An improved coupled PDE system applied to the inverse image denoising problem

Abdelmajid El Hakoume; Lekbir Afraites; Amine Laghrib; Abdelmajid El Hakoume; Lekbir Afraites; Amine Laghrib

doi:10.3934/era.2022134

Electronic Research Archive

2022, Volume 30, Issue 7: 2618-2642. doi: 10.3934/era.2022134

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Research article

An improved coupled PDE system applied to the inverse image denoising problem

EMI FST Béni-Mellal, Université Sultan Moulay Slimane, Maroc

Received: 24 July 2021 Revised: 15 December 2021 Accepted: 19 February 2022 Published: 16 May 2022

The problem of interest in this paper is the mathematical and numerical analysis of a new non-variational model based on a high order non-linear PDE system resulting from image denoising. This model is motivated by involving the decomposition approach of $ H^{-1} $ norm suggested by Guo et al. ^[1,2] which is more appropriate to represent the small details in the textured image. Our model is based on a diffusion tensor that improves the behavior of the Perona-Malik diffusion directions in homogeneous regions and the Weickert model near tiny edges with a high diffusion order. A rigorous analysis of the existence and uniqueness of the weak solution of the proposed reaction-diffusion model is cheked in a suitable functional framework, using the Schauder fixed point theorem. Finally, we carry out a numerical result to show the effectiveness of our model by comparing the results obtained with some competitive models.
- image denoising,
- high order PDE system,
- anisotropic diffusion tensor,
- fixed point theorem
Citation: Abdelmajid El Hakoume, Lekbir Afraites, Amine Laghrib. An improved coupled PDE system applied to the inverse image denoising problem[J]. Electronic Research Archive, 2022, 30(7): 2618-2642. doi: 10.3934/era.2022134

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Abstract

The problem of interest in this paper is the mathematical and numerical analysis of a new non-variational model based on a high order non-linear PDE system resulting from image denoising. This model is motivated by involving the decomposition approach of $ H^{-1} $ norm suggested by Guo et al. ^[1,2] which is more appropriate to represent the small details in the textured image. Our model is based on a diffusion tensor that improves the behavior of the Perona-Malik diffusion directions in homogeneous regions and the Weickert model near tiny edges with a high diffusion order. A rigorous analysis of the existence and uniqueness of the weak solution of the proposed reaction-diffusion model is cheked in a suitable functional framework, using the Schauder fixed point theorem. Finally, we carry out a numerical result to show the effectiveness of our model by comparing the results obtained with some competitive models.

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