On the well-posedness of a tensor-based second order PDE with bilateral term for image super-resolution

El Mourabit Idriss; Abdelilah Hakim; Amine Laghrib

doi:10.3934/eect.2022047

Article Contents

2023, Volume 12, Issue 2: 703-731. Doi: 10.3934/eect.2022047

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On the well-posedness of a tensor-based second order PDE with bilateral term for image super-resolution

1.
MIMSC, EST Essaouira, Université Cadi Ayyad, Morocco
2.
LAMAI, FST Marrakech, Université Cadi Ayyad, Morocco
3.
EMI FST Béni-Mellal, Université Sultan Moulay Slimane, Maroc

^*Corresponding author: Amine Laghrib
^*Corresponding author: Amine Laghrib

Received: November 2021

Revised: June 2022

Early access: September 2022

Published: April 2023

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Abstract

In this paper, we introduce a second-order equation based on a diffusion tensor combined with a bilateral total variation (BTV) term that takes the benefit from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edge and the BTV term in reducing blur. Compared to the other partial differential equations (PDE) used in the SR context, the proposed super-resolution (SR) PDE can efficiently preserve image components such as corners and flat regions with less apparition of blur in homogeneous regions. Moreover, since the SR approaches are always ill-posed, a mathematical study of the existence and uniqueness of the solution is also checked in a Sobolev space. From the numerical experiments we can observe that the proposed PDE can efficiently improve the quality of the high-resolution (HR) image. Surprisingly, the apparition of blur in the restored image is less compared to the other methods. In addition to visual evaluation, two-based metrics are computed to confirm the improvements offered by the proposed PDE.

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Mathematics Subject Classification: Primary: 65K10, 90C26; Secondary: 68U10.

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Figure 1. Results of different denoising models (Butterfly image)

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Figure 2. The denoising result of the restored (Motif) image while the Gaussian noise is considered with $ \sigma^2 = 0.02 $

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Figure 3. The denoising result of the restored (Motif) image while the Gaussian noise is considered with $ \sigma^2 = 0.05 $

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Figure 4. Comparisons of different SR methods for the (Fish image)

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Figure 5. Comparisons of different SR methods (Face image)

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Figure 6. Comparisons of different SR methods (Women image)

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Figure 7. Comparisons of different SR methods (Rock image)

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Figure 8. Comparisons of different SR methods (Butterfly image)

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Figure 9. The PSNR values with respect to the iteration number for each simulated test

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Figure 10. Comparisons of different SR methods (wheel sequence)

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Figure 11. Comparisons of different SR methods (Whitthard image)

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Figure 12. Comparisons of different SR methods (Calendar sequence)

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Table 1. The PSNR and SSIM table for the first example when $ \sigma^2 = 0.02 $


Method	PSNR	SSIM

Noisy imge	18.94	0.1736
ADPDE	29.95	0.8653
WCPM	31.81	0.8497
AFPDE	31.59	0.9539
FOPDE	31.28	0.8473
NWPDE	32.31	0.9577
MTPDE	34.14	0.9387
Our	35.30	0.9647

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