# Tensor Completion Based on Triple Tubal Nuclear Norm
## 4 TriTNN-Based Tensor Completion
### An ADMM Solver to Problem (21)
We conduct tensor completion experiments on color images, videos and a dataset for autonomous vehicle. For an estimation tensor \(\hat{\boldsymbol{\mathcal{L}}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), its quality is evaluated by the peak signal-to-noise ratio (PSNR) computed by the definition:

\[PSNR=10\log_{10}\big{(}\frac{n_{1}n_{2}n_{3}\|\boldsymbol{\mathcal{L}}^{*}\|_ {\infty}^{2}}{\|\hat{\boldsymbol{\mathcal{L}}}-\boldsymbol{\mathcal{L}}^{*}\| _{\mathrm{F}}^{2}}\big{)},\]

where \(\boldsymbol{\mathcal{L}}^{*}\) is the underlying tensor. The higher the PSNR value is, the better the recovery performance will be. ### Color Image Inpainting

Color images in row \(\times\) column \(\times\) channel are naturally expressed in 3D tensor form. Image inpainting aims at reconstructing a color image from a small fraction of its entries. In this experiment, twelve test images of size \(256\times 256\times 3\) are used; see Figure 5. Given an image \(\boldsymbol{\mathcal{M}}\) of size \(d_{1}\times d_{2}\times 3\), we randomly sample 30% of its pixels and add i.i.d. Gaussian noise with standard deviation \(\sigma=0.1\sigma_{0}\), where \(\sigma_{0}=\|\boldsymbol{\mathcal{M}}\|_{\mathrm{F}}/\sqrt{3d_{1}d_{2}}\) is the rescaled magnitude of \(\boldsymbol{\mathcal{M}}\). The weight parameters \(\boldsymbol{\alpha}\) of SNN are chosen to satisfy \(\alpha_{1}:\alpha_{2}:\alpha_{3}=1:1:0.01\) as suggested in [2]. Parameter \(\tau=8e3\) and \(\lambda_{1},\lambda_{2},\lambda_{3}\) in Problem (22) are chosen to satisfy \(\lambda_{1}:\lambda_{2}:\lambda_{3}=1:0.01:0.01\). Parameters of other algorithms are tuned for better performances in most cases. We also employ the structural similarity index measure (SSIM) [43] to measure the quality of inpainted color images. The higher the SSIM value is, the better the inpainting performance will be. Given a color image, we test ten times and report the averaged PSNR and SSIM values. The inpainting results of five images are shown in Figure 6 for qualitative comparison. We can see that the proposed TriTNN-based model obtains better visual performances. For quantitative comparison, the PSNR and SSIM values on the twelve images of seven algorithms are reported in Figure 7. It can be seen that the proposed TriTNN-based outperforms the competitors in most cases. Figure 5: Twelve color images used in the experiments. Figure 6: Examples of color image inpainting. (**a**) is the observed noisy incomplete image (Obs.) with sampling ratio 0.3 and noise level \(\sigma=0.1\sigma_{0}\); (**b–g**) show the inpainting results of nuclear norm-based models: SNN [2], latent tensor nuclear norm-based model (LatentNN) [21], SquareNN [23], tensor tubal nuclear norm (TNN) [31], t-TNN [16] and the proposed model, respectively. The corresponding PSNR and SSIM values are listed in (**k**). The highest PSNR and SSIM indicating the best inpainting performance are highlighted in bold. It is suggested to be viewed in color as a pdf file with a 4\(\times\) zoom in. TriTNN, triple tubal nuclear norm. ### Video Inpainting

The video inpainting task aims at imputing the missing pixels of a video. The performance competition is carried out on five widely-used YUVvideos (They are available from [[https://sites.google.com/site/subudhibadri/fewhelpfulldownloads](https://sites.google.com/site/subudhibadri/fewhelpfulldownloads)]([https://sites.google.com/site/subudhibadri/fewhelpfulldownloads](https://sites.google.com/site/subudhibadri/fewhelpfulldownloads))): salesman_qcif, silent_qcif, suzie_qcif, tempete_cif and waterfall_cif. Due to the computational limitation, we use the first 30 frames of Y components in each video. This results in three tensors sized \(144\times 176\times 32\) and two tensors sized \(288\times 352\times 32\). For each video, we uniformly sample 10% of the entries and conduct the video inpainting experiments. The weight parameters \(\alpha\) of SNN are chosen to satisfy \(\alpha_{1}:\alpha_{2}:\alpha_{3}=1:1:1\) as suggested in [2].