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A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement
1. | LMA FST Béni-Mellal, Université Sultan Moulay Slimane, Morocco |
2. | Faculté Polydisciplinaire Ouarzazate, Morocco |
3. | LAMAI, FST Marrakech, Université Cadi Ayyad, Morocco |
The multiframe super-resolution (SR) techniques are considered as one of the active research fields. More precisely, the construction of the desired high resolution (HR) image with less artifacts in the SR models, which are always ill-posed problems, requires a great care. In this paper, we propose a new fourth-order equation based on a diffusive tensor that takes the benefit from the diffusion model of Perona-Malik in the flat regions and the Weickert model near boundaries with a high diffusion order. As a result, the proposed SR approach can efficiently preserve image features such as corner and texture much better with less blur near edges. The existence and uniqueness of the proposed partial differential equation (PDE) are also demonstrated in an appropriate functional space. Finally, the given experimental results show the effectiveness of the proposed PDE compared to some competitive methods in both visually and quantitatively.
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show all references
References:
[1] |
Super-Resolution Imaging, Digital Imaging and Computer Vision, CRC Press, 2010. Google Scholar |
[2] |
L. Afraites, A. Atlas, F. Karami and D. Meskine, Some class of parabolic systems applied to image processing, Discrete Contin. Dyn. Syst. Ser B, 21 2016), 1671–1687.
doi: 10.3934/dcdsb.2016017. |
[3] |
A. Aftabizadeh,
Existence and uniqueness theorems for fourth-order boundary value problems, Journal of Mathematical Analysis and Applications, 116 (1986), 415-426.
doi: 10.1016/S0022-247X(86)80006-3. |
[4] |
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, volume 17., Siam, 2014.
doi: 10.1137/1.9781611973488. |
[5] |
S. Baker and T. Kanade,
Limits on super-resolution and how to break them, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1167-1183.
doi: 10.1109/CVPR.2000.854852. |
[6] |
E. Beretta, M. Bertsch and R. Dal Passo,
Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Archive for Rational Mechanics and Analysis, 129 (1995), 175-200.
doi: 10.1007/BF00379920. |
[7] |
F. Bernis and A. Friedman,
Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[8] |
S. Borman and R. L. Stevenson, Super-resolution from image sequences-a review, In Circuits and Systems, 1998. Proceedings. 1998 Midwest Symposium on, pages 374–378. IEEE, 1998.
doi: 10.1109/MWSCAS.1998.759509. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[10] |
F. E. Browder,
A new generalization of the schauder fixed point theorem, Mathematische Annalen, 174 (1967), 285-290.
doi: 10.1007/BF01364275. |
[11] |
A. Buades, B. Coll and J.-M. Morel, A non-local algorithm for image denoising, In Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, IEEE, 2 (2005), 60–65. Google Scholar |
[12] |
J. Chaparova,
Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations, Journal of Mathematical Analysis and Applications, 273 (2002), 121-136.
doi: 10.1016/S0022-247X(02)00216-0. |
[13] |
J. Chen, J. Nunez-Yanez and A. Achim,
Video super-resolution using generalized gaussian markov random fields, IEEE Signal Processing Letters, 19 (2012), 63-66.
doi: 10.1109/LSP.2011.2178595. |
[14] |
W. Dong, L. Zhang, G. 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. |
[15] |
I. El Mourabit, M. El Rhabi, A. Hakim, A. 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. |
[16] |
S. Farsiu, M. Elad and P. Milanfar,
Multiframe demosaicing and super-resolution of color images, IEEE transactions on image processing, 15 (2006), 141-159.
doi: 10.1109/TIP.2005.860336. |
[17] |
S. Farsiu, M. D. Robinson, M. Elad and P. Milanfar,
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doi: 10.1109/TIP.2004.834669. |
[18] |
M. Fernández-Suárez and A. Y. Ting, Fluorescent probes for super-resolution imaging in living cells, Nature Reviews. Molecular Cell Biology, 9 (2008), 929. Google Scholar |
[19] |
M. Gao and S. Qin,
High performance super-resolution reconstruction of multi-frame degraded images with local weighted anisotropy and successive regularization., Optik-International Journal for Light and Electron Optics, 126 (2015), 4219-4227.
doi: 10.1016/j.ijleo.2015.08.119. |
[20] |
H. Greenspan,
Super-resolution in medical imaging, The Computer Journal, 52 (2009), 43-63.
doi: 10.1093/comjnl/bxm075. |
[21] |
H. Greenspan, G. Oz, N. Kiryati and S. Peled,
MRI inter-slice reconstruction using super-resolution, International Conference on Medical Image Computing and Computer-Assisted Intervention, 2208 (2001), 1204-1206.
doi: 10.1007/3-540-45468-3_164. |
[22] |
M. R. Hajiaboli,
An anisotropic fourth-order diffusion filter for image noise removal, International Journal of Computer Vision, 92 (2011), 177-191.
doi: 10.1007/s11263-010-0330-1. |
[23] |
G. Han and Z. Xu,
Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Analysis: Theory, Methods & Applications, 68 (2008), 3646-3656.
doi: 10.1016/j.na.2007.04.007. |
[24] |
Y. He, K.-H. Yap, L. Chen and L.-P. Chau, Blind super-resolution image reconstruction using a maximum a posteriori estimation, In Image Processing, 2006 IEEE International Conference on, IEEE, 2006, 1729–1732.
doi: 10.1109/ICIP.2006.312715. |
[25] |
Y. He, K.-H. Yap, L. Chen and L.-P. Chau,
A nonlinear least square technique for simultaneous image registration and super-resolution, IEEE Transactions on Image Processing, 16 (2007), 2830-2841.
doi: 10.1109/TIP.2007.908074. |
[26] |
T. Hermosilla, E. Bermejo, A. Balaguer and L. A. Ruiz,
Non-linear fourth-order image interpolation for subpixel edge detection and localization, Image and Vision Computing, 26 (2008), 1240-1248.
doi: 10.1016/j.imavis.2008.02.012. |
[27] |
D. Holland, D. Boyd and P. Marshall, Updating topographic mapping in great britain using imagery from high-resolution satellite sensors, ISPRS Journal of Photogrammetry and Remote Sensing, 60 (2006), 212-223. Google Scholar |
[28] |
A. Kanemura, S.-I. Maeda and S. Ishii, Superresolution with compound markov random fields via the variational em algorithm, Neural Networks, 22 (2000), 1025-1034. Google Scholar |
[29] |
A. Laghrib, A. Ben-Loghfyry, A. 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. |
[30] |
A. Laghrib, A. Ghazdali, A. 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. |
[31] |
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), 1-10.
doi: 10.1186/s13640-015-0075-4. |
[32] |
A. Laghrib, A. 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. |
[33] |
X. Li, Y. Hu, X. Gao, D. Tao and B. Ning,
A multi-frame image super-resolution method, Signal Processing, 90 (2010), 405-414.
doi: 10.1016/j.sigpro.2009.05.028. |
[34] |
Y. Li,
Positive solutions of fourth-order boundary value problems with two parameters, Journal of Mathematical Analysis and Applications, 281 (2003), 477-484.
doi: 10.1016/S0022-247X(03)00131-8. |
[35] |
J. Lions, Quelques Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod Paris, 1969. |
[36] |
J. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications III, Dunod, 1968. |
[37] |
B. Maiseli, C. Wu, J. Mei, Q. 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. |
[38] |
B. J. Maiseli, N. Ally and H. Gao, A noise-suppressing and edge-preserving multiframe super-resolution image reconstruction method, Signal Processing: Image Communication, 34 (2015), 1-13. Google Scholar |
[39] |
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. |
[40] |
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Image | PSNR | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 27.00 | 28.10 | 28.93 | 28.50 | 29.66 | |
Lion | 26.90 | 26.95 | 27.42 | 27.88 | 28.33 | |
Penguin | 31.77 | 31.50 | 31.85 | 32.09 | 33.00 | |
Man | 29.55 | 29.03 | 29.80 | 30.40 | 31.10 | |
Surf | 30.01 | 30.05 | 30.88 | 30.91 | 31.44 | |
Zebra | 29.96 | 29.92 | 30.61 | 30.84 | 31.22 |
Image | PSNR | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 27.00 | 28.10 | 28.93 | 28.50 | 29.66 | |
Lion | 26.90 | 26.95 | 27.42 | 27.88 | 28.33 | |
Penguin | 31.77 | 31.50 | 31.85 | 32.09 | 33.00 | |
Man | 29.55 | 29.03 | 29.80 | 30.40 | 31.10 | |
Surf | 30.01 | 30.05 | 30.88 | 30.91 | 31.44 | |
Zebra | 29.96 | 29.92 | 30.61 | 30.84 | 31.22 |
Image | SSIM | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 0.786 | 0.783 | 0.796 | 0.802 | 0.812 | |
Lion | 0.805 | 0.807 | 0.837 | 0.845 | 0.866 | |
Penguin | 0.848 | 0.833 | 0.846 | 0.860 | 0.888 | |
Man | 0.776 | 0.772 | 0.800 | 0.819 | 0.836 | |
Surf | 0.802 | 0.789 | 0.816 | 0.840 | 0.855 | |
Zebra | 0.804 | 0.802 | 0.811 | 0.823 | 0.859 |
Image | SSIM | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 0.786 | 0.783 | 0.796 | 0.802 | 0.812 | |
Lion | 0.805 | 0.807 | 0.837 | 0.845 | 0.866 | |
Penguin | 0.848 | 0.833 | 0.846 | 0.860 | 0.888 | |
Man | 0.776 | 0.772 | 0.800 | 0.819 | 0.836 | |
Surf | 0.802 | 0.789 | 0.816 | 0.840 | 0.855 | |
Zebra | 0.804 | 0.802 | 0.811 | 0.823 | 0.859 |
Image | IFC | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 1.712 | 1.700 | 1.820 | 1.844 | 1.901 | |
Lion | 1.777 | 1.770 | 1.778 | 1.830 | 1.860 | |
Penguin | 1.825 | 1.802 | 1.933 | 1.930 | 1.964 | |
Man | 1.790 | 1.768 | 1.900 | 1.893 | 1.992 | |
Surf | 1.788 | 1.785 | 1.801 | 1.825 | 1.885 | |
Zebra | 1.800 | 1.802 | 1.863 | 1.860 | 1.933 |
Image | IFC | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | ||
Build | 1.712 | 1.700 | 1.820 | 1.844 | 1.901 | |
Lion | 1.777 | 1.770 | 1.778 | 1.830 | 1.860 | |
Penguin | 1.825 | 1.802 | 1.933 | 1.930 | 1.964 | |
Man | 1.790 | 1.768 | 1.900 | 1.893 | 1.992 | |
Surf | 1.788 | 1.785 | 1.801 | 1.825 | 1.885 | |
Zebra | 1.800 | 1.802 | 1.863 | 1.860 | 1.933 |
Image | Size | SR algorithms | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | |||
Build | 9.84 | 8.26 | 12.02 | 9.88 | 24.96 | 26.44 | |
Lion | 6.82 | 13.64 | 6.62 | 7.34 | 6.61 | 52.60 | |
Penguin | 8.31 | 14.48 | 7.16 | 7.62 | 8.08 | 48.33 | |
Man | 10.66 | 9.75 | 13.92 | 9.84 | 25.93 | 26.10 | |
Surf | 11.05 | 11.18 | 14.22 | 10.86 | 25.80 | 25.93 | |
Zebra | 9.86 | 9.14 | 12.22 | 10.32 | 20.24 | 22.02 | |
Barbara | 10.87 | 9.96 | 13.88 | 10.66 | 26.60 | 27.10 | |
Average time | 13.77 | 20.06 | 12.33 | 14.10 | 14.66 | 58.68 |
Image | Size | SR algorithms | |||||
TV reg. [39] | Adaptive reg. [67] | GDP SR [73] | The PDE in [15] | Our Method | |||
Build | 9.84 | 8.26 | 12.02 | 9.88 | 24.96 | 26.44 | |
Lion | 6.82 | 13.64 | 6.62 | 7.34 | 6.61 | 52.60 | |
Penguin | 8.31 | 14.48 | 7.16 | 7.62 | 8.08 | 48.33 | |
Man | 10.66 | 9.75 | 13.92 | 9.84 | 25.93 | 26.10 | |
Surf | 11.05 | 11.18 | 14.22 | 10.86 | 25.80 | 25.93 | |
Zebra | 9.86 | 9.14 | 12.22 | 10.32 | 20.24 | 22.02 | |
Barbara | 10.87 | 9.96 | 13.88 | 10.66 | 26.60 | 27.10 | |
Average time | 13.77 | 20.06 | 12.33 | 14.10 | 14.66 | 58.68 |
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