# American Institute of Mathematical Sciences

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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##### References:
The original seven images
The First six LR images used for the super-resolution process
Comparisons of different SR methods (Build image)
Comparisons of different SR methods (Lion image)
Comparisons of different SR methods (Penguin image)
Comparisons of different SR methods (Man image)
Comparisons of different SR methods (Surf image)
Comparisons of different SR methods (Zebra image)
Comparisons of different SR methods (Barbara image)
Comparisons of different SR methods (Book image)
Comparisons of different SR methods (Disk image)
Comparisons of different SR methods (Paint image)
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 $\mathbf{32.18}$
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 $\mathbf{0.890}$
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 $\mathbf{2.200}$
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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