This paper is devoted to the mathematical and numerical study of a new proposed model based on a fractional diffusion equation coupled with a nonlinear regularization of the Total Variation operator. This model is primarily intended to introduce a weak norm in the fidelity term, where this norm is considered more appropriate for capturing very oscillatory characteristics interpreted as a texture. Furthermore, our proposed model profits from the benefits of a variable exponent used to distinguish the features of the image. By using Faedo-Galerkin method, we prove the well-posedness (existence and uniqueness) of the weak solution for the proposed model. Based on the alternating direction implicit method of Peaceman-Rachford and the approximations of the Gr$ \ddot{u} $nwald-Letnikov operators, we develop the numerical discretization of our fractional diffusion equation. Experimental results claim that our model provides high-quality results in cartoon-texture-edges decomposition and image denoising. In particular, our model can successfully reduce the staircase phenomenon during the image denoising. Furthermore, small details, texture and fine structures still maintained in the restored image. Finally, we compare our numerical results with the existing models in the literature.

中文翻译：

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非线性分数阶反应扩散系统在图像降噪和分解中的应用

本文致力于基于分数扩散方程和总变化算子的非线性正则化的新模型的数学和数值研究。该模型主要旨在在保真度方面引入一个弱规范，在该规范中，该规范被认为更适合于捕获被解释为纹理的非常振荡的特征。此外，我们提出的模型受益于用于区分图像特征的可变指数的好处。通过使用Faedo-Galerkin方法，我们证明了所提出模型的弱解的适定性（存在性和唯一性）。基于Peaceman-Rachford的交替方向隐式方法和Gr $ \ ddot {u} $ nwald-Letnikov算子的逼近，我们发展了分数扩散方程的数值离散化。实验结果表明，我们的模型在卡通纹理边缘分解和图像降噪方面提供了高质量的结果。特别地，我们的模型可以成功地减少图像去噪过程中的阶梯现象。此外，小细节，纹理和精细结构仍保留在还原的图像中。最后，我们将数值结果与文献中的现有模型进行比较。