September  2021, 10(3): 633-655. doi: 10.3934/eect.2020084

A nonlocal Weickert type PDE applied to multi-frame super-resolution

1. 

LAMAI, FST Marrakech, Université Cadi Ayyad, Maroc

2. 

FP Ouarzazate, Université Ibn Zohr, Maroc

3. 

LAMAI, FST Marrakech, Université Cadi Ayyad, Maroc

4. 

LMA FST Béni-Mellal, Université Sultan Moulay Slimane, Maroc

* Corresponding author: F. Ait Bella

Received  December 2019 Revised  May 2020 Published  September 2021 Early access  August 2020

In this paper, we propose a nonlocal Weickert type PDE for the multiframe super-resolution task. The proposed PDE can not only preserve singularities and edges while smoothing, but also can keep safe the texture much better. This PDE is based on the nonlocal setting of the anisotropic diffusion behavior by constructing a nonlocal term of Weickert type, which is known by its coherence enhancing diffusion tensor properties. A mathematical study concerning the well-posedness of the nonlocal PDE is also investigated with an appropriate choice of the functional space. This PDE has demonstrated its efficiency by combining the diffusion process of Perona-Malik in the flat regions and the anisotropic diffusion of the Weickert model near strong edges, as well as the ability of the non-local term to preserve the texture. The elaborated experimental results give a great insight into the effectiveness of the proposed nonlocal PDE compared to some PDEs, visually and quantitatively.

Citation: Fatimzehrae Ait Bella, Aissam Hadri, Abdelilah Hakim, Amine Laghrib. A nonlocal Weickert type PDE applied to multi-frame super-resolution. Evolution Equations and Control Theory, 2021, 10 (3) : 633-655. doi: 10.3934/eect.2020084
References:
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S. Baker and T. Kanade, Limits on super-resolution and how to break them, IEEE Transactions on Pattern Analysis & Machine Intelligence, (2002), 1167–1183.

[2]

T. BroxO. Kleinschmidt and D. Cremers, Efficient nonlocal means for denoising of textural patterns, IEEE Transactions on Image Processing, 17 (2008), 1083-1092.  doi: 10.1109/TIP.2008.924281.

[3]

A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, In Proc. Sympos. Pure Math, 4 (1961), 33-49. 

[4]

K. DabovA. FoiV. 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.

[5]

W. Dong, T. Huang, G. Shi, Y. Ma and X. Li, Robust tensor approximation with laplacian scale mixture modeling for multiframe image and video denoising, IEEE Journal of Selected Topics in Signal Processing, 12 (2018), 1435–1448, . doi: 10.1109/JSTSP.2018.2873047.

[6]

W. DongL. ZhangG. Shi and X. Wu, Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization, IEEE Transactions on Image Processing, 20 (2011), 1838-1857.  doi: 10.1109/TIP.2011.2108306.

[7]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.  doi: 10.1137/110833294.

[8]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Mathematical Models and Methods in Applied Sciences, 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.

[9]

I. El MourabitM. El RhabiA. HakimA. Laghrib and E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction, Signal Processing, 132 (2017), 51-65.  doi: 10.1016/j.sigpro.2016.09.014.

[10]

S. FarsiuD. RobinsonM. Elad and P. Milanfar, Advances and challenges in super-resolution, International Journal of Imaging Systems and Technology, 14 (2004), 47-57. 

[11]

S. FarsiuM. D. RobinsonM. Elad and P. Milanfar, Fast and robust multiframe super resolution, IEEE Transactions on Image Processing, 13 (2004), 1327-1344.  doi: 10.1109/TIP.2004.834669.

[12]

M. Fernández-Suárez and A. Y. Ting, Fluorescent probes for super-resolution imaging in living cells, Nature Reviews Molecular Cell Biology, 9 (2008), 929-943.  doi: 10.1038/nrm2531.

[13]

W. T FreemanT. R. Jones and E. C. Pasztor, Example-based super-resolution, IEEE Computer Graphics and Applications, 22 (2002), 56-65.  doi: 10.1109/38.988747.

[14]

B. HuangH. Babcock and X. Zhuang, Breaking the diffraction barrier: Super-resolution imaging of cells, Cell, 143 (2010), 1047-1058.  doi: 10.1016/j.cell.2010.12.002.

[15]

D. G. S. Bagon Michal Irani, Super-resolution from a single image, In Proceedings of the IEEE International Conference on Computer Vision, Kyoto, Japan, pages 349–356, 2009.

[16]

K. Iwamoto, T. Yoshida and M. Ikehara, Super-resolutions based on shock filter and non local means with bm3d for congruity, In TENCON 2014-2014 IEEE Region 10 Conference, pages 1–6. IEEE, 2014. doi: 10.1109/TENCON.2014.7022420.

[17]

A. KappelerS. YooQ. Dai and A. K. Katsaggelos, Video super-resolution with convolutional neural networks, IEEE Transactions on Computational Imaging, 2 (2016), 109-122.  doi: 10.1109/TCI.2016.2532323.

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K. In Kim and Y. Kwon, Single-image super-resolution using sparse regression and natural image prior, IEEE Transactions on Pattern Analysis and Machine Intelligence, 32 (2010), 1127-1133. 

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A. LaghribA. Ben-LoghfyryA. Hadri and A. Hakim, A nonconvex fractional order variational model for multi-frame image super-resolution, Signal Processing: Image Communication, 67 (2018), 1-11.  doi: 10.1016/j.image.2018.05.011.

[20]

A. LaghribM. EzzakiM. El RhabiA. HakimP. Monasse and S. Raghay, Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution, Computer Vision and Image Understanding, 168 (2018), 50-63.  doi: 10.1016/j.cviu.2017.08.007.

[21]

A. LaghribA. GhazdaliA. Hakim and S. Raghay, A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration, Computers & Mathematics with Applications, 72 (2016), 2535-2548.  doi: 10.1016/j.camwa.2016.09.013.

[22]

A. LaghribA. Hadri and A. Hakim, An edge preserving high-order pde for multiframe image super-resolution, Journal of the Franklin Institute, 356 (2019), 5834-5857.  doi: 10.1016/j.jfranklin.2019.02.032.

[23]

A. Laghrib, A. Hakim and S. Raghay, A combined total variation and bilateral filter approach for image robust super resolution, EURASIP Journal on Image and Video Processing, 2015 (2015), 19. doi: 10.1186/s13640-015-0075-4.

[24]

A. LaghribA. Hakim and S. Raghay, An iterative image super-resolution approach based on bregman distance, Signal Processing: Image Communication, 58 (2017), 24-34.  doi: 10.1016/j.image.2017.06.006.

[25]

X. Liu and L. Huang, A new nonlocal total variation regularization algorithm for image denoising, Mathematics and Computers in Simulation, 97 (2014), 224-233.  doi: 10.1016/j.matcom.2013.10.001.

[26]

B. MaiseliC. WuJ. MeiQ. Liu and H. Gao, A robust super-resolution method with improved high-frequency components estimation and aliasing correction capabilities, Journal of the Franklin Institute, 351 (2014), 513-527.  doi: 10.1016/j.jfranklin.2013.09.007.

[27]

B. Jacob MaiseliN. Ally and H. Gao, A noise-suppressing and edge-preserving multiframe super-resolution image reconstruction method, Signal Processing: Image Communication, 34 (2015), 1-13. 

[28]

A. Marquina and S. J. Osher, Image super-resolution by tv-regularization and bregman iteration, Journal of Scientific Computing, 37 (2008), 367-382.  doi: 10.1007/s10915-008-9214-8.

[29]

D. Mitzel, T. Pock, T. Schoenemann and D. Cremers, Video super resolution using duality based tv-l 1 optical flow, In Joint Pattern Recognition Symposium, pages 432–441. Springer, 2009. doi: 10.1007/978-3-642-03798-6_44.

[30]

S. C. ParkM. K. Park and M. G. Kang, Super-resolution image reconstruction: A technical overview, IEEE Signal Processing Magazine, 20 (2003), 21-36.  doi: 10.1109/MSP.2003.1203207.

[31]

M. ProtterM. EladH. Takeda and P. Milanfar, Generalizing the nonlocal-means to super-resolution reconstruction, IEEE Transactions on Image Processing, 18 (2009), 36-51.  doi: 10.1109/TIP.2008.2008067.

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[33]

F. ShiJ. ChengL. WangP.-T. Yap and D. Shen, Lrtv: Mr image super-resolution with low-rank and total variation regularizations, IEEE Transactions on Medical Imaging, 34 (2015), 2459-2466.  doi: 10.1109/TMI.2015.2437894.

[34]

R. Tsai, Multiframe image restoration and registration, Advance Computer Visual and Image Processing, 1 (1984), 317-339. 

[35]

M. Unger, T. Pock, M. Werlberger and H. Bischof, A convex approach for variational super-resolution, In Joint Pattern Recognition Symposium, pages 313–322. Springer, 2010. doi: 10.1007/978-3-642-15986-2_32.

[36]

Z. Wang, E. P. Simoncelli and A. C. Bovik, Multiscale structural similarity for image quality assessment, In The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2 (2003), 1398–1402. IEEE. doi: 10.1109/ACSSC.2003.1292216.

[37]

J. Weickert, Coherence-enhancing diffusion filtering, International Journal of Computer Vision, 31 (1999), 111-127.  doi: 10.1023/A:1008009714131.

[38]

J. YangJ. WrightT. S. Huang and Y. Ma, Image super-resolution via sparse representation, IEEE Transactions on Image Processing, 19 (2010), 2861-2873.  doi: 10.1109/TIP.2010.2050625.

[39]

W. YangX. ZhangY. TianW. WangJ.-H. Xue and Q. Liao, Deep learning for single image super-resolution: A brief review, IEEE Transactions on Multimedia, 21 (2019), 3106-3121.  doi: 10.1109/TMM.2019.2919431.

[40]

Q. YuanL. Zhang and H. Shen, Multiframe super-resolution employing a spatially weighted total variation model, IEEE Transactions on Circuits and Systems for Video Technology, 22 (2011), 379-392.  doi: 10.1109/TCSVT.2011.2163447.

[41]

F. C. ZanacchiZ. LavagninoM. P. DonnorsoA. Del BueL. FuriaM. Faretta and A. Diaspro, Live-cell 3d super-resolution imaging in thick biological samples, Nature Methods, 8 (2011), 1047-1049.  doi: 10.1038/nmeth.1744.

[42]

E. Zeidler, Nonlinear Functional Analysis Vol.1: Fixed-Point Theorems, Springer-Verlag Berlin and Heidelberg GmbH and Co. K, springer edition, 1986.

[43]

WL Zeng and XB Lu, A robust variational approach to super-resolution with nonlocal tv regularisation term, The Imaging Science Journal, 61 (2013), 268-278.  doi: 10.1179/1743131X11Y.0000000064.

[44]

X. ZhuP. JinX. Wang and N. Ai, Multi-frame image super-resolution reconstruction via low-rank fusion combined with sparse coding, Multimedia Tools and Applications, 78 (2019), 7143-7154.  doi: 10.1007/s11042-018-6495-2.

show all references

References:
[1]

S. Baker and T. Kanade, Limits on super-resolution and how to break them, IEEE Transactions on Pattern Analysis & Machine Intelligence, (2002), 1167–1183.

[2]

T. BroxO. Kleinschmidt and D. Cremers, Efficient nonlocal means for denoising of textural patterns, IEEE Transactions on Image Processing, 17 (2008), 1083-1092.  doi: 10.1109/TIP.2008.924281.

[3]

A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, In Proc. Sympos. Pure Math, 4 (1961), 33-49. 

[4]

K. DabovA. FoiV. 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.

[5]

W. Dong, T. Huang, G. Shi, Y. Ma and X. Li, Robust tensor approximation with laplacian scale mixture modeling for multiframe image and video denoising, IEEE Journal of Selected Topics in Signal Processing, 12 (2018), 1435–1448, . doi: 10.1109/JSTSP.2018.2873047.

[6]

W. DongL. ZhangG. Shi and X. Wu, Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization, IEEE Transactions on Image Processing, 20 (2011), 1838-1857.  doi: 10.1109/TIP.2011.2108306.

[7]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696.  doi: 10.1137/110833294.

[8]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Mathematical Models and Methods in Applied Sciences, 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.

[9]

I. El MourabitM. El RhabiA. HakimA. Laghrib and E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction, Signal Processing, 132 (2017), 51-65.  doi: 10.1016/j.sigpro.2016.09.014.

[10]

S. FarsiuD. RobinsonM. Elad and P. Milanfar, Advances and challenges in super-resolution, International Journal of Imaging Systems and Technology, 14 (2004), 47-57. 

[11]

S. FarsiuM. D. RobinsonM. Elad and P. Milanfar, Fast and robust multiframe super resolution, IEEE Transactions on Image Processing, 13 (2004), 1327-1344.  doi: 10.1109/TIP.2004.834669.

[12]

M. Fernández-Suárez and A. Y. Ting, Fluorescent probes for super-resolution imaging in living cells, Nature Reviews Molecular Cell Biology, 9 (2008), 929-943.  doi: 10.1038/nrm2531.

[13]

W. T FreemanT. R. Jones and E. C. Pasztor, Example-based super-resolution, IEEE Computer Graphics and Applications, 22 (2002), 56-65.  doi: 10.1109/38.988747.

[14]

B. HuangH. Babcock and X. Zhuang, Breaking the diffraction barrier: Super-resolution imaging of cells, Cell, 143 (2010), 1047-1058.  doi: 10.1016/j.cell.2010.12.002.

[15]

D. G. S. Bagon Michal Irani, Super-resolution from a single image, In Proceedings of the IEEE International Conference on Computer Vision, Kyoto, Japan, pages 349–356, 2009.

[16]

K. Iwamoto, T. Yoshida and M. Ikehara, Super-resolutions based on shock filter and non local means with bm3d for congruity, In TENCON 2014-2014 IEEE Region 10 Conference, pages 1–6. IEEE, 2014. doi: 10.1109/TENCON.2014.7022420.

[17]

A. KappelerS. YooQ. Dai and A. K. Katsaggelos, Video super-resolution with convolutional neural networks, IEEE Transactions on Computational Imaging, 2 (2016), 109-122.  doi: 10.1109/TCI.2016.2532323.

[18]

K. In Kim and Y. Kwon, Single-image super-resolution using sparse regression and natural image prior, IEEE Transactions on Pattern Analysis and Machine Intelligence, 32 (2010), 1127-1133. 

[19]

A. LaghribA. Ben-LoghfyryA. Hadri and A. Hakim, A nonconvex fractional order variational model for multi-frame image super-resolution, Signal Processing: Image Communication, 67 (2018), 1-11.  doi: 10.1016/j.image.2018.05.011.

[20]

A. LaghribM. EzzakiM. El RhabiA. HakimP. Monasse and S. Raghay, Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution, Computer Vision and Image Understanding, 168 (2018), 50-63.  doi: 10.1016/j.cviu.2017.08.007.

[21]

A. LaghribA. GhazdaliA. Hakim and S. Raghay, A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration, Computers & Mathematics with Applications, 72 (2016), 2535-2548.  doi: 10.1016/j.camwa.2016.09.013.

[22]

A. LaghribA. Hadri and A. Hakim, An edge preserving high-order pde for multiframe image super-resolution, Journal of the Franklin Institute, 356 (2019), 5834-5857.  doi: 10.1016/j.jfranklin.2019.02.032.

[23]

A. Laghrib, A. Hakim and S. Raghay, A combined total variation and bilateral filter approach for image robust super resolution, EURASIP Journal on Image and Video Processing, 2015 (2015), 19. doi: 10.1186/s13640-015-0075-4.

[24]

A. LaghribA. Hakim and S. Raghay, An iterative image super-resolution approach based on bregman distance, Signal Processing: Image Communication, 58 (2017), 24-34.  doi: 10.1016/j.image.2017.06.006.

[25]

X. Liu and L. Huang, A new nonlocal total variation regularization algorithm for image denoising, Mathematics and Computers in Simulation, 97 (2014), 224-233.  doi: 10.1016/j.matcom.2013.10.001.

[26]

B. MaiseliC. WuJ. MeiQ. Liu and H. Gao, A robust super-resolution method with improved high-frequency components estimation and aliasing correction capabilities, Journal of the Franklin Institute, 351 (2014), 513-527.  doi: 10.1016/j.jfranklin.2013.09.007.

[27]

B. Jacob MaiseliN. Ally and H. Gao, A noise-suppressing and edge-preserving multiframe super-resolution image reconstruction method, Signal Processing: Image Communication, 34 (2015), 1-13. 

[28]

A. Marquina and S. J. Osher, Image super-resolution by tv-regularization and bregman iteration, Journal of Scientific Computing, 37 (2008), 367-382.  doi: 10.1007/s10915-008-9214-8.

[29]

D. Mitzel, T. Pock, T. Schoenemann and D. Cremers, Video super resolution using duality based tv-l 1 optical flow, In Joint Pattern Recognition Symposium, pages 432–441. Springer, 2009. doi: 10.1007/978-3-642-03798-6_44.

[30]

S. C. ParkM. K. Park and M. G. Kang, Super-resolution image reconstruction: A technical overview, IEEE Signal Processing Magazine, 20 (2003), 21-36.  doi: 10.1109/MSP.2003.1203207.

[31]

M. ProtterM. EladH. Takeda and P. Milanfar, Generalizing the nonlocal-means to super-resolution reconstruction, IEEE Transactions on Image Processing, 18 (2009), 36-51.  doi: 10.1109/TIP.2008.2008067.

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[33]

F. ShiJ. ChengL. WangP.-T. Yap and D. Shen, Lrtv: Mr image super-resolution with low-rank and total variation regularizations, IEEE Transactions on Medical Imaging, 34 (2015), 2459-2466.  doi: 10.1109/TMI.2015.2437894.

[34]

R. Tsai, Multiframe image restoration and registration, Advance Computer Visual and Image Processing, 1 (1984), 317-339. 

[35]

M. Unger, T. Pock, M. Werlberger and H. Bischof, A convex approach for variational super-resolution, In Joint Pattern Recognition Symposium, pages 313–322. Springer, 2010. doi: 10.1007/978-3-642-15986-2_32.

[36]

Z. Wang, E. P. Simoncelli and A. C. Bovik, Multiscale structural similarity for image quality assessment, In The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2 (2003), 1398–1402. IEEE. doi: 10.1109/ACSSC.2003.1292216.

[37]

J. Weickert, Coherence-enhancing diffusion filtering, International Journal of Computer Vision, 31 (1999), 111-127.  doi: 10.1023/A:1008009714131.

[38]

J. YangJ. WrightT. S. Huang and Y. Ma, Image super-resolution via sparse representation, IEEE Transactions on Image Processing, 19 (2010), 2861-2873.  doi: 10.1109/TIP.2010.2050625.

[39]

W. YangX. ZhangY. TianW. WangJ.-H. Xue and Q. Liao, Deep learning for single image super-resolution: A brief review, IEEE Transactions on Multimedia, 21 (2019), 3106-3121.  doi: 10.1109/TMM.2019.2919431.

[40]

Q. YuanL. Zhang and H. Shen, Multiframe super-resolution employing a spatially weighted total variation model, IEEE Transactions on Circuits and Systems for Video Technology, 22 (2011), 379-392.  doi: 10.1109/TCSVT.2011.2163447.

[41]

F. C. ZanacchiZ. LavagninoM. P. DonnorsoA. Del BueL. FuriaM. Faretta and A. Diaspro, Live-cell 3d super-resolution imaging in thick biological samples, Nature Methods, 8 (2011), 1047-1049.  doi: 10.1038/nmeth.1744.

[42]

E. Zeidler, Nonlinear Functional Analysis Vol.1: Fixed-Point Theorems, Springer-Verlag Berlin and Heidelberg GmbH and Co. K, springer edition, 1986.

[43]

WL Zeng and XB Lu, A robust variational approach to super-resolution with nonlocal tv regularisation term, The Imaging Science Journal, 61 (2013), 268-278.  doi: 10.1179/1743131X11Y.0000000064.

[44]

X. ZhuP. JinX. Wang and N. Ai, Multi-frame image super-resolution reconstruction via low-rank fusion combined with sparse coding, Multimedia Tools and Applications, 78 (2019), 7143-7154.  doi: 10.1007/s11042-018-6495-2.

Figure 1.  Some possible configurations for $ \Omega $ and $ \Omega_I $
Figure 2.  The denoising result of the restored "Square" image whit Gaussian noise (parameter $ \sigma^2 = 0.03 $)
Figure 3.  The denoising result of the restored "Cercle" image whith the Gaussian noise (parameter $ \sigma = 0.04 $)
Figure 4.  The denoising result of the restored "Motif" image when the impulse noise is neglected (parameter $ \sigma = 0.03 $)
Figure 5.  The super-resolution results of our model when the Gaussian noise is considered with $ \sigma^2 = 0.03 $ and comparisons with other SR models for the "Butterfly" image
Figure 6.  Super-resolution results of our model when $ \sigma^2 = 0.04 $, and comparisons with other models for the "Penguin" image
Figure 7.  The super-resolution results of our model when the Gaussian noise is considered with $ \sigma^2 = 0.05 $ and comparisons with other models for the "Build" image
Figure 8.  The super-resolution results of the sequence "Cash-box". The btained PSNR values for these methods are depicated such as : SRCN (32.77), LRSC (33.27), SBM3D (33.88), Our (33.79)
Figure 9.  The super-resolution results of the sequence "Satellite". The btained PSNR values for these methods are depicated such as : SRCN (25.11), LRSC (25.29), SBM3D (25.58), Our (26.08)
Figure 10.  The super-resolution results of the video sequence "Text"
Figure 11.  The super-resolution results of the video sequence "Bar-code" image
Figure 12.  The super-resolution results of the video sequence "Wheel"
Table 1.  The PSNR results of different SR methods for selected images. Note that we used a benchmark of $ 30 $ images in our tests and present only ten in this table
Image $ \sigma^2 $ noise SR method
NPDE NTPDE NLPDE FOPDE TALS Our
Butterfly $ \sigma^2=0.03 $ 29.33 29.11 30.12 31.02 31.06 32.54
Penguin $ \sigma^2=0.04 $ 27.88 28.29 28.52 29.04 29.21 29.90
Build $ \sigma^2=0.05 $ 25.33 25.06 25.83 26.52 27.44 28.03
Barbara $ \sigma^2=0.06 $ 24.66 25.60 26.42 26.74 28.03 28.10
Pirate $ \sigma^2=0.02 $ 30.12 30.87 31.02 31.86 32.10 32.53
Lena $ \sigma^2=0.04 $ 29.44 30.12 30.06 31.17 30.54 31.50
Cameraman $ \sigma^2=0.01 $ 30.06 30.97 30.52 31.11 31.74 32.86
Baboon $ \sigma^2=0.02 $ 30.10 30.22 31.08 31.16 31.99 32.40
Fly $ \sigma^2=0.03 $ 29.44 29.70 30.16 30.22 30.76 31.05
Horses $ \sigma^2=0.01 $ 29.88 30.44 30.77 30.49 31.42 32.20
Image $ \sigma^2 $ noise SR method
NPDE NTPDE NLPDE FOPDE TALS Our
Butterfly $ \sigma^2=0.03 $ 29.33 29.11 30.12 31.02 31.06 32.54
Penguin $ \sigma^2=0.04 $ 27.88 28.29 28.52 29.04 29.21 29.90
Build $ \sigma^2=0.05 $ 25.33 25.06 25.83 26.52 27.44 28.03
Barbara $ \sigma^2=0.06 $ 24.66 25.60 26.42 26.74 28.03 28.10
Pirate $ \sigma^2=0.02 $ 30.12 30.87 31.02 31.86 32.10 32.53
Lena $ \sigma^2=0.04 $ 29.44 30.12 30.06 31.17 30.54 31.50
Cameraman $ \sigma^2=0.01 $ 30.06 30.97 30.52 31.11 31.74 32.86
Baboon $ \sigma^2=0.02 $ 30.10 30.22 31.08 31.16 31.99 32.40
Fly $ \sigma^2=0.03 $ 29.44 29.70 30.16 30.22 30.76 31.05
Horses $ \sigma^2=0.01 $ 29.88 30.44 30.77 30.49 31.42 32.20
Table 2.  The SSIM results of different SR methods for selected images. Note that we used a benchmark of $ 30 $ images in our tests and present only ten in this table
Image $ \sigma^2 $ noise SR method
NPDE NTPDE NLPDE FOPDE TALS Our
Butterfly $ \sigma^2=0.03 $ 0.822 0.829 0.839 0.877 0.888 0.901
Penguin $ \sigma^2=0.04 $ 0.785 0.796 0.806 0.828 0.843 0.882
Build $ \sigma^2=0.05 $ 0.683 0.694 0.714 0.730 0.764 0.800
Barbara $ \sigma^2=0.06 $ 0.626 0.660 0.691 0.674 0.708 0.755
Pirate $ \sigma^2=0.02 $ 0.839 0.847 0.855 0.891 0.886 0.922
Lena $ \sigma^2=0.04 $ 0.760 0.775 0.793 0.811 0.826 0.829
Cameraman $ \sigma^2=0.01 $ 0.881 0.902 0.933 0.948 0.942 0.967
Baboon $ \sigma^2=0.02 $ 0.826 0.836 0.842 0.859 0.870 0.889
Fly $ \sigma^2=0.03 $ 0.781 0.767 0.807 0.819 0.826 0.867
Horses $ \sigma^2=0.01 $ 0.869 0.890 0.917 0.923 0.948 0.968
Image $ \sigma^2 $ noise SR method
NPDE NTPDE NLPDE FOPDE TALS Our
Butterfly $ \sigma^2=0.03 $ 0.822 0.829 0.839 0.877 0.888 0.901
Penguin $ \sigma^2=0.04 $ 0.785 0.796 0.806 0.828 0.843 0.882
Build $ \sigma^2=0.05 $ 0.683 0.694 0.714 0.730 0.764 0.800
Barbara $ \sigma^2=0.06 $ 0.626 0.660 0.691 0.674 0.708 0.755
Pirate $ \sigma^2=0.02 $ 0.839 0.847 0.855 0.891 0.886 0.922
Lena $ \sigma^2=0.04 $ 0.760 0.775 0.793 0.811 0.826 0.829
Cameraman $ \sigma^2=0.01 $ 0.881 0.902 0.933 0.948 0.942 0.967
Baboon $ \sigma^2=0.02 $ 0.826 0.836 0.842 0.859 0.870 0.889
Fly $ \sigma^2=0.03 $ 0.781 0.767 0.807 0.819 0.826 0.867
Horses $ \sigma^2=0.01 $ 0.869 0.890 0.917 0.923 0.948 0.968
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