In this work, we introduce an innovative fractional nonlinear parabolic model using a time-fractional order derivative, specifically employing the Caputo sense for fractional differentiation. This model aims to enhance traditional super-resolution models, particularly in the context of multi-frame image super-resolution. Additionally, we incorporate a regularized Perona–Malik diffusion mechanism to control the speed and direction of diffusion at each image location. We begin our study by exploring the theoretical solvability of our proposed model. Firstly, we employ the Faedo–Galerkin approach to establish the existence and uniqueness of a weak solution for an auxiliary fractional super-resolution model. Subsequently, we use the Schauder fixed point method to demonstrate the existence and uniqueness of a weak solution for our model. To validate the effectiveness of our model in the multi-frame super-resolution (SR) context, we conduct numerical experiments on images featuring diverse characteristics, including corners and edges, while applying various warping, decimation, and blurring matrices to the low-resolution (LR) images. We start the evaluation by introducing an adaptive discrete scheme tailored to the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise. Additionally, we perform simulations on real data (videos). The obtained high-resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming competitive models both visually and quantitatively.

中文翻译：

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一种基于 Caputo 时间分数阶导数的正则化非线性扩散的新型多帧图像超分辨率模型


在这项工作中，我们引入了一种使用时间分数阶导数的创新分数非线性抛物线模型，特别是使用 Caputo sense 进行分数阶微分。该模型旨在增强传统的超分辨率模型，特别是在多帧图像超分辨率的背景下。此外，我们采用了正则化的 Perona-Malik 扩散机制来控制每个图像位置的扩散速度和方向。我们从探索我们提出的模型的理论可解性开始我们的研究。首先，我们采用 Faedo-Galerkin 方法来确定辅助分数阶超分辨率模型的弱解的存在性和唯一性。随后，我们使用 Schauder 不动点方法来证明我们模型弱解的存在性和唯一性。为了验证我们的模型在多帧超分辨率 （SR） 上下文中的有效性，我们对具有不同特征的图像进行了数值实验，包括角落和边缘，同时将各种翘曲、抽取和模糊矩阵应用于低分辨率 （LR） 图像。我们通过引入为所提出的模型量身定制的自适应离散方案来开始评估。为了证明我们方法的稳健性，我们将图像置于不同程度的噪声中。此外，我们还对真实数据（视频）进行模拟。获得的高分辨率 （HR） 结果显示出显着的效率和抗噪声稳健性，在视觉和定量上都优于竞争模型。