# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 5 Experiments
### Image Recovery  
In this part, we focus on noisy color image recovery. A color image with size \(N_{1}\times N_{2}\) and 3 channel, i.e., red, green and blue, can be regarded as a third order tensor \(\mathcal{X}^{N_{1}\times N_{2}\times N_{3}}\) and each frontal slice of \(\mathcal{X}\) is the corresponding channel of the color image. Please note that each channel of the color image is usually not a low rank matrix, but it is observed that the top singular values dominate the main information [19; 47]. Thus we can reconstruct the noisy image in low rank structure.  
We perform the experiments on Berkeley Segmentation Dataset [48], which contains 200 color image for training. For each image, we randomly set 10% of pixels to the random values in [0; 255], that is, we corrupt the image on 3 channels simultaneously which is more challenging than make corruptions on only one channel. We compare our t-Gamma algorithms with RPCA [4], Gamma [26], SNN [49] and TRPCA [20], in which RPCA minimizes a weighted combination of the nuclear norm and of the \(l_{1}\) norm in matrix form, Gamma replaces the nuclear norm by matrix Gamma quasi-norm, SNN uses the sum of the nuclear norm for all mode-\(i\) matricization of the given tensor, and TRPCA extends the RPCA from the matrix to tensor case. For RPCA and Gamma, we conduct the algorithms on each channel, i.e., regarding each channel of the image as a matrix, and then combine the channel recovery result into a color image. The parameter \(\lambda\) in RPCA and Gamma is set to \(\lambda=\frac{1}{\sqrt{\max\{N_{1},N_{2}\}}}\) as suggested in [4]. However, when the parameter \(\gamma\) in Gamma is set to 0.01 as stated in [26], it cannot perform well. We empirically set the \(\gamma=\sqrt{\min\{N_{1},N_{2}\}}\) in Gamma and t-Gamma. For SNN, we set \(\lambda=[15,15,1.5]\) as suggested in [20]. The parameters \(\lambda\) in TRPCA and t-Gamma is set to \(\lambda=\frac{1}{\sqrt{\max\{N_{1},N_{2}\}N_{3}\}}\). By using the Peak Signal to Noise Ratio (PSNR) to evaluate the recovery results and PSNR is defined as  
\[PSNR=10\log_{10}\left(\frac{\|\mathcal{M}\|_{\infty}^{2}}{\frac{1}{N_{1}N_{2} N_{3}}\|\hat{\mathcal{X}}-\mathcal{M}\|_{F}^{2}}\right).\]  
In Figure 1, we give the comparison of PSNR value on 200 images and each subfigure contains 50 results. The sample recovery performance are shown in Figure 2. From Figure 1, we can observe that the proposed t-Gamma algorithm perform better on almost every image and can achieve around 2 dB improvement in many cases. Table 1 shows the average recovery performance obtained by these five algorithms on Berkeley Segmentation Dataset. In Figure 2 we can visually conclude that our algorithm yields a better result than the compare algorithms which can illustrate the merit of the our approach.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **RPCA** & **Gamma** & **SNN** & **TRPCA** & **t-Gamma** \\ \hline Average PSNR & 25.5843 & 27.0762 & 27.6329 & 29.1157 & 30.7896 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average PSNR obtained by five Algorithms on 200 images.  
Figure 1: Comparison of the PSNR values with RPCA [4], Gamma [26], SNN [49], TRPCA [20] and t-Gamma.  
Figure 2: Recovery results on 5 sample images. The first two columns are original and corrupted images reps. The third to seventh columns are recovery images obtained by RPCA, Gamma, SNN, TRPCA and t-Gamma resp. (**a**) Original; (**b**) Corrupted; (**c**) RPCA [4]; (**d**) Gamma [26]; (**e**) SNN [49]; (**f**) TRPCA [20]; (**g**) t-Gamma.