# Low Tensor Rank Constrained Image Inpainting Using a Novel Arrangement Scheme
## 4 Experimental Results and Analyses
### Runtime and Complexity Analysis  
From a high-dimensional curse perspective, converting an image to a higher-order tensor can result in increased complexity, which inevitably leads to a longer runtime. We compared our methods (UnfoldingLRTV, tSVDLRTV, and TTLRTV) with the traditional MatrixLR method and the close STDC, HaLRTC, FBCP, SPCQV, and LRTV methods in terms of running time, as shown in Table 7.  
UnfoldingLRTV methods: In the first step, the QA scheme was used to decompose a single image into several small graphs. Because of the similarity of these small graphs, the QA tensor could be reduced to a matrix with a low-rank structure in an unfolding  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{4}{c}{**Runtime (s)**} \\ \cline{2-5}  & \multicolumn{1}{c}{_House_} & _Lena_ & _Airplane_ & _Boats_ \\ \cline{2-5}  & _Random_ **50\%** & _Lines_ & _Random Lines_ & _Random **80\%** \\ \hline
**MatrixLR** & **4.95** & **0.17** & **0.16** & **5.01** \\
**STDC** & 5.43 & 5.13 & 5.17 & 5.16 \\
**HaLRTC** & 8.00 & 0.88 & 0.84 & 6.84 \\
**FBCP** & 188.32 & 86.45 & 132.09 & 219.33 \\
**SPCTV** & 19.25 & 16.37 & 16.03 & 17.69 \\
**LRTV** & 19.08 & 20.17 & 21.04 & 21.05 \\
**TTLRTV** & 145.5 & 143.2 & 142.6 & 142.3 \\
**tSVDLRTV** & 15.23 & 15.07 & 15.17 & 15.14 \\
**UnfoldingLRTV** & **9.49** & **8.53** & **8.69** & **8.72** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Runtimes(s) of the different methods.  
way. Ignoring TV constraints, the unfoldingLRTV method only needed to solve the low-rank matrix completion problem of an unfolding matrix, so the running time was similar to the traditional MatrixLR method, and the accuracy was higher than the traditional MatrixLR method.  
The tSVDLRTV methods: As the color image was highly unbalanced in the size of the three dimensions, which is not suitable for the low tubal rank constraint, we used the QA scheme to rearrange an image into a third-order tensor with a more balanced size of every dimension. Then, TNN was used to constrain the low tubal rank of the rearranged tensor. Due to the fast Fourier scheme, it is necessary to perform a low-rank matrix constraint on each frontal slice after the threeh more efficient low t-dimensional Fourier transform. At this time, the SVD decomposition process will increase the time consumption.  
TTLRTV methods: TT multi-rank is the combination of the rank of each TT matrix. The TTLRTV method essentially completes the same data amount N-1 times, where N is the order of the QA tensor. Therefore, although the TITRTV method is effective, it is necessarily more computationally expensive than the low matrix rank completion method.  
In summary, among the three methods, the UnfoldingLRTV method achieved the best performance both in accuracy and runtime. The TTLRTV method reached better accuracy, but it was time-consuming. The tSVDLRTV method showed moderate performance both in runtime and accuracy.  
All in all, the above three methods can exploit the potential low-rank prior of an image and can be successfully used for image inpainting problems, demonstrating that the three flexible QA schemes are perfect for exploring the low-rank prior of an image.