January  2020, 25(1): 415-442. doi: 10.3934/dcdsb.2019188

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

* Corresponding author: A.laghrib

Received  August 2018 Revised  February 2019 Published  September 2019

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.

Citation: Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188
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show all references

References:
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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. Google Scholar

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[7]

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[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. Google Scholar

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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. Google Scholar

[13]

J. ChenJ. 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. Google Scholar

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[16]

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[17]

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[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. Google Scholar

[20]

H. Greenspan, Super-resolution in medical imaging, The Computer Journal, 52 (2009), 43-63. doi: 10.1093/comjnl/bxm075. Google Scholar

[21]

H. GreenspanG. OzN. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

Y. HeK.-H. YapL. 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. Google Scholar

[26]

T. HermosillaE. BermejoA. 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. Google Scholar

[27]

D. HollandD. 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. KanemuraS.-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. 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. Google Scholar

[30]

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. Google Scholar

[31]

A. LaghribA. 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. Google Scholar

[32]

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. Google Scholar

[33]

X. LiY. HuX. GaoD. 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. Google Scholar

[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. Google Scholar

[35]

J. Lions, Quelques Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod Paris, 1969. Google Scholar

[36]

J. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications III, Dunod, 1968. Google Scholar

[37]

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. Google Scholar

[38]

B. J. 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. 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. Google Scholar

[40]

L. MinX. Yang and D. Ye, Well-posedness for a fourth order nonlinear equation related to image processing, Nonlinear Analysis: Real World Applications, 17 (2014), 192-202. doi: 10.1016/j.nonrwa.2013.11.005. Google Scholar

[41]

K. Nelson, A. Bhatti and S. Nahavandi, Performance evaluation of multi-frame super-resolution algorithms, In Digital Image Computing Techniques and Applications (DICTA), 2012 International Conference on, IEEE, 2012, 1–8.Google Scholar

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Figure 1.  The original seven images
Figure 2.  The First six LR images used for the super-resolution process
Figure 3.  Comparisons of different SR methods (Build image)
Figure 4.  Comparisons of different SR methods (Lion image)
Figure 5.  Comparisons of different SR methods (Penguin image)
Figure 6.  Comparisons of different SR methods (Man image)
Figure 7.  Comparisons of different SR methods (Surf image)
Figure 8.  Comparisons of different SR methods (Zebra image)
Figure 9.  Comparisons of different SR methods (Barbara image)
Figure 10.  Comparisons of different SR methods (Book image)
Figure 11.  Comparisons of different SR methods (Disk image)
Figure 12.  Comparisons of different SR methods (Paint image)
Table 1.  The PSNR table
Image PSNR
$ BTV $ reg. [17] 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 $ \mathbf{31.08} $
Lion 26.90 26.95 27.42 27.88 28.33 $ \mathbf{29.51} $
Penguin 31.77 31.50 31.85 32.09 33.00 $ \mathbf{34.02} $
Man 29.55 29.03 29.80 30.40 31.10 $ \mathbf{32.55} $
Surf 30.01 30.05 30.88 30.91 31.44 $ \mathbf{32.12} $
Zebra 29.96 29.92 30.61 30.84 31.22 $ \mathbf{32.18} $
Image PSNR
$ BTV $ reg. [17] 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 $ \mathbf{31.08} $
Lion 26.90 26.95 27.42 27.88 28.33 $ \mathbf{29.51} $
Penguin 31.77 31.50 31.85 32.09 33.00 $ \mathbf{34.02} $
Man 29.55 29.03 29.80 30.40 31.10 $ \mathbf{32.55} $
Surf 30.01 30.05 30.88 30.91 31.44 $ \mathbf{32.12} $
Zebra 29.96 29.92 30.61 30.84 31.22 $ \mathbf{32.18} $
Table 2.  The SSIM table
Image SSIM
$ BTV $ reg. [17] 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 $ \mathbf{0.830} $
Lion 0.805 0.807 0.837 0.845 0.866 $ \mathbf{0.884} $
Penguin 0.848 0.833 0.846 0.860 0.888 $ \mathbf{0.900} $
Man 0.776 0.772 0.800 0.819 0.836 $ \mathbf{0.874} $
Surf 0.802 0.789 0.816 0.840 0.855 $ \mathbf{0.872} $
Zebra 0.804 0.802 0.811 0.823 0.859 $ \mathbf{0.890} $
Image SSIM
$ BTV $ reg. [17] 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 $ \mathbf{0.830} $
Lion 0.805 0.807 0.837 0.845 0.866 $ \mathbf{0.884} $
Penguin 0.848 0.833 0.846 0.860 0.888 $ \mathbf{0.900} $
Man 0.776 0.772 0.800 0.819 0.836 $ \mathbf{0.874} $
Surf 0.802 0.789 0.816 0.840 0.855 $ \mathbf{0.872} $
Zebra 0.804 0.802 0.811 0.823 0.859 $ \mathbf{0.890} $
Table 3.  The IFC table
Image IFC
$ BTV $ reg. [17] 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 $ \mathbf{1.980} $
Lion 1.777 1.770 1.778 1.830 1.860 $ \mathbf{1.889} $
Penguin 1.825 1.802 1.933 1.930 1.964 $ \mathbf{2.111} $
Man 1.790 1.768 1.900 1.893 1.992 $ \mathbf{2.155} $
Surf 1.788 1.785 1.801 1.825 1.885 $ \mathbf{2.090} $
Zebra 1.800 1.802 1.863 1.860 1.933 $ \mathbf{2.200} $
Image IFC
$ BTV $ reg. [17] 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 $ \mathbf{1.980} $
Lion 1.777 1.770 1.778 1.830 1.860 $ \mathbf{1.889} $
Penguin 1.825 1.802 1.933 1.930 1.964 $ \mathbf{2.111} $
Man 1.790 1.768 1.900 1.893 1.992 $ \mathbf{2.155} $
Surf 1.788 1.785 1.801 1.825 1.885 $ \mathbf{2.090} $
Zebra 1.800 1.802 1.863 1.860 1.933 $ \mathbf{2.200} $
Table 4.  CPU times (in seconds) of different super-resolution methods and the proposed method when the magnification factor is $ 4 $
Image Size SR algorithms
$ BTV $ reg. [17] TV reg. [39] Adaptive reg. [67] GDP SR [73] The PDE in [15] Our Method
Build $ 472\times 312 $ 9.84 8.26 12.02 9.88 24.96 26.44
Lion $ 472\times 312 $ 6.82 13.64 6.62 7.34 6.61 52.60
Penguin $ 472\times 312 $ 8.31 14.48 7.16 7.62 8.08 48.33
Man $ 312\times 472 $ 10.66 9.75 13.92 9.84 25.93 26.10
Surf $ 504\times 504 $ 11.05 11.18 14.22 10.86 25.80 25.93
Zebra $ 472\times 312 $ 9.86 9.14 12.22 10.32 20.24 22.02
Barbara $ 512\times 512 $ 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
$ BTV $ reg. [17] TV reg. [39] Adaptive reg. [67] GDP SR [73] The PDE in [15] Our Method
Build $ 472\times 312 $ 9.84 8.26 12.02 9.88 24.96 26.44
Lion $ 472\times 312 $ 6.82 13.64 6.62 7.34 6.61 52.60
Penguin $ 472\times 312 $ 8.31 14.48 7.16 7.62 8.08 48.33
Man $ 312\times 472 $ 10.66 9.75 13.92 9.84 25.93 26.10
Surf $ 504\times 504 $ 11.05 11.18 14.22 10.86 25.80 25.93
Zebra $ 472\times 312 $ 9.86 9.14 12.22 10.32 20.24 22.02
Barbara $ 512\times 512 $ 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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