the clearer and more accurate background segmentation. These quantitative comparison illustrates the superior of t-Gamma tensor quasi-norm as the surrogate of the rank of tensor.  
Figure 6: Background modeling: two example frames from Lobby sequences are shown. (**a**) Original; (**b**) RPCA [4]; (**c**) Gamma [26]; (**d**) SNN [49]; (**e**) TRPCA [20]; (**f**) t-Gamma.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{3}{c}{**RPCA**} & \multicolumn{3}{c}{**Gamma**} & \multicolumn{3}{c}{**SNN**} & \multicolumn{3}{c}{**TNN**} & \multicolumn{3}{c}{**t-Gamma**} \\ \cline{2-13}  & **Recall** & **Precision** & **F-Measure** & **Recall** & **Precision** & **F-Measure** & **Recall** & **Precision** & **F-Measure** & **Recall** & **Precision** & **F-Measure** \\ \hline highway & 0.8642 & 0.9109 & 0.8869 & 0.8837 & 0.9067 & 0.895 & 0.6982 & 0.9041 & 0.7879 & 0.8235 & 0.902 & 0.861 & 0.8805 & 0.909 & **0.8961** \\ \hline pedestrians & 0.9983 & 0.9419 & 0.9692 & 0.9982 & 0.9275 & 0.9616 & 0.9955 & 0.9285 & 0.9608 & 0.9983 & 0.9258 & 0.9607 & 0.9982 & 0.9309 & **0.9634** \\ \hline PETS2006 & 0.8151 & 0.7131 & 0.7607 & 0.8326 & 0.7678 & 0.7989 & 0.288 & 0.6974 & 0.4077 & 0.8033 & 0.6853 & 0.7396 & 0.8301 & 0.7805 & **0.8045** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Quantitative Evaluation for algorithm on CDNet Datasets  
Figure 7: Visual results for CDNet dataset. Row 1 and row 2 are samples of _highway_, row 3 and row 4 are samples of _palestrians_; row 5 and row 6 are samples of _PETS2006_. (a) Original; (b) Ground True; (c) RPCA [4]; (d) Gamma [26]; (e) SNN [49]; (f) TRCA [20]; (g) t-Gamma