# Accelerated Tensor Robust Principal Component Analysis via Factorized Tensor Norm Minimization
## 5 Experimental Results
### Background Subtraction  
Another popular application of TRPCA is background subtraction. Since the video sequences can be modeled as a combination of a static background (low-rank component) and moving objects (sparse component), the background subtraction problem can be effectively addressed using TRPCA algorithms [26]. To evaluate the performance of the algorithms, three video sequences are used, which include the \"highway\", \"Bus station\", and \"park\". We extract 300 frames from the video sequences with frame sizes of \(240\times 320\), \(240\times 360\), and \(288\times 352\), respectively. To assess the accuracy of background subtraction, we compute precision, recall, and F-score based on the foreground detection results and corresponding ground-truth data. As in the color image recovery experiment in Section 5.1, we set the stopping criterion \(\epsilon=1.0e-4\) for all algorithms. The hyperparameters for each algorithms are chosen empirically to achieve the highest possible F-score value. Tables 5-7 summarize the performance comparison of the algorithms. Note that the precision, recall, and F-score values presented in Tables 5-7 are averaged over all video frames. Figure 5a-c illustrate example frames from the video sequences, along with their ground-truth background subtraction results and the outputs from each algorithm. From the results, TRPCAAL consistently achieved the fastest execution time while maintaining competitive or the best F-score values across all video sequences, demonstrating strong overall performance in both speed and accuracy. Unlike the experimental results in color image recovery, N-TRPCA produced unsatisfactory results in both accuracy and execution time. While IRCUR achieved slightly higher F-score values in some cases, it suffered from extremely high execution times, making it impractical for real-time or large-sized video sequences. Therefore, TRPCAAL emerges as the most attractive choice, providing accurate results with minimal computational cost in background subtraction. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Noise** & & **Iterations** & **Execution Time** & **PSNR (dB)** & **FSIM** \\ \hline
10\% & noisy image & - & - & 18.6444 & 0.8690 \\  & **TRPCAAL** & 8.00 & **0.3435** & **37.7254** & **0.9915** \\  & FTTNN & 26.00 & **1.0196** & 33.9594 & 0.9725 \\  & IRCUR & 6.80 & 16.1834 & 33.5131 & 0.9698 \\  & TRPCA\_TNN & 58.05 & 4.3278 & 35.6128 & 0.9838 \\  & N-TRPCA & - & 19.4259 & **42.1522** & **0.9976** \\  & EATP-DCT & 24.10 & 1.4197 & 35.2786 & 0.9740 \\ \hline
20\% & noisy image & - & - & 15.6322 & 0.7935 \\  & **TRPCAAL** & 13.15 & **0.5223** & **34.3571** & **0.9830** \\  & FTTNN & 25.00 & **0.8425** & 29.5879 & 0.9186 \\  & IRCUR & 5.90 & 10.6101 & 28.1123 & 0.9047 \\  & TRPCA\_TNN & 57.25 & 4.2798 & 33.0207 & 0.9730 \\  & N-TRPCA & - & 19.4864 & **38.8358** & **0.9940** \\  & EATP-DCT & 27.35 & 1.5598 & 33.9489 & 0.9716 \\ \hline
40\% & noisy image & - & - & 12.6272 & 0.6913 \\  & **TRPCAAL** & 17.10 & **0.5034** & **27.0085** & **0.9195** \\  & FTTNN & 24.00 & **0.7682** & 23.6106 & 0.8241 \\  & IRCUR & 19.20 & 18.1779 & 24.3123 & 0.8327 \\  & TRPCA\_TNN & 56.60 & 4.3993 & 24.7224 & 0.9139 \\  & N-TRPCA & - & 20.2492 & **31.5374** & **0.9640** \\  & EATP-DCT & 44.80 & 2.3762 & 20.3844 & 0.8389 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results for the “house” image with 10%, 20%, and 40% noise levels. The top two results in execution time and accuracy are bolded. \begin{table}
\begin{tabular}{l l l l l l l} \hline
**Noise** & & **Iterations** & **Execution Time** & **Precision** & **Recall** & **F-Score** \\ \hline
**TRPCAAL** & 4 & **4.2106** & 0.3719 & 0.5477 & **0.4356** \\ FTTNN & 25 & **36.8798** & 0.3322 & 0.1521 & 0.2047 \\ IRCUR & 24 & 585.0544 & 0.1704 & 0.6491 & **0.2565** \\ TRPCA\_TNN & 43 & 93.0426 & 0.1195 & 0.1549 & 0.1310 \\ N-TRPCA - & - & 287.7277 & 0.7393 & 0.0243 & 0.0470 \\ EATP-DCT & 26 & 38.5858 & 0.1635 & 0.5586 & 0.2455 \\ \hline \end{tabular}
\end{table}