# Tensor Completion Based on Triple Tubal Nuclear Norm
## 4 TriTNN-Based Tensor Completion
### An ADMM Solver to Problem (21)
Parameters \(\tau=2,\rho=5\)e-\(5\) and \(\lambda_{1},\lambda_{2},\lambda_{3}\) are chosen to satisfy \(\lambda_{1}:\lambda_{2}:\lambda_{3}=1:1:1\) for the proposed model. Parameters of other algorithms are tuned for better performances in most cases. Given a video, we test ten times and report the averaged PSNR value. The qualitative comparison is shown in Figure 8. The PSNR values are reported in Table 1. It can be seen that the TriTNN-based model outperforms the others. The proposed TriTNN has better performances than TNN and t-TNN, since the TriTNN exploits the row, column and tube redundancy simultaneously, whereas TNN or t-TNN only exploit one type of redundancy. The superiority of TriTNN over SNN, LatentNN and SquareNN may be explained by the fact that the circulant block

Figure 7: Quantitative evaluation of algorithms on color images for Uniform-0.3: the image is sampled uniformly with ratio \(p=0.3\) and corrupted with noise level \(\sigma=0.1\sigma_{0}\). **(a)** PSNR values; **(b)** SSIM values. matricization of a tensor coded in TriTNN makes use of more information than directly unfolding along each mode. ### A Dataset for Autonomous Driving

Environment perception for autonomous driving has attracted more and more attention in computer vision. In this subsection, experiments on a dataset collected for autonomous driving are performed. The dataset (a collection of Frame No. 165-No. 244 in Scenario B and Scenario B; additional sensor data available at [[http://www.mrt.kit.edu/z/publ/download/velodynetracking/dataset.html](http://www.mrt.kit.edu/z/publ/download/velodynetracking/dataset.html)]([http://www.mrt.kit.edu/z/publ/download/velodynetracking/dataset.html](http://www.mrt.kit.edu/z/publ/download/velodynetracking/dataset.html))) has 80 frames of gray images and LiDAR point cloud data acquired by a Velodyne HDL-64E LiDAR. The image sequence is resized to be a tensor of size \(128\times 256\times 80\), and the LiDAR data are resampled, transformed and formatted to be two tensors of size \(64\times 436\times 80\) representing the distance data and the intensity data, respectively. Given a tensor to complete \(\mathbf{\mathcal{T}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), experiments are carried out with seven different observation settings where the sampling ratio \(p\) varies from 0.1-0.7. The observed entries are further corrupted by i.i.d. Gaussian noise with standard deviation \(\sigma=0.2\sigma_{0}\), where \(\sigma_{0}=\|\mathbf{\mathcal{T}}\|_{F}/\sqrt{d_{1}d_{2}d_{3}}\) is the normalized magnitude of \(\mathbf{\mathcal{T}}\). The weight parameters \(\mathbf{\alpha}\) of SNN are chosen to satisfy \(\alpha_{1}:\alpha_{2}:\alpha_{3}=1:1:1\). Parameters \(\tau=2,\rho=5\)e-\(5\) and \(\lambda_{1},\lambda_{2},\lambda_{3}\) are chosen to satisfy \(\lambda_{1}:\lambda_{2}:\lambda_{3}=1:1:1\) for the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Video** & **SNN**[2] & **LatentNN**[21] & **SquareNN**[23] & **TNN**[31] & **t-TNN**[16] & **TriTNN** \\ \hline salesman\_qcif & 21.10 & 24.21 & 20.31 & 25.56 & 25.73 & **26.18** \\ silent\_qcif & 24.07 & 23.95 & 22.98 & 27.93 & 28.03 & **28.47** \\ suzie\_qcif & 25.79 & 24.63 & 24.48 & 27.92 & 29.47 & **29.70** \\ tempete\_cif & 19.10 & 18.45 & 19.23 & 20.65 & 21.21 & **21.46** \\ waterfall\_cif & 24.10 & 22.39 & 24.06 & 26.71 & 27.44 & **28.22** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantitative evaluation of algorithms in PSNR values for YUV video inpainting: each video is sampled uniformly with ratio \(p=0.1\). Figure 8: Examples of YUVvideo inpainting. (**a**) is the first frame of each video; (**b**) is the observed incomplete frame (Obs.) with 90% missing entries; (**c–h**) show the inpainting results of nuclear norm-based models: SNN [2], LatentNN [21], SquareNN [23], TNN [31], t-TNN [16] and the proposed model TriTNN-based model, respectively. It is suggested to be viewed as a pdf file with a 4\(\times\) zoom in.