# Low Tensor Rank Constrained Image Inpainting Using a Novel Arrangement Scheme
## 4 Experimental Results and Analyses  
We applied the above methods 1-3 to solve image inpainting problems. For simplicity, methods 1-3, which only exploit low unfolding matrix rank, low tensor tubal rank, and low tensor train rankness, are denoted as UfoldingLR, TTLR, and tSVDLR methods, respectively. The methods that enhance the low rank and total variation constraints simultaneously are denoted as UnfoldingLRTV, tSVDLRTV, and TTLRTV methods, respectively. The low matrix rank completion method, which is solved by model (17) and the ADMM algorithm, is denoted as the MatrixLR method. \[\min_{X}\,rank(X)\,\,\text{subject to}\,\,X(i,\,\,j)=Y(i,\,j),\,\,\forall(i,\,j)\in\Omega \tag{17}\]  
The method that only exploits sparsity in the gradient domain and is solved by the ADMM algorithm is denoted as the TV method. In addition, the following close methods were applied for comparison; some of their codes are available online. STDC: This method exploits the images into three-factor matrices and one core tensor for image inpainting [7,21,22] ([[http://doi.ieeecomputersociety.org/10.1109/TPAMI.2013.164](http://doi.ieeecomputersociety.org/10.1109/TPAMI.2013.164)]([http://doi.ieeecomputersociety.org/10.1109/TPAMI.2013.164](http://doi.ieeecomputersociety.org/10.1109/TPAMI.2013.164)), accessed on 12 December 2019). HaLRTC: This method constrains the low rankness of the three mode-n matrices caused by the decomposition of a color image for inpainting, which is solved by the ADMM [23,24] ([[https://www.cs.rochester.edu/u/jliu/publications.html](https://www.cs.rochester.edu/u/jliu/publications.html)]([https://www.cs.rochester.edu/u/jliu/publications.html](https://www.cs.rochester.edu/u/jliu/publications.html)), accessed on 18 December 2019). SPCTV ([[https://sites.google.com/site/yokotatsu/home/software](https://sites.google.com/site/yokotatsu/home/software)]([https://sites.google.com/site/yokotatsu/home/software](https://sites.google.com/site/yokotatsu/home/software)), accessed on 12 January 2020). The smooth PARAFAC tensor completion and total variation method [25] uses the PD (PARAFAC decomposition, a derivation of Tucker decomposition) framework and constrains the TV on every factor matrix of PD. LRTV: This method combines the constraints of the low rankness of every mode-n matrix and the TV regularization on every mode-n matrix for color image inpainting [26,27] ([[https://sites.google.com/site/yokotatsu/home/software/lrtv_pds](https://sites.google.com/site/yokotatsu/home/software/lrtv_pds)]([https://sites.google.com/site/yokotatsu/home/software/lrtv_pds](https://sites.google.com/site/yokotatsu/home/software/lrtv_pds)), accessed on 16 April 2021). FBCP: Inpainting methods based on Bayesian tensor completion [28, 29, 30, 31] ([[https://github.com/qbzhao/BCPF](https://github.com/qbzhao/BCPF)]([https://github.com/qbzhao/BCPF](https://github.com/qbzhao/BCPF)), accessed on 10 March 2021). TTC and TTCTV: The tensor completion method with tensor train and with total variation regularized tensor trains, respectively [50] ([[https://github.com/IRENEKO/TTC](https://github.com/IRENEKO/TTC)]([https://github.com/IRENEKO/TTC](https://github.com/IRENEKO/TTC)), accessed on 9 March 2021). CFN-RTC: This robust tensor completion method combines capped Frobenius norm and TR decomposition and adds dimensions through simple data reshaping for processing image [33] ([[https://github.com/Li-X-P/Code-of-Robust-Tensor-Completion](https://github.com/Li-X-P/Code-of-Robust-Tensor-Completion)]([https://github.com/Li-X-P/Code-of-Robust-Tensor-Completion](https://github.com/Li-X-P/Code-of-Robust-Tensor-Completion)), accessed on 10 March 2021).