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

Amine Laghrib; Abdelkrim Chakib; Aissam Hadri; Abdelilah Hakim

doi:10.3934/dcdsb.2019188

Article Contents

2020, Volume 25, Issue 1: 415-442. Doi: 10.3934/dcdsb.2019188

This issue Previous Article Bi-center problem and Hopf cyclicity of a Cubic Liénard system Next Article On the forward dynamical behavior of nonautonomous systems

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
^* Corresponding author: A.laghrib

Received: August 2018

Revised: February 2019

Early access: September 2019

Published: January 2020

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Abstract

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:

Mathematics Subject Classification: Primary: 68U10; Secondary: 94A08.

Citation:

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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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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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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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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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