# American Institute of Mathematical Sciences

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  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 & Control Theory, doi: 10.3934/eect.2020084
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Some possible configurations for $\Omega$ and $\Omega_I$
The denoising result of the restored "Square" image whit Gaussian noise (parameter $\sigma^2 = 0.03$)
The denoising result of the restored "Cercle" image whith the Gaussian noise (parameter $\sigma = 0.04$)
The denoising result of the restored "Motif" image when the impulse noise is neglected (parameter $\sigma = 0.03$)
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
Super-resolution results of our model when $\sigma^2 = 0.04$, and comparisons with other models for the "Penguin" image
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
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)
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)
The super-resolution results of the video sequence "Text"
The super-resolution results of the video sequence "Bar-code" image
The super-resolution results of the video sequence "Wheel"
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
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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