# Accelerated Tensor Robust Principal Component Analysis via Factorized Tensor Norm Minimization
## 5 Experimental Results
### Color Image Recovery  
Color images may contain sparse and impulse noise due to sensor malfunction, external interference, or transmission errors caused by network issues. Such noise significantly degrades the performance of image processing and computer vision algorithms, necessitating the separation of sparse noise from the image as a pre-processing step. One effective solution for recovering color images is to apply TRPCA models, which approximate the clean image as a low-rank tensor under sparse noise corruption. In this section, we compare the performance of various TRPCA algorithms in the context of color image recovery. To evaluate their effectiveness, we use three popular test images: \"peppers\", \"airplane\", and \"house\", shown in Figures 1a, 2a, and 3a, respectively. The image sizes are \(128\times 128,256\times 256\), and \(512\times 512\). Impulse noise with random magnitudes and positions is added to the test images at different noise levels (10%, 20%, and 40%). Note that the noise level indicates the proportion of image pixels corrupted by impulse noise relative to the total number of pixels. Example images corrupted with 20% noise are shown in Figures 1b, 2b, and 3b. To compare the performance, we measured the execution time, the number of iterations to converge, and the peak signal-to-noise ratio (PSNR), defined as  
\[\text{PSNR}=10\text{log}_{10}\Bigg{(}\frac{||\mathcal{X}||_{\infty}^{2}}{ \frac{1}{n_{1}n_{2}n_{3}}||\mathcal{X}-\mathcal{L}||_{F}^{2}}\Bigg{)},\]  
where \(\mathcal{X}\) represents the original image and \(\mathcal{L}\) is the low-rank approximated image recovered by the algorithms. The PSNR value quantifies the overall differences between the original and recovered image. For further evaluation of recovery accuracy, we use the feature similarity index (FSIM) [25], which emphasizes structural similarity between the original and recovered image. For all algorithms, we set the stopping criterion \(\epsilon=1e-4\) and the maximum iteration number as 500. The stopping criterion is defined as  
\[\frac{\|\mathcal{L}_{k}-\mathcal{L}_{k-1}\|_{F}}{\|\mathcal{L}_{k-1}\|_{F}} \leq\epsilon.\]  
The hyperparameters is empirically chosen, for example \(r,\lambda\) and \(\mu\) for TRPCAAL. Figure 4 illustrates PSNR and execution time as functions of \(\mu\) and \(r\) using the \"airplane\" image with 20% random noise. The trends in the plots indicates that the choice of \(r\) is more important to the accuracy of TRPCAAL, while its effect on execution time is minimal. This implies that the proposed algorithm maintains computational stability. Throughout the additional experiments, we found that varying \(\lambda\) had a negligible impact on performance, so we fixed \(\lambda=0.1\). For the other algorithms, we empirically selected the hyperparameters to optimize recovery accuracy. Tables 1-3 present the experimental results for all algorithms. In the tables, the top two performing results in each category are highlighted for easier comparison. Additionally, example outputs with 20% random noise are shown in Figures 1-3. From the results, we observe that N-TRPCA consistently achieves the highest accuracy in terms of PSNR and FSIM across most scenarios in the color image recovery application. However, as evident from the data, its execution time is significantly longer than that of the other algorithms. Apart from N-TRPCA, TRPCAAL generally ranks as the second most accurate algorithm, except in the cases of the \"peppers\" image with 20% and 40% noise, where FTTNN and TRPCA_TNN show slightly better accuracy. Nevertheless, even in these cases, the performance of TRPCAAL is only marginally lower. What makes TRPCAAL stand out is its execution time--it is the fastest across all test cases. Notably, TRPCAAL's computational efficiency becomes increasingly apparent as image size increases. For example,when recovering a \(128\times 128\) image, TRPCAAL is approximately 33 times faster than N-TRPCA, the slowest among the tested algorithms. This gap widens to 43 times for a \(512\times 512\) image.