A nonlocal Weickert type PDE applied to multi-frame super-resolution

Fatimzehrae Ait Bella; Aissam Hadri; Abdelilah Hakim; Amine Laghrib

doi:10.3934/eect.2020084

Article Contents

2021, Volume 10, Issue 3: 633-655. Doi: 10.3934/eect.2020084

This issue Previous Article Results on controllability of non-densely characterized neutral fractional delay differential system Next Article On some damped 2 body problems

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
^* Corresponding author: F. Ait Bella

Received: December 2019

Revised: May 2020

Early access: August 2020

Published: September 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Some possible configurations for $ \Omega $ and $ \Omega_I $

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Figure 2. The denoising result of the restored "Square" image whit Gaussian noise (parameter $ \sigma^2 = 0.03 $)

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Figure 3. The denoising result of the restored "Cercle" image whith the Gaussian noise (parameter $ \sigma = 0.04 $)

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Figure 4. The denoising result of the restored "Motif" image when the impulse noise is neglected (parameter $ \sigma = 0.03 $)

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Figure 5. 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

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Figure 6. Super-resolution results of our model when $ \sigma^2 = 0.04 $, and comparisons with other models for the "Penguin" image

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Figure 7. 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

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Figure 8. 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)

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Figure 9. 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)

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Figure 10. The super-resolution results of the video sequence "Text"

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Figure 11. The super-resolution results of the video sequence "Bar-code" image

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Figure 12. The super-resolution results of the video sequence "Wheel"

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Table 1. 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

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Table 2. 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

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