American Institute of Mathematical Sciences

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

A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement

Amine Laghrib 1,, , Abdelkrim Chakib 1, , Aissam Hadri 2, and  Abdelilah Hakim 3,

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  January 2020 Early access  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.

Keywords: Multi-frame super-resolution, fourth-order PDE, fixed point theorem, tensor diffusion.
Mathematics Subject Classification: Primary: 68U10; Secondary: 94A08.
Citation: Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete and 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.

[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.

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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.

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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.

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F. E. Browder, A new generalization of the schauder fixed point theorem, Mathematische Annalen, 174 (1967), 285-290.  doi: 10.1007/BF01364275.

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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.

[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.

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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.

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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]

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[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. 

[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. 

[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.

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

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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. 

[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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Figure 1.  The original seven images
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Figure 2.  The First six LR images used for the super-resolution process
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Figure 3.  Comparisons of different SR methods (Build image)
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Figure 4.  Comparisons of different SR methods (Lion image)
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Figure 5.  Comparisons of different SR methods (Penguin image)
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Figure 6.  Comparisons of different SR methods (Man image)
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Figure 7.  Comparisons of different SR methods (Surf image)
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Figure 8.  Comparisons of different SR methods (Zebra image)
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Figure 9.  Comparisons of different SR methods (Barbara image)
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Figure 10.  Comparisons of different SR methods (Book image)
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Figure 11.  Comparisons of different SR methods (Disk image)
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Figure 12.  Comparisons of different SR methods (Paint image)
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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} $
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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} $
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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} $
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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
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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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