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Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems
Tensor train rank minimization with nonlocal self-similarity for tensor completion
1. | School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China |
2. | Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong |
3. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China |
The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($ \rm{order} >3 $). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with ket augmentation, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.
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show all references
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J. A. Bengua, H. N. Phiem, H. D. Tuan and M. N. Do,
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[2] |
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M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester,
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J.-F. Cai, E. J. Cand$\grave{e}$s and Z. Shen,
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R. H. Chan, M. Tao and X. Yuan,
Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers, SIAM Journal on Imaging Sciences, 6 (2013), 680-697.
doi: 10.1137/110860185. |
[7] |
Y. Chang, L.-X. Yan and S. Zhong, Hyper-laplacian regularized unidirectional low-rank tensor recovery for multispectral image denoising, IEEE Conference on Computer Vision and Pattern Recognition, (2017), 5901–5909.
doi: 10.1109/CVPR.2017.625. |
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Y. Chen, C. Hsu and H. M. Liao, Simultaneous tensor decomposition and completion using factor priors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 20 (2014), 577-591. Google Scholar |
[9] |
L.-B. Cui, X.-Q. Zhang and S.-L. Wu,
A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with $\mathcal{M}$-tensors, Computational and Applied Mathematics, 39 (2020), 1-16.
doi: 10.1007/s40314-020-01194-8. |
[10] |
K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian,
Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095.
doi: 10.1109/TIP.2007.901238. |
[11] |
M. Ding, T.-Z. Huang and T.-H. Ma, Cauchy noise removal using group-based low-rank prior, Applied Mathematics and Computation, 372 (2020), 124971, 15 pp.
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[12] |
M. Ding, T.-Z. Huang, T.-Y. Ji, X.-L. Zhao and J.-H. Yang,
Low-rank tensor completion using matrix factorization based on tensor train rank and total variation, Journal of Scientific Computing, 81 (2019), 941-964.
doi: 10.1007/s10915-019-01044-8. |
[13] |
M. Ding, T.-Z. Huang, S. Wang, J.-J. Mei and X.-L. Zhao,
Total variation with overlapping group sparsity for deblurring images under Cauchy noise, Applied Mathematics and Computation, 341 (2019), 128-147.
doi: 10.1016/j.amc.2018.08.014. |
[14] |
Y. Du, G. Han, Y. Quan, Z. Yu, H. Wong, C. L. P. Chen and J. Zhang,
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doi: 10.1109/TCYB.2018.2853122. |
[15] |
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[16] |
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doi: 10.1190/geo2014-0467.1. |
[17] |
X. Fu, K.-J. Huang, B. Yang, W. K. Ma and N. D. Sidiropoulos,
Robust volume minimization-based matrix factorization for remote sensing and document clustering, IEEE Transactions on Signal Processing, 64 (2016), 6254-6268.
doi: 10.1109/TSP.2016.2602800. |
[18] |
S. Gandy, B. Recht and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization, Inverse Problems, 27 (2011), 025010, 19pp.
doi: 10.1088/0266-5611/27/2/025010. |
[19] |
T. Goldstein, B. O'Donoghue, S. Setzer and R. Baraniuk,
Fast alternating direction optimization methods, SIAM Journal on Imaging Sciences, 7 (2014), 1588-1623.
doi: 10.1137/120896219. |
[20] |
L. Grasedyck, M. Kluge and S. Krämer, Alternating least squares tensor completion in the TT-format, preprint, arXiv: 1509.00311. Google Scholar |
[21] |
S.-H. Gu, L. Zhang, W.-M. Zuo and X.-C. Feng, Weighted nuclear norm minimization with application to image denoising, IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862–2869.
doi: 10.1109/CVPR.2014.366. |
[22] |
B.-S. He and X. Yuan,
On the O(1/n) convergence rate of the douglas-rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709.
doi: 10.1137/110836936. |
[23] |
W. He, H.-Y. Zhang, L.-P. Zhang and H.-F. Shen,
Total-variation-regularized low-rank matrix factorization for hyperspectral image restoration, IEEE Transactions on Geoscience and Remote Sensing, 54 (2016), 178-188.
doi: 10.1109/TGRS.2015.2452812. |
[24] |
C. J. Hillar and L. H. Lim, Most tensor problems are NP-hard, Journal of the ACM, 60 (2013), Art. 45, 39 pp.
doi: 10.1145/2512329. |
[25] |
Y.-M. Huang, H.-Y. Yan, Y.-W. Wen and X. Yang,
Rank minimization with applications to image noise removal, Information Sciences, 429 (2018), 147-163.
doi: 10.1016/j.ins.2017.10.047. |
[26] |
T.-X. Jiang, T.-Z. Huang, X.-L. Zhao and L.-J. Deng, Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm, Journal of Computational and Applied Mathematics, 372 (2020), 112680, 15pp.
doi: 10.1016/j.cam.2019.112680. |
[27] |
T.-X. Jiang, M. K. Ng, X.-L. Zhao and T.-Z. Huang,
Framelet representation of tensor nuclear norm for third-order tensor completion, IEEE Transactions on Image Processing, 29 (2020), 7233-7244.
doi: 10.1109/TIP.2020.3000349. |
[28] |
T. G. Kolda, B. W. Bader and J. P. Kenny, Higher-order Web link analysis using multilinear algebra, IEEE International Conference on Data Mining, (2005), 242–249.
doi: 10.1109/ICDM.2005.77. |
[29] |
T. G. Kolda and B. W. Bader,
Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.
doi: 10.1137/07070111X. |
[30] |
M. E. Kilmer, K. Braman, N. Hao and R. C. Hoover,
Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148-172.
doi: 10.1137/110837711. |
[31] |
N. Komodakis, Image inpainting, IEEE Conference on Computer Vision and Pattern Recognition, 1 (2006), 442-452. Google Scholar |
[32] |
R.-J. Lai and J. Li,
Manifold based low-rank regularization for image restoration and semi-supervised learning, Journal of Scientific Computing, 74 (2018), 1241-1263.
doi: 10.1007/s10915-017-0492-x. |
[33] |
J. I. Latorre, Image Compression and Entanglement, Computer Science, 2005. Google Scholar |
[34] |
F. Li, M. K. Ng and R. J. Plemmons,
Coupled segmentation and denoising/deblurring models for hyperspectral material identification, Numerical Linear Algebra with Applications, 19 (2012), 153-173.
doi: 10.1002/nla.750. |
[35] |
Y.-P. Liu, Z. Long and C. Zhu,
Image completion using low tensor tree rank and total variation minimization, IEEE Transactions on Multimedia, 21 (2019), 338-350.
doi: 10.1109/TMM.2018.2859026. |
[36] |
Y.-Y. Liu, F.-H. Shang, L.-C. Jiao, J. Cheng and H. Cheng,
Trace norm regularized CANDECOMP/PARAFAC decomposition with missing data, IEEE Transactions on Cybernetics, 45 (2015), 2437-2448.
doi: 10.1109/TCYB.2014.2374695. |
[37] |
Y.-P. Liu, Z. Long, H.-Y. Huang and C. Zhu,
Low CP rank and tucker rank tensor completion for estimating missing components in image data, IEEE Transactions on Circuits and Systems for Video Technology, 30 (2020), 944-954.
doi: 10.1109/TCSVT.2019.2901311. |
[38] |
J. Liu, P. Musialski, P. Wonka and J. Ye,
Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 208-220.
doi: 10.1109/TPAMI.2012.39. |
[39] |
C.-Y. Lu, J.-S. Feng, Y.-D. Chen, W. Liu, Z.-C. Lin and S.-C. Yan, Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization, IEEE Conference on Computer Vision and Pattern Recognition, (2016), 5249–5257.
doi: 10.1109/CVPR.2016.567. |
[40] |
C.-Y. Lu, J.-S. Feng, Z.-C. Lin and S.-C. Yan, Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements, International Joint Conference on Artificial Intelligence, 2018.
doi: 10.24963/ijcai.2018/347. |
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Image | SR | 0.1 | 0.2 | 0.3 | 0.4 | ||||||||
Method | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | |
lena | HaLRTC | 19.29 | 0.4151 | 5.52 | 23.10 | 0.6047 | 4.65 | 25.68 | 0.7311 | 8.65 | 28.00 | 0.8205 | 7.83 |
tSVD | 19.55 | 0.3500 | 87.65 | 23.33 | 0.5572 | 88.40 | 26.08 | 0.7033 | 88.80 | 28.60 | 0.8066 | 87.16 | |
SiLRTC-TT | 21.67 | 0.5954 | 39.17 | 24.80 | 0.7366 | 27.36 | 27.01 | 0.8226 | 20.32 | 28.90 | 0.8782 | 16.11 | |
TMac-TT | 24.25 | 0.6829 | 67.99 | 27.22 | 0.8097 | 65.13 | 28.87 | 0.8584 | 27.99 | 30.22 | 0.8902 | 20.38 | |
NL-TT | 26.52 | 0.8124 | 192.95 | 30.09 | 0.8970 | 121.74 | 32.02 | 0.9309 | 90.28 | 33.87 | 0.9528 | 76.41 | |
airplane | HaLRTC | 19.80 | 0.4621 | 6.80 | 23.18 | 0.6437 | 4.69 | 25.62 | 0.7614 | 3.94 | 27.97 | 0.8399 | 7.36 |
tSVD | 19.87 | 0.4196 | 84.57 | 23.30 | 0.6139 | 87.18 | 25.86 | 0.7387 | 87.63 | 28.25 | 0.8258 | 87.57 | |
SiLRTC-TT | 20.81 | 0.6072 | 32.20 | 23.42 | 0.7361 | 25.32 | 25.62 | 0.8213 | 20.04 | 27.55 | 0.8768 | 17.06 | |
TMac-TT | 22.46 | 0.6766 | 7.62 | 25.81 | 0.8105 | 60.23 | 27.67 | 0.8622 | 25.36 | 28.97 | 0.8915 | 16.93 | |
NL-TT | 24.33 | 0.7840 | 299.77 | 28.33 | 0.8929 | 109.30 | 30.29 | 0.9268 | 85.13 | 31.99 | 0.9489 | 107.19 | |
monarch | HaLRTC | 17.12 | 0.4381 | 8.20 | 19.59 | 0.6069 | 4.25 | 21.89 | 0.7404 | 3.66 | 24.20 | 0.8271 | 3.33 |
tSVD | 17.14 | 0.3372 | 91.07 | 19.98 | 0.5462 | 87.39 | 22.60 | 0.6980 | 88.03 | 25.23 | 0.8023 | 89.33 | |
SiLRTC-TT | 17.95 | 0.5784 | 38.01 | 20.32 | 0.7196 | 30.94 | 22.38 | 0.8100 | 26.44 | 24.39 | 0.8702 | 22.52 | |
TMac-TT | 19.21 | 0.6621 | 185.13 | 22.45 | 0.7912 | 104.59 | 24.86 | 0.8505 | 76.80 | 27.24 | 0.9046 | 81.50 | |
NL-TT | 22.22 | 0.8307 | 587.32 | 25.42 | 0.9140 | 379.66 | 27.95 | 0.9496 | 306.90 | 30.74 | 0.9729 | 246.85 | |
lena | HaLRTC | 17.54 | 0.2942 | 6.37 | 20.97 | 0.4651 | 7.06 | 23.59 | 0.6144 | 5.38 | 25.88 | 0.7272 | 4.01 |
tSVD | 17.88 | 0.2570 | 88.13 | 20.85 | 0.4186 | 99.70 | 23.29 | 0.5676 | 93.58 | 25.50 | 0.6857 | 90.83 | |
SiLRTC-TT | 20.90 | 0.5462 | 47.73 | 23.61 | 0.6830 | 33.84 | 25.69 | 0.7732 | 25.55 | 27.35 | 0.8353 | 19.25 | |
TMac-TT | 21.62 | 0.5629 | 15.30 | 24.60 | 0.7193 | 41.10 | 26.22 | 0.7764 | 17.78 | 27.55 | 0.8392 | 51.87 | |
NL-TT | 23.94 | 0.7351 | 223.34 | 27.45 | 0.8459 | 172.99 | 29.33 | 0.8928 | 93.35 | 31.38 | 0.9259 | 72.19 | |
airplane | HaLRTC | 17.81 | 0.3050 | 5.93 | 20.77 | 0.4847 | 6.32 | 23.15 | 0.6214 | 4.71 | 25.29 | 0.7289 | 4.14 |
tSVD | 17.97 | 0.2900 | 85.67 | 20.66 | 0.4588 | 85.71 | 22.97 | 0.5926 | 84.86 | 25.06 | 0.7029 | 89.59 | |
SiLRTC-TT | 20.20 | 0.5570 | 39.31 | 22.49 | 0.6809 | 28.91 | 24.33 | 0.7661 | 23.86 | 26.09 | 0.8298 | 20.87 | |
TMac-TT | 21.06 | 0.6169 | 35.50 | 23.15 | 0.7114 | 21.76 | 24.41 | 0.7729 | 19.68 | 26.17 | 0.8416 | 45.12 | |
NL-TT | 22.45 | 0.7255 | 175.78 | 25.25 | 0.8210 | 103.25 | 27.29 | 0.8749 | 81.90 | 29.24 | 0.9149 | 67.92 | |
monarch | HaLRTC | 16.04 | 0.3424 | 6.13 | 18.28 | 0.5031 | 6.17 | 20.12 | 0.6363 | 4.62 | 21.93 | 0.7401 | 4.03 |
tSVD | 16.33 | 0.2786 | 84.71 | 18.21 | 0.4312 | 86.92 | 19.90 | 0.5620 | 115.32 | 21.65 | 0.6791 | 88.34 | |
SiLRTC-TT | 17.46 | 0.5472 | 110.83 | 19.48 | 0.6695 | 34.32 | 21.19 | 0.7606 | 41.57 | 22.83 | 0.8290 | 25.77 | |
TMac-TT | 15.12 | 0.3466 | 142.62 | 18.66 | 0.6710 | 144.85 | 21.74 | 0.7739 | 73.79 | 23.49 | 0.8282 | 32.73 | |
NL-TT | 18.07 | 0.6564 | 147.54 | 22.33 | 0.8462 | 142.52 | 24.53 | 0.9086 | 108.21 | 26.25 | 0.9391 | 89.87 |
Image | SR | 0.1 | 0.2 | 0.3 | 0.4 | ||||||||
Method | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | |
lena | HaLRTC | 19.29 | 0.4151 | 5.52 | 23.10 | 0.6047 | 4.65 | 25.68 | 0.7311 | 8.65 | 28.00 | 0.8205 | 7.83 |
tSVD | 19.55 | 0.3500 | 87.65 | 23.33 | 0.5572 | 88.40 | 26.08 | 0.7033 | 88.80 | 28.60 | 0.8066 | 87.16 | |
SiLRTC-TT | 21.67 | 0.5954 | 39.17 | 24.80 | 0.7366 | 27.36 | 27.01 | 0.8226 | 20.32 | 28.90 | 0.8782 | 16.11 | |
TMac-TT | 24.25 | 0.6829 | 67.99 | 27.22 | 0.8097 | 65.13 | 28.87 | 0.8584 | 27.99 | 30.22 | 0.8902 | 20.38 | |
NL-TT | 26.52 | 0.8124 | 192.95 | 30.09 | 0.8970 | 121.74 | 32.02 | 0.9309 | 90.28 | 33.87 | 0.9528 | 76.41 | |
airplane | HaLRTC | 19.80 | 0.4621 | 6.80 | 23.18 | 0.6437 | 4.69 | 25.62 | 0.7614 | 3.94 | 27.97 | 0.8399 | 7.36 |
tSVD | 19.87 | 0.4196 | 84.57 | 23.30 | 0.6139 | 87.18 | 25.86 | 0.7387 | 87.63 | 28.25 | 0.8258 | 87.57 | |
SiLRTC-TT | 20.81 | 0.6072 | 32.20 | 23.42 | 0.7361 | 25.32 | 25.62 | 0.8213 | 20.04 | 27.55 | 0.8768 | 17.06 | |
TMac-TT | 22.46 | 0.6766 | 7.62 | 25.81 | 0.8105 | 60.23 | 27.67 | 0.8622 | 25.36 | 28.97 | 0.8915 | 16.93 | |
NL-TT | 24.33 | 0.7840 | 299.77 | 28.33 | 0.8929 | 109.30 | 30.29 | 0.9268 | 85.13 | 31.99 | 0.9489 | 107.19 | |
monarch | HaLRTC | 17.12 | 0.4381 | 8.20 | 19.59 | 0.6069 | 4.25 | 21.89 | 0.7404 | 3.66 | 24.20 | 0.8271 | 3.33 |
tSVD | 17.14 | 0.3372 | 91.07 | 19.98 | 0.5462 | 87.39 | 22.60 | 0.6980 | 88.03 | 25.23 | 0.8023 | 89.33 | |
SiLRTC-TT | 17.95 | 0.5784 | 38.01 | 20.32 | 0.7196 | 30.94 | 22.38 | 0.8100 | 26.44 | 24.39 | 0.8702 | 22.52 | |
TMac-TT | 19.21 | 0.6621 | 185.13 | 22.45 | 0.7912 | 104.59 | 24.86 | 0.8505 | 76.80 | 27.24 | 0.9046 | 81.50 | |
NL-TT | 22.22 | 0.8307 | 587.32 | 25.42 | 0.9140 | 379.66 | 27.95 | 0.9496 | 306.90 | 30.74 | 0.9729 | 246.85 | |
lena | HaLRTC | 17.54 | 0.2942 | 6.37 | 20.97 | 0.4651 | 7.06 | 23.59 | 0.6144 | 5.38 | 25.88 | 0.7272 | 4.01 |
tSVD | 17.88 | 0.2570 | 88.13 | 20.85 | 0.4186 | 99.70 | 23.29 | 0.5676 | 93.58 | 25.50 | 0.6857 | 90.83 | |
SiLRTC-TT | 20.90 | 0.5462 | 47.73 | 23.61 | 0.6830 | 33.84 | 25.69 | 0.7732 | 25.55 | 27.35 | 0.8353 | 19.25 | |
TMac-TT | 21.62 | 0.5629 | 15.30 | 24.60 | 0.7193 | 41.10 | 26.22 | 0.7764 | 17.78 | 27.55 | 0.8392 | 51.87 | |
NL-TT | 23.94 | 0.7351 | 223.34 | 27.45 | 0.8459 | 172.99 | 29.33 | 0.8928 | 93.35 | 31.38 | 0.9259 | 72.19 | |
airplane | HaLRTC | 17.81 | 0.3050 | 5.93 | 20.77 | 0.4847 | 6.32 | 23.15 | 0.6214 | 4.71 | 25.29 | 0.7289 | 4.14 |
tSVD | 17.97 | 0.2900 | 85.67 | 20.66 | 0.4588 | 85.71 | 22.97 | 0.5926 | 84.86 | 25.06 | 0.7029 | 89.59 | |
SiLRTC-TT | 20.20 | 0.5570 | 39.31 | 22.49 | 0.6809 | 28.91 | 24.33 | 0.7661 | 23.86 | 26.09 | 0.8298 | 20.87 | |
TMac-TT | 21.06 | 0.6169 | 35.50 | 23.15 | 0.7114 | 21.76 | 24.41 | 0.7729 | 19.68 | 26.17 | 0.8416 | 45.12 | |
NL-TT | 22.45 | 0.7255 | 175.78 | 25.25 | 0.8210 | 103.25 | 27.29 | 0.8749 | 81.90 | 29.24 | 0.9149 | 67.92 | |
monarch | HaLRTC | 16.04 | 0.3424 | 6.13 | 18.28 | 0.5031 | 6.17 | 20.12 | 0.6363 | 4.62 | 21.93 | 0.7401 | 4.03 |
tSVD | 16.33 | 0.2786 | 84.71 | 18.21 | 0.4312 | 86.92 | 19.90 | 0.5620 | 115.32 | 21.65 | 0.6791 | 88.34 | |
SiLRTC-TT | 17.46 | 0.5472 | 110.83 | 19.48 | 0.6695 | 34.32 | 21.19 | 0.7606 | 41.57 | 22.83 | 0.8290 | 25.77 | |
TMac-TT | 15.12 | 0.3466 | 142.62 | 18.66 | 0.6710 | 144.85 | 21.74 | 0.7739 | 73.79 | 23.49 | 0.8282 | 32.73 | |
NL-TT | 18.07 | 0.6564 | 147.54 | 22.33 | 0.8462 | 142.52 | 24.53 | 0.9086 | 108.21 | 26.25 | 0.9391 | 89.87 |
Method | HaLRTC | tSVD | SiLRTC-TT | TMac-TT | NL-TT | ||||||||||
Image | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time |
house | 36.44 | 0.9707 | 6.98 | 36.14 | 0.9681 | 86.86 | 38.52 | 0.9793 | 14.05 | 38.03 | 0.9740 | 11.93 | 45.34 | 0.9906 | 26.46 |
facade | 12.95 | 0.5681 | 0.12 | 12.95 | 0.5681 | 83.16 | 28.14 | 0.9062 | 28.01 | 27.50 | 0.8947 | 9.24 | 29.60 | 0.9357 | 416.18 |
sailboat | 26.49 | 0.8700 | 4.63 | 26.69 | 0.8696 | 86.10 | 26.53 | 0.8838 | 39.29 | 26.40 | 0.8995 | 8.44 | 27.86 | 0.9370 | 171.55 |
barbara | 32.44 | 0.9580 | 4.83 | 32.44 | 0.9579 | 86.28 | 33.99 | 0.9681 | 20.24 | 33.29 | 0.9654 | 6.87 | 37.56 | 0.9867 | 40.36 |
peppers | 31.64 | 0.9595 | 2.22 | 31.53 | 0.9551 | 85.85 | 32.59 | 0.9676 | 24.55 | 32.77 | 0.9651 | 9.27 | 36.33 | 0.9862 | 46.86 |
Average | 27.99 | 0.8653 | 3.76 | 27.95 | 0.8638 | 85.65 | 31.95 | 0.9410 | 25.23 | 31.60 | 0.9397 | 9.15 | 35.34 | 0.9672 | 140.28 |
Method | HaLRTC | tSVD | SiLRTC-TT | TMac-TT | NL-TT | ||||||||||
Image | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time |
house | 36.44 | 0.9707 | 6.98 | 36.14 | 0.9681 | 86.86 | 38.52 | 0.9793 | 14.05 | 38.03 | 0.9740 | 11.93 | 45.34 | 0.9906 | 26.46 |
facade | 12.95 | 0.5681 | 0.12 | 12.95 | 0.5681 | 83.16 | 28.14 | 0.9062 | 28.01 | 27.50 | 0.8947 | 9.24 | 29.60 | 0.9357 | 416.18 |
sailboat | 26.49 | 0.8700 | 4.63 | 26.69 | 0.8696 | 86.10 | 26.53 | 0.8838 | 39.29 | 26.40 | 0.8995 | 8.44 | 27.86 | 0.9370 | 171.55 |
barbara | 32.44 | 0.9580 | 4.83 | 32.44 | 0.9579 | 86.28 | 33.99 | 0.9681 | 20.24 | 33.29 | 0.9654 | 6.87 | 37.56 | 0.9867 | 40.36 |
peppers | 31.64 | 0.9595 | 2.22 | 31.53 | 0.9551 | 85.85 | 32.59 | 0.9676 | 24.55 | 32.77 | 0.9651 | 9.27 | 36.33 | 0.9862 | 46.86 |
Average | 27.99 | 0.8653 | 3.76 | 27.95 | 0.8638 | 85.65 | 31.95 | 0.9410 | 25.23 | 31.60 | 0.9397 | 9.15 | 35.34 | 0.9672 | 140.28 |
Image | SR | 0.05 | 0.1 | 0.2 | ||||||
Method | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | |
toy | HaLRTC | 20.14 | 0.6519 | 8.42 | 23.99 | 0.7790 | 6.69 | 28.91 | 0.8994 | 7.27 |
tSVD | 25.89 | 0.7680 | 264.95 | 30.34 | 0.8844 | 271.40 | 36.57 | 0.9602 | 300.93 | |
SiLRTC-TT | 22.36 | 0.7138 | 379.68 | 25.81 | 0.8392 | 188.36 | 30.44 | 0.9433 | 513.80 | |
TMac-TT | 27.28 | 0.8329 | 232.52 | 32.37 | 0.9317 | 143.12 | 35.74 | 0.9669 | 50.91 | |
NL-TT | 29.58 | 0.9243 | 1414.34 | 34.44 | 0.9730 | 803.33 | 38.72 | 0.9899 | 557.72 | |
feathers | HaLRTC | 20.66 | 0.6422 | 9.96 | 24.26 | 0.7720 | 14.14 | 28.81 | 0.8876 | 11.07 |
tSVD | 25.15 | 0.6886 | 274.69 | 29.29 | 0.8266 | 340.60 | 34.82 | 0.9265 | 264.27 | |
SiLRTC-TT | 22.86 | 0.7196 | 247.87 | 26.32 | 0.8417 | 232.09 | 31.11 | 0.9411 | 374.05 | |
TMac-TT | 27.29 | 0.7611 | 58.00 | 32.12 | 0.9190 | 216.36 | 36.63 | 0.9631 | 62.90 | |
NL-TT | 29.61 | 0.9102 | 1080.65 | 34.76 | 0.9699 | 860.40 | 39.56 | 0.9879 | 607.91 | |
superballs | HaLRTC | 23.28 | 0.7661 | 20.48 | 28.63 | 0.8621 | 9.71 | 34.10 | 0.9426 | 11.93 |
tSVD | 28.24 | 0.7636 | 267.41 | 32.39 | 0.8663 | 270.40 | 38.20 | 0.9564 | 270.92 | |
SiLRTC-TT | 26.27 | 0.8290 | 289.90 | 29.79 | 0.9087 | 157.46 | 34.03 | 0.9651 | 379.94 | |
TMac-TT | 29.97 | 0.8343 | 60.71 | 33.90 | 0.9346 | 63.73 | 40.19 | 0.9803 | 109.14 | |
NL-TT | 32.93 | 0.9507 | 1150.35 | 37.25 | 0.9812 | 666.11 | 42.67 | 0.9939 | 409.49 |
Image | SR | 0.05 | 0.1 | 0.2 | ||||||
Method | PSNR | SSIM | Time | PSNR | SSIM | Time | PSNR | SSIM | Time | |
toy | HaLRTC | 20.14 | 0.6519 | 8.42 | 23.99 | 0.7790 | 6.69 | 28.91 | 0.8994 | 7.27 |
tSVD | 25.89 | 0.7680 | 264.95 | 30.34 | 0.8844 | 271.40 | 36.57 | 0.9602 | 300.93 | |
SiLRTC-TT | 22.36 | 0.7138 | 379.68 | 25.81 | 0.8392 | 188.36 | 30.44 | 0.9433 | 513.80 | |
TMac-TT | 27.28 | 0.8329 | 232.52 | 32.37 | 0.9317 | 143.12 | 35.74 | 0.9669 | 50.91 | |
NL-TT | 29.58 | 0.9243 | 1414.34 | 34.44 | 0.9730 | 803.33 | 38.72 | 0.9899 | 557.72 | |
feathers | HaLRTC | 20.66 | 0.6422 | 9.96 | 24.26 | 0.7720 | 14.14 | 28.81 | 0.8876 | 11.07 |
tSVD | 25.15 | 0.6886 | 274.69 | 29.29 | 0.8266 | 340.60 | 34.82 | 0.9265 | 264.27 | |
SiLRTC-TT | 22.86 | 0.7196 | 247.87 | 26.32 | 0.8417 | 232.09 | 31.11 | 0.9411 | 374.05 | |
TMac-TT | 27.29 | 0.7611 | 58.00 | 32.12 | 0.9190 | 216.36 | 36.63 | 0.9631 | 62.90 | |
NL-TT | 29.61 | 0.9102 | 1080.65 | 34.76 | 0.9699 | 860.40 | 39.56 | 0.9879 | 607.91 | |
superballs | HaLRTC | 23.28 | 0.7661 | 20.48 | 28.63 | 0.8621 | 9.71 | 34.10 | 0.9426 | 11.93 |
tSVD | 28.24 | 0.7636 | 267.41 | 32.39 | 0.8663 | 270.40 | 38.20 | 0.9564 | 270.92 | |
SiLRTC-TT | 26.27 | 0.8290 | 289.90 | 29.79 | 0.9087 | 157.46 | 34.03 | 0.9651 | 379.94 | |
TMac-TT | 29.97 | 0.8343 | 60.71 | 33.90 | 0.9346 | 63.73 | 40.19 | 0.9803 | 109.14 | |
NL-TT | 32.93 | 0.9507 | 1150.35 | 37.25 | 0.9812 | 666.11 | 42.67 | 0.9939 | 409.49 |
Image | SR | 0.1 | 0.2 | 0.3 | 0.4 | ||||
Method | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |
lena | LDMM | 22.20 | 0.6202 | 26.52 | 0.7944 | 27.74 | 0.8457 | 29.95 | 0.8879 |
WNLL | 26.24 | 0.8043 | 28.08 | 0.8534 | 29.29 | 0.8855 | 30.35 | 0.9071 | |
MLR | 26.47 | 0.8052 | 29.03 | 0.8747 | 30.74 | 0.9119 | 32.14 | 0.9333 | |
NL-TT | 26.52 | 0.8124 | 30.09 | 0.8970 | 32.02 | 0.9309 | 33.87 | 0.9528 | |
airplane | LDMM | 20.12 | 0.6230 | 24.03 | 0.7940 | 25.88 | 0.8532 | 28.59 | 0.8988 |
WNLL | 23.75 | 0.7675 | 25.76 | 0.8264 | 27.00 | 0.8629 | 28.03 | 0.8891 | |
MLR | 24.14 | 0.7750 | 26.78 | 0.8595 | 28.62 | 0.9015 | 29.92 | 0.9260 | |
NL-TT | 24.33 | 0.7840 | 28.33 | 0.8929 | 30.29 | 0.9268 | 31.99 | 0.9489 | |
monarch | LDMM | 18.61 | 0.6196 | 19.01 | 0.6463 | 22.34 | 0.8515 | 25.55 | 0.9176 |
WNLL | 20.54 | 0.7584 | 22.61 | 0.8262 | 23.78 | 0.8619 | 24.93 | 0.8916 | |
MLR | 20.95 | 0.8030 | 23.73 | 0.8884 | 25.76 | 0.9291 | 27.48 | 0.9525 | |
NL-TT | 22.22 | 0.8307 | 25.42 | 0.9140 | 27.95 | 0.9496 | 30.74 | 0.9729 |
Image | SR | 0.1 | 0.2 | 0.3 | 0.4 | ||||
Method | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |
lena | LDMM | 22.20 | 0.6202 | 26.52 | 0.7944 | 27.74 | 0.8457 | 29.95 | 0.8879 |
WNLL | 26.24 | 0.8043 | 28.08 | 0.8534 | 29.29 | 0.8855 | 30.35 | 0.9071 | |
MLR | 26.47 | 0.8052 | 29.03 | 0.8747 | 30.74 | 0.9119 | 32.14 | 0.9333 | |
NL-TT | 26.52 | 0.8124 | 30.09 | 0.8970 | 32.02 | 0.9309 | 33.87 | 0.9528 | |
airplane | LDMM | 20.12 | 0.6230 | 24.03 | 0.7940 | 25.88 | 0.8532 | 28.59 | 0.8988 |
WNLL | 23.75 | 0.7675 | 25.76 | 0.8264 | 27.00 | 0.8629 | 28.03 | 0.8891 | |
MLR | 24.14 | 0.7750 | 26.78 | 0.8595 | 28.62 | 0.9015 | 29.92 | 0.9260 | |
NL-TT | 24.33 | 0.7840 | 28.33 | 0.8929 | 30.29 | 0.9268 | 31.99 | 0.9489 | |
monarch | LDMM | 18.61 | 0.6196 | 19.01 | 0.6463 | 22.34 | 0.8515 | 25.55 | 0.9176 |
WNLL | 20.54 | 0.7584 | 22.61 | 0.8262 | 23.78 | 0.8619 | 24.93 | 0.8916 | |
MLR | 20.95 | 0.8030 | 23.73 | 0.8884 | 25.76 | 0.9291 | 27.48 | 0.9525 | |
NL-TT | 22.22 | 0.8307 | 25.42 | 0.9140 | 27.95 | 0.9496 | 30.74 | 0.9729 |
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